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## Tree Diagrams

Simplifying complexity.

By the Mind Tools Content Team

We might associate tree diagrams with high school math; as simple tools commonly used to calculate probabilities or chart a series of events. However, they can be useful in the workplace too. They can help you analyze options, solve problems, and brainstorm different ideas with your team.

In this article, we'll look at what tree diagrams are, and discuss the many powerful ways you can apply them.

## What Is a Tree Diagram?

A tree diagram is a visual depiction of relationships that starts with a central node, or "trunk." This is the problem that needs solving or the idea you are analyzing. Each possible solution or event has its own "branch," which comes off the trunk to the top or bottom right-hand side. Additional decisions, consequences or effects split off from each of these "second layer branches," giving the diagram a tree-like structure.

You can use tree diagrams to break down categories or events into finer and finer levels of detail. This helps simplify complex problems (and their proposed solutions), and makes it easier for you to get an overview of your options.

## Figure 1 – Example Tree Diagram

Among other uses, tree diagrams can help you to:

- Find the root cause of a problem.
- Outline the steps needed to solve a problem or implement a plan.
- Identify the true scope of a project.
- Explain steps or details to others.
- Brainstorm possible outcomes.

Tree diagrams also help shift your thinking from the "big picture" to the subtleties of an issue. When you first draw the diagram, you look at the issue in its broadest sense. Your focus narrows when you flesh it out, and drill down to the finer detail. This shift in perspective is especially helpful when you're faced with large or particularly complex problems.

Mariella's boss has asked her team to improve its performance by 20 percent. She draws a tree diagram to break the objective down into meaningful sub-goals, so team members can understand exactly what they need to do.

She begins by writing the primary goal first; this is the "trunk" of the diagram.

- Improve performance by 20 percent.

Next, she and her team discuss how they can meet this goal. The team creates a new "branch" in the diagram for each idea to help it become more productive.

- Cross-train team members.
- Get better organized.
- Find more customers.

Each of these ideas is broken down further, so team members are clear about what they need to do. So, the group keeps brainstorming, and adds the following additional branches:

- Cross-train team members: Job rotation, job shadowing, job sharing.
- Get organized: Simplify client filing system, reduce clutter in the office, time-management training.
- Find more customers: Increase cold calling, social networking outreach, sponsor community events.

Mariella and her team keep expanding the tree diagram, using the final branches to assign tasks to specific people and to set deadlines. She then puts a printout in the break room to remind everyone what they're working on, and who is responsible for each task.

## Types of Tree Diagram

There are many different types of tree diagram, and each has a specific application.

## Making Decisions

Decision Trees are a common form of tree diagram. They can help you make the right choice when you're faced with several possible options. With them, you look at each alternative and investigate its potential outcome to determine which one offers the best balance between risk and reward.

There are always consequences when you make a decision. Sometimes they are positive, other times they're not. The Futures Wheel tool uses a type of tree diagram to help you explore them.

You can also use a tree diagram when carrying out an Impact Analysis to identify the possible negative consequences of a proposed change.

## Solving Problems

If you're faced with a complex problem, it can be challenging to identify its root cause and come up with an effective solution. Specialist tree diagrams can help you simplify this task.

Cause and Effect Analysis uses "fishbone diagrams," which are effectively horizontal tree diagrams. They enable you to discover the root cause of a problem, identify bottlenecks , and analyze why a particular process isn't working.

Sometimes you'll need to come up with several possible solutions to a problem, and this is when Concept Fans – based on the structure of tree diagrams – are useful. They let you take both a micro and a macro look at your problem, and generate as many options as you need to solve it.

Critical to Quality Trees are similar – they help you identify your customers' needs, and explore ways to meet those needs.

## Predicting Behavior

You can use tree diagrams to predict behavior. For example, they're commonly used in game theory to predict how your competitors might react in a given situation, or how a negotiation could play out. In this context, you use "game trees" to chart "players'" every possible move.

## Taking Notes

Tree diagrams can be useful for taking notes. For example, Mind Mapping is a useful tool for summarizing information, for consolidating large chunks of information, for making connections, and for creative problem solving. Mind Maps show facts, as well as the overall structure of a subject and the relative importance of individual parts of it.

## How to Create a Tree Diagram

This is a relatively simple task, and there are several ways to do it.

To draw a tree diagram by hand, start on the left-hand side of your paper (for a horizontal tree), or at the top (for a vertical tree). Write the problem or issue you're addressing in a square or circle.

Then, drill down to the next level of detail. For example, if you're trying to find the root cause of a problem, think about what could be causing the issue. Draw lines out for each possible cause and label it appropriately.

Once you've brainstormed all the possibilities at this level, look at each idea in turn. Using our problem-solving example, you can then drill down to the next level of detail by asking "What" or "Why" questions. For example, "What do we need to do to make this happen?" or "Why does this happen?"

## Using Software

You can draw tree diagrams using packages like Microsoft® Word. There are many free templates that you can download from the Internet, which automate and simplify the process.

You can also download software, like SmartDraw ™, or add-ins for Microsoft Excel, such as TreePlan ™, to create comprehensive and professional-looking diagrams.

The advantage to using software is that you can share what you create with others easily, and quickly make changes or add more branches. It also allows you to create high quality, professional diagrams, and seamlessly insert them into presentations or reports.

## Using Online Apps

There are also web-based applications, such as draw.io ™ and Creately ™, which help you create and save your tree diagram online. These applications are useful for team collaboration, especially when you're working with a home-based or virtual team.

Use tree diagrams to solve problems and make decisions by breaking information down into finer levels of detail. They help you to simplify complex problems, and make it easier for you to visualize all your options. Tree diagrams also help you find the root cause of a problem, break down large goals, and explain steps to others.

To draw a tree diagram, start by writing your problem or issue on the left-hand side of the page; this is the "trunk" of your tree. Next, identify the tasks that you need to complete to accomplish your goal; each should have its own "branch" off the central trunk. Look at them individually, and create further branches until you can't simplify or drill down any further.

## Apply This to Your Life

- Think carefully about any problems or situations you're facing now at work. Draw a tree diagram to brainstorm possible solutions.
- If you have a decision to make, use a tree diagram to explore your options and identify the consequences of each one.

"Microsoft Word" is a trademark of Microsoft (see www.microsoft.com ); "SmartDraw" is a trademark of SmartDraw.com (see www.smartdraw.com ); "TreePlan" is a trademark of TreePlan Software (see www.treeplan.com ); "draw.io" is a trademark of www.draw.io (see www.draw.io ); "Creately" is a trademark of Cinergix Pty Ltd (see www.cinergix.com ).

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## How to make a problem tree: Practical example

There are many methods and methodologies based on the tree analogy. The problem tree and the objective tree are examples of this. On this occasion we deal with the first of them.

It is today’s topic for business management with Ingenio Empresa.

Another post on the Logframe methodology:

- Example of a logical framework
- Stakeholder analysis
- Problem tree
- Objective tree
- Analysis of alternatives
- Analytical project structure
- Project Narrative
- Project indicators
- Means of verification
- Project assumptions

## What is the problem tree

Also known as the tree method, problem tree technique, situational analysis or problem analysis, this tool allows us to map or diagram the problem. The structure of a problem tree is:

At the roots are the causes of the problem The trunk represents the main problem In the leaves and branches are the effects or consequences.

It is a way to represent the problem, achieving at a glance to understand what is happening (main problem), why it is happening (causes) and what is causing it (effects or consequences), which allows us to do several things in the project planning, as you will see below in the advantages.

## Advantages of the problem tree

What is the purpose of a problem tree? The 4 most important and that summarize all the others are:

- It allows us to break down the problem, the causes and its effects, improving its analysis.
- There is a better understanding of the problem by disaggregating it into causes and consequences.
- It is linked to other research and analysis tools such as the Vester matrix or solution tree.
- Facilitates the realization of other important components of a research or project in its planning stage, e.g. stakeholder analysis, risk analysis and objectives.

## The problem tree in project planning

The Japanese often tell us that an identified problem already constitutes 90% of the solution, which is why the logical framework methodology gives so much importance to problem analysis. The planning work we do with the problem tree and the objective tree is a very important part of problem identification.

Having made this introduction, it is important that you know the importance of using the problem tree in conjunction with other tools, where what we seek is to:

- Obtain significant data to describe the problem
- Determine the causes and effects
- Elaborate project objectives

You will understand it better when we start to elaborate a problem tree step by step:

## How to make a problem tree step by step

- Analyze the situation: Yes, you know there is a problem situation, but analyze it. What is happening, why is it happening and what is triggering it. Collect data that will allow you to understand the problem situation. This in itself will give you a lot of input for the next step.
- Identify the main problems of the situation you have analyzed: Any technique to generate ideas will be useful. A team brainstorming, defining by consensus what the main problem is, is usually a good alternative. However, if the problem is much more technical and requires many experts and discussions, since it is complex to differentiate causes from effects, try the Vester matrix. This alone will allow you to prioritize the main problem, and will get you a few steps ahead by giving you causes and effects of the main problem.
- Determine the effects and causes of the main problem: You already have the trunk of the tree, now identify the causes (roots) and the effects or consequences (leaves or branches). Again, it is better if this is done as a team, seeking to reach a consensus. If in step 2 you elaborated the Vester matrix, you will already have this step quite clear.
- Draw the tree: Simple. We will see how in the example below.
- Go deeper into causes and effects: Solving the core problem will be much easier as you determine the root causes and effects. That is, if you have already determined a cause, is it possible that this cause is brought about by something else in turn? Draw a line and go as deep as possible.

With this done, all that remains is to move from the problem tree to the objective tree , taking causes to means and consequences to ends.

## Example of a problem tree

Based on the steps described above, let’s look at an example of a problem tree:

In step 1 : Colusa Inc is a web hosting company. In the last semester, it has been presenting a 35% increase in complaints and claims from its customers. Colusa Inc made a classification of the reasons for the complaints by analyzing their frequency. In addition to this, telephone and e-mail interviews were conducted with customers who had reported complaints, which allowed us to further refine the classification.

In step 2 we identify the problems: From this example and depending on the classification made, we could obtain several problems depending on which one is affecting us the most. In this case we would be choosing a specific option, however in this example we are going to work with the general option. Consequently, the main problem is: Increase of 35% in customer complaints and claims in the last quarter of the year by hiring hosting services.

In step 3 we will identify the causes and effects. With the information collected above, we are already well on our way. The three main types of complaints are:

- After-sales service is bad: The personnel sent do not know what they are doing and are sometimes rude (this includes telephone support).
- Poor quality of the product: It does not work when installed, the website crashes frequently or does not have enough hosting capacity.
- The product went up too much in price

And as you can see, these are direct causes.

What we have so far from the problem tree is the following:

The first to be identified is usually the causes. Now we identify the effects. In addition to thinking about all the possible effects that the main problem may have, we will consider the 1 to 1 effects of each cause, where, for example, a poor after-sales service will reduce the number of purchases made by customers. The result is as follows:

This in itself would already be a problem tree, but if we want to make this tool useful, we must get to the bottom of the causes and effects. In the example below, we show up to the second level: This would be the finished problem tree .

Image rights:

Featured image in the post is from: Freepik

## 9 thoughts on “How to make a problem tree: Practical example”

Very useful way to analyze issues. Thanks.

Solution tree for the problem

This elaboration has helped, but ‘strengthening of competence’ as one of the causes baffles me a little. It appears more of a solution to the problem than a cause. I will be grateful if I can get a clarification on it.

Hi Evans. The cause focuses on the strengthening of the other companies that provide a similar service to the one provided by Colusa, which leads to a problem if the other companies capture more customers. Hence “Strengthening of competition” is a cause.

Well you are right in that the way it is stated it seems to focus on strengthening the competence of the staff, or at least that’s what I thought with your comment. We could change the wording to specify that it is about business competence.

Thank you for this. i have been helped

Excellent learning material simple and straightforward

Thanks Fidele.

Good learning materia simple and understandable

What about the solution tree

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## How to use 5 Whys Tree Diagram for Root Cause Analysis?

When there are issues such as unsatisfied customers, decreasing market share, poor quality, etc. you have to understand the root cause of the issue. Only by addressing the root cause can a problem be fixed. Root Cause Analysis (RCA) is a method of problem-solving used for identifying the root causes of faults or problems. It is widely used in a wide variety of industries.

## What is A Root Cause?

All trees have a trunk, branches, and leaves. These are parts that immediately spring to mind since they are the ones we can see. The part we don’t see is the root system that anchors the tree to the ground. This root system often occupies a far larger area than the tree itself. It also continues to grow even if the tree’s branches have had a hard pruning. That’s why the system of drilling down to get to the heart of an issue is called Root Cause Analysis.

## Why Root Cause Analysis?

Root cause analysis is an important step to enable companies to make the right changes to prevent faults from happening over and over again. There are three ways of dealing with recurrent problems. We can:

- Ignore them.
- Perform a temporary fix.
- Get to the bottom of why they’re happening in the first place.

If we take the first option, the problem will never be solved and could escalate. If we take option two, it is the equivalent of painting over a stain or sticking a piece of tape over a leak and hoping it will hold – you’re treating the symptoms, but not the cause.

Taking the third option, i.e. analyzing the root cause, is the most time consuming, but should allow you to take steps to ensure that the problem never occurs in the future.

## How to use the 5 WHY analysis?

Here are some easy steps to effectively perform a 5 WHY analysis:

- Write down the problem. Writing helps you to formalize the problem and describe it completely. If you work with a team, it also helps the team to focus on the same problem.
- Ask yourself why did the problem happen and write down the answer.
- Ask yourself – looking at your answer – again why did the problem happen, and write down the answer.
- And again, ask yourself – looking at your answer – why did the problem happen.
- Ask yourself this question as often as necessary until the team agrees that the problem’s real root cause is identified. This may take fewer or more times than 5 WHYS.

## Performing Root Cause Analysis with Multiple 5 Whys using Tree Diagram

Not like a single 5 whys analysis hand one cause at a time, the tree diagram can be used to narrow down and eliminate possible causes in a diagram, ideally to one or more addressable root causes to be considered at one single diagram.

Edit this Diagram

## How to Create a 5 Whys Tree Diagram?

The Five Whys exercise is a questioning technique for going beyond symptoms of problems to identify the underlying or root causes of a problem. To facilitate the Five Whys Tree process, follow these steps:

- Below it, list the possible causes of the problem by asking the question “Why?” or “Why is that true?” or “Why is that happening?”
- For each of the causes, again ask the question “Why?”, and list the responses below.
- Continue this process at least 5 times or until you have reached the source of the problem, the lowest level cause stakeholders can do something about, or the response “That is just the way it is, or that is just what happened.”

For example: If your problem is that your car won’t start, the cause could be that it has no fuel. The root cause of this could then be that you forgot to fill up the tank, and the corrective action is, of course, to find some fuel.

Root Cause Analysis (RCA) can be decomposed into 4 steps:

- Identify and describe clearly the problem – Write down the specific problem. Writing the issue helps you formalize the problem and describe it completely. It also helps a team focus on the same problem.
- Identify any issues that contributed to the problem – Ask Why the problem happens and write the answer down below the problem
- Determine root causes – If the answer you just provided doesn’t identify the root cause of the problem that you wrote down in Step 1, ask Why again and write that answer down. Repeat and until the team is in agreement that the problem’s root cause is identified. Again, this may take fewer or more times than five Whys.
- Identify recommendations for the recurrence of problems in the future and implement the necessary solutions

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Planning tools: problem tree analysis.

Toolkit/guidelines

Problem tree analysis is central to many forms of project planning and is well developed among development agencies. Problem tree analysis (also called Situational analysis or just Problem analysis) helps to find solutions by mapping out the anatomy of cause and effect around an issue in a similar way to a Mind map, but with more structure. This brings several advantages:

- The problem can be broken down into manageable and definable chunks. This enables a clearer prioritisation of factors and helps focus objectives.
- There is more understanding of the problem and its often interconnected and even contradictory causes. This is often the first step in finding win-win solutions.
- It identifies the constituent issues and arguments, and can help establish who and what the political actors and processes are at each stage.
- It can help establish whether further information, evidence or resources are needed to make a strong case, or build a convincing solution.
- Present issues - rather than apparent, future or past issues - are dealt with and identified.
- The process of analysis often helps build a shared sense of understanding, purpose and action.

Problem tree analysis is best carried out in a small focus group of about six to eight people using flip chart paper or an overhead transparency. It is important that factors can be added as the conversation progresses. The first step is to discuss and agree the problem or issue to be analysed. Do not worry if it seems like a broad topic because the problem tree will help break it down. The problem or issue is written in the centre of the flip chart and becomes the 'trunk' of the tree. This becomes the 'focal problem'. The wording does not need to be exact as the roots and branches will further define it, but it should describe an actual issue that everyone feels passionately about.

Next, the group identify the causes of the focal problem - these become the roots - and then identify the consequences, which become the branches. These causes and consequences can be created on post-it notes or cards, perhaps individually or in pairs, so that they can be arranged in a cause-and-effect logic.

The heart of the exercise is the discussion, debate and dialogue that is generated as factors are arranged and re-arranged, often forming sub-dividing roots and branches (like a Mind map). Take time to allow people to explain their feelings and reasoning, and record related ideas and points that come up on separate flip chart paper under titles such as solutions, concerns and decisions.

## Discussion questions might include:

- Does this represent the reality? Are the economic, political and socio-cultural dimensions to the problem considered?
- Which causes and consequences are getting better, which are getting worse and which are staying the same?
- What are the most serious consequences? Which are of most concern? What criteria are important to us in thinking about a way forward?
- Which causes are easiest / most difficult to address? What possible solutions or options might there be? Where could a policy change help address a cause or consequence, or create a solution?
- What decisions have we made, and what actions have we agreed?

The Problem tree is closely linked to the Objectives tree, another key tool in the project planner's repertoire, and well used by development agencies. The Problem tree can be converted into an objectives tree by rephrasing each of the problems into positive desirable outcomes - as if the problem had already been treated. In this way, root causes and consequences are turned into root solutions, and key project or influencing entry points are quickly established. These objectives may well be worded as objectives for change. These can then feed into a Force field analysis which provides a useful next step.

## Example: HIV/AIDS programme in Kenya

As part of designing an HIV/AIDS activity in Kenya, a DFID design team needed to have a deeper understanding of various issues and constraints related to the epidemic. Before moving to a large log frame workshop the team decided to conduct focus group interviews with potential target groups and service providers. Through the focus groups the team gained a much deeper understanding of HIV/AIDS-related problems, constraints and opportunities. At the same time, participants in the groups learned much about common problems they themselves were facing and their possible solutions. Counselling and testing groups discovered they all faced a critical issue about how to protect the confidentiality of HIV-positive clients. Through the discussion they were able to exchange ideas of how to achieve this. Some had a policy focus and helped understand where changes in government practise and legislation could help. These issues were brought into the log frame workshop, where they were integrated in the design through an activity output dealing with improved counselling and testing services.

This tool first appeared in the ODI Toolkit, Successful Communication, A Toolkit for Researchers and Civil Society Organisations .

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## Guide to understanding tree diagrams

## What is a tree diagram?

A tree diagram allows users to visualize possible outcomes and probabilities for a given situation. Tree diagrams, also called decision trees, are particularly useful in charting the outcomes of dependent events, where if one element changes, it impacts the entire outcome. Tracking and analyzing cause-and-effect scenarios is much easier when you have a visual aid such as a tree diagram.

In a tree diagram, each "branch" of the tree connects an idea or a step in the process to a possible outcome. Outcomes are commonly referred to as "nodes" on a tree diagram. The resulting diagram resembles a tree with many options and outcomes that branch off from the original idea.

Tree diagrams are versatile and useful for decision-making and other tasks across various fields and industries, including marketing, software development, logistics, project management, and more.

## Examples of when to use a tree diagram

Tree diagrams are popular for brainstorming, problem-solving, and other idea-generation exercises. For more information about when to use a tree diagram, check out these examples:

## Brainstorming possible outcomes of a scenario

Tree diagrams are designed to help you thoroughly explore the possible outcomes of different problems and scenarios. Marketing and UX professionals can use tree diagrams to explore user experience paths and the various possible outcomes along different customer journeys.

## Problem-solving and root cause analysis

Tree diagrams are a helpful tool for problem-solving , including troubleshooting and root cause analysis. Software developers can use tree diagrams to analyze coding issues and find fixes during website and app development projects.

## Anticipating potential workflow issues

Before you launch a new process, product, or service, you can use a tree diagram to anticipate potential issues along the way. For example, supply chain managers can use tree diagrams to map out resource allocation and plans for a new product launch , exploring various options and identifying potential bottlenecks and roadblocks before beginning the project.

## Indicating the hierarchy of tasks

The tree diagram's hierarchical format is useful for project planning, especially when your project involves many different tasks and subtasks. Project managers can use a tree diagram to specify the task hierarchy for a project, helping project participants understand and agree upon each task's order and priority.

## Benefits of tree diagrams

Tree diagrams help you explore potential outcomes in an organized, visual way. Some key benefits of tree diagrams include:

## Better decision making

Tree diagrams provide a visual framework for decision-making, giving you an organized and systematic way to explore your options. This helps you gather and analyze all the data you need to make the best decision and get the best outcome.

## Enhanced troubleshooting

The tree diagram format provides a visual aid for evaluating problems and trying different fixes. When you create a tree diagram for your problem and add a branch for each new idea as you troubleshoot, you're more likely to find a solution quickly and easily.

## Streamlined workflows

Tree diagrams make it easy to record and replicate workflows, helping you streamline processes and get your team on the same page about the best way to do certain tasks and projects. With your workflows documented and streamlined, your team will enjoy greater unity, harmony, and productivity.

## How to make a tree diagram

Create a tree diagram by following these simple steps:

- Choose your main concept, idea, or topic. This could be a problem you need to solve, a project you're starting, or another topic.
- Place your main concept at the top of your diagram. Tree diagrams are hierarchical, so you should always start with your biggest, broadest idea and get more specific as you go.
- Create the first branches. Your first level of branches will be ideas or steps that would come immediately after or are immediately related to the main concept.
- Keep adding branches. Add more ideas based on your first layer of branches and continue branching off until you reach a conclusion or outcome of each path.
- Finish your tree diagram. Once you've exhausted all ideas, you should have enough possible outcomes mapped out to assist you in solving your problem, making your decision, pursuing your project, or moving forward with whatever situation inspired your tree diagram.
- Make adjustments as needed. If the situation in question changes, you may need to tweak your tree diagram. Typically, the branches in a tree diagram rely heavily on the ideas they connect to — so if one element changes, it will likely have a ripple effect on the rest of the chart.
- Share your insights. Distribute your tree diagram or present your insights to team members and stakeholders.

## Why use MindManager to make tree diagrams

Using a tree diagram software like MindManager allows you to create a flexible, digital tree map that you can easily edit and share with your team. Key features and benefits of MindManager include:

- User-friendly, intuitive interface
- Extensive image library — over 700 topic images, icons, and symbols to add to your tree diagrams
- Convenient file storage, retrieval, and sharing
- Powerful integrations with file storage apps like Box and OneDrive
- Google Docs integration via Zapier
- Numerous templates, tools, and features to facilitate brainstorming and strategic planning
- Google Chrome extension — MindManager Snap — to easily collect and import text, links, and images from the web
- Ability to add rich data — links, images, and documents — directly to your diagrams and charts

Professionals across all types of industries and roles use MindManager to organize ideas, visualize complex concepts, and collaborate across departments. With MindManager, you can collaborate and communicate with your team in new and unexpected ways.

## Tree diagram templates

MindManager comes pre-installed with tree diagram templates. To use these templates:

- Open MindManager
- Click NEW in the navigation menu
- Select the template you want to use
- A preview screen will appear - check to see if you'd like to use your selected template
- Select 'Create Map'
- Customize the template for your specific project

## Tree diagram FAQs

What are the different types of tree diagrams.

One of the main types of tree diagrams is the cause and effect tree diagram, which is used to map potential outcomes for a given decision or situation. Other types of tree diagrams include:

- Y to x tree diagrams
- Functional tree diagrams
- Abstract syntax tree diagrams

## Is a tree diagram the same as a decision tree?

Yes, a tree diagram is the same thing as a decision tree . Tree diagrams are sometimes referred to as decision trees because of how they help individuals map out different options and related outcomes.

## Let your ideas branch out — make a tree diagram

Tree diagrams provide a visual framework for exploring the different possible outcomes for a problem, project, or workflow. Professionals across various industries use tree diagrams for brainstorming, decision making, problem-solving, troubleshooting, workflow management, and more.

## Visualize more with MindManager

Ready to make a tree diagram? Try MindManager today

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## Other types of maps and charts

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## Easy Problem Solving

## Tree Diagram

What is a tree diagram.

It's a diagram that presents a hierarchical view from the goal/question until a certain level.

A tree diagram is a great visual and thinking tool and is used for a variety of reasons - task mapping, decision, probability tree, to show a logical break down of an argument.

We will concentrate on the simpler - task mapping usage

It is a task mapping tool for implementation. You start with a goal and then break into detailed actions, that would be required for implementation

See variations below for other uses in continuous improvement and project area

## When to use a Tree Diagram

To breakdown any goal / task into smaller, manageable pieces

If a goal looks overwhelming use it

To align the team with the goal

To align the actions with goals

Visually present logical linkages of tasks

Helps expand thinking and make it real

## How to make a Tree Diagram

Get the right team members, who know about the work

Goal: Start with the goal statement, on a board, flip-chart

e.g. Goal statement - could be improvement ideas that you want to implement; any project

Means: next comes, brainstorming to find major tasks, which are "means" to achieve it.

e.g. for party planning goal - it could be catering, music, venue...

use affinity diagram technique to arrive at "means"

Stay broad, don't jump into the lowest tasks.

Means can be drawn to right / below the goal statement

How: Post major headers are there, you know break down them to the actual task level. Each post it for a "means", goes below / side of it.

This would take you to level 3 of the tree ( goal- means - how)

Don't stop at 3, some tasks have more break down levels.

ideally, level 3/4/5 should be in a state to be handed over to a single person/ team to do

If the team present can't break it down due to lack of expertise, the task can be given to another team to break down e.g. if your team is not into technology - you may invite IT expert and then he/she take the box to his / her team to breakdown

Once done - review for completeness and logical flow.

have you forgotten something

do we really need to do this

we will get the desired results in the time we are thinking

Draw lines connecting the tasks

Now your tree is completed, handover the tasks to appropriate people/groups and move to schedule planning

PDPC: Process Decision Program Chart - its used to consider various risks in implementation and planning their responses/ contingencies.

Reach till level 2 - major tasks (Means) - ask what could go wrong, to make level 3

At level 4, write possible, feasible responses

choose the most effective responses and update your overall plan

WBS: work break down the structure - this used for project planning and trick is to break down to a manageable task level

CTQ /KPI drill down tree : Critical to Quality / Key Performance Indicator tree - used to start from a broad CTQ/ KPI Or a customer desire and then break it down to the area/KPIs that has most impact and you want to work on to do the project ( see example)

## Example of a Tree Diagram, Sample CTQ Tree

Venue and budget pre decided – so they become constraints to work with

Invite – can have further breakdown – design and send invite

Red ones are risk measures ( refer to PDPC in variations)

## CTQ/KPI Tree

- Demo Videos
- Interactive Product Tours
- Request Demo

## How to use 5 Whys Tree Diagram for Root Cause Analysis?

When there are issues such as unsatisfied customers, decreasing market share, poor quality, etc. you have to understand the root cause of the issue. Only by addressing the root cause can a problem be fixed. Root Cause Analysis (RCA) is a method of problem-solving used for identifying the root causes of faults or problems. It is widely used in a wide variety of industries.

## What is A Root Cause?

All trees have a trunk, branches, and leaves. These are parts that immediately spring to mind since they are the ones we can see. The part we don’t see is the root system that anchors the tree to the ground. This root system often occupies a far larger area than the tree itself. It also continues to grow even if the tree’s branches have had a hard pruning. That’s why the system of drilling down to get to the heart of an issue is called Root Cause Analysis.

## Why Root Cause Analysis?

Root cause analysis is an important step to enable companies to make the right changes to prevent faults from happening over and over again. There are three ways of dealing with recurrent problems. We can:

- Ignore them.
- Perform a temporary fix.
- Get to the bottom of why they’re happening in the first place.

If we take the first option, the problem will never be solved and could escalate. If we take option two, it is the equivalent of painting over a stain or sticking a piece of tape over a leak and hoping it will hold – you’re treating the symptoms, but not the cause.

Taking the third option, i.e. analyzing the root cause, is the most time consuming, but should allow you to take steps to ensure that the problem never occurs in the future.

## How to use the 5 WHY analysis?

Here are some easy steps to effectively perform a 5 WHY analysis:

- Write down the problem. Writing helps you to formalize the problem and describe it completely. If you work with a team, it also helps the team to focus on the same problem.
- Ask yourself why did the problem happen and write down the answer.
- Ask yourself – looking at your answer – again why did the problem happen, and write down the answer.
- And again, ask yourself – looking at your answer – why did the problem happen.
- Ask yourself this question as often as necessary until the team agrees that the problem’s real root cause is identified. This may take fewer or more times than 5 WHYS.

## Performing Root Cause Analysis with Multiple 5 Whys using Tree Diagram

Not like a single 5 whys analysis hand one cause at a time, the tree diagram can be used to narrow down and eliminate possible causes in a diagram, ideally to one or more addressable root causes to be considered at one single diagram.

Edit this Diagram

## How to Create a 5 Whys Tree Diagram?

The Five Whys exercise is a questioning technique for going beyond symptoms of problems to identify the underlying or root causes of a problem. To facilitate the Five Whys Tree process, follow these steps:

- Below it, list the possible causes of the problem by asking the question “Why?” or “Why is that true?” or “Why is that happening?”
- For each of the causes, again ask the question “Why?”, and list the responses below.
- Continue this process at least 5 times or until you have reached the source of the problem, the lowest level cause stakeholders can do something about, or the response “That is just the way it is, or that is just what happened.”

For example: If your problem is that your car won’t start, the cause could be that it has no fuel. The root cause of this could then be that you forgot to fill up the tank, and the corrective action is, of course, to find some fuel.

Root Cause Analysis (RCA) can be decomposed into 4 steps:

- Identify and describe clearly the problem – Write down the specific problem. Writing the issue helps you formalize the problem and describe it completely. It also helps a team focus on the same problem.
- Identify any issues that contributed to the problem – Ask Why the problem happens and write the answer down below the problem
- Determine root causes – If the answer you just provided doesn’t identify the root cause of the problem that you wrote down in Step 1, ask Why again and write that answer down. Repeat and until the team is in agreement that the problem’s root cause is identified. Again, this may take fewer or more times than five Whys.
- Identify recommendations for the recurrence of problems in the future and implement the necessary solutions

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- What is decision tree analysis? 5 steps ...

## What is decision tree analysis? 5 steps to make better decisions

Decision tree analysis involves visually outlining the potential outcomes, costs, and consequences of a complex decision. These trees are particularly helpful for analyzing quantitative data and making a decision based on numbers. In this article, we’ll explain how to use a decision tree to calculate the expected value of each outcome and assess the best course of action. Plus, get an example of what a finished decision tree will look like.

Have you ever made a decision knowing your choice would have major consequences? If you have, you know that it’s especially difficult to determine the best course of action when you aren’t sure what the outcomes will be.

## What is a decision tree?

A decision tree is a flowchart that starts with one main idea and then branches out based on the consequences of your decisions. It’s called a “decision tree” because the model typically looks like a tree with branches.

These trees are used for decision tree analysis, which involves visually outlining the potential outcomes, costs, and consequences of a complex decision. You can use a decision tree to calculate the expected value of each outcome based on the decisions and consequences that led to it. Then, by comparing the outcomes to one another, you can quickly assess the best course of action. You can also use a decision tree to solve problems, manage costs, and reveal opportunities.

## Decision tree symbols

A decision tree includes the following symbols:

Alternative branches: Alternative branches are two lines that branch out from one decision on your decision tree. These branches show two outcomes or decisions that stem from the initial decision on your tree.

Decision nodes: Decision nodes are squares and represent a decision being made on your tree. Every decision tree starts with a decision node.

Chance nodes: Chance nodes are circles that show multiple possible outcomes.

End nodes: End nodes are triangles that show a final outcome.

A decision tree analysis combines these symbols with notes explaining your decisions and outcomes, and any relevant values to explain your profits or losses. You can manually draw your decision tree or use a flowchart tool to map out your tree digitally.

## What is decision tree analysis used for?

You can use decision tree analysis to make decisions in many areas including operations, budget planning, and project management . Where possible, include quantitative data and numbers to create an effective tree. The more data you have, the easier it will be for you to determine expected values and analyze solutions based on numbers.

For example, if you’re trying to determine which project is most cost-effective, you can use a decision tree to analyze the potential outcomes of each project and choose the project that will most likely result in highest earnings.

## How to create a decision tree

Follow these five steps to create a decision tree diagram to analyze uncertain outcomes and reach the most logical solution.

## 1. Start with your idea

Begin your diagram with one main idea or decision. You’ll start your tree with a decision node before adding single branches to the various decisions you’re deciding between.

For example, if you want to create an app but can’t decide whether to build a new one or upgrade an existing one, use a decision tree to assess the possible outcomes of each.

In this case, the initial decision node is:

Create an app

The three options—or branches—you’re deciding between are:

Building a new scheduling app

Upgrading an existing scheduling app

Building a team productivity app

## 2. Add chance and decision nodes

After adding your main idea to the tree, continue adding chance or decision nodes after each decision to expand your tree further. A chance node may need an alternative branch after it because there could be more than one potential outcome for choosing that decision.

For example, if you decide to build a new scheduling app, there’s a chance that your revenue from the app will be large if it’s successful with customers. There’s also a chance the app will be unsuccessful, which could result in a small revenue. Mapping both potential outcomes in your decision tree is key.

## 3. Expand until you reach end points

Keep adding chance and decision nodes to your decision tree until you can’t expand the tree further. At this point, add end nodes to your tree to signify the completion of the tree creation process.

Once you’ve completed your tree, you can begin analyzing each of the decisions.

## 4. Calculate tree values

Ideally, your decision tree will have quantitative data associated with it. The most common data used in decision trees is monetary value.

For example, it’ll cost your company a specific amount of money to build or upgrade an app. It’ll also cost more or less money to create one app over another. Writing these values in your tree under each decision can help you in the decision-making process .

You can also try to estimate expected value you’ll create, whether large or small, for each decision. Once you know the cost of each outcome and the probability it will occur, you can calculate the expected value of each outcome using the following formula:

Expected value (EV) = (First possible outcome x Likelihood of outcome) + (Second possible outcome x Likelihood of outcome) - Cost

Calculate the expected value by multiplying both possible outcomes by the likelihood that each outcome will occur and then adding those values. You’ll also need to subtract any initial costs from your total.

## 5. Evaluate outcomes

Once you have your expected outcomes for each decision, determine which decision is best for you based on the amount of risk you’re willing to take. The highest expected value may not always be the one you want to go for. That’s because, even though it could result in a high reward, it also means taking on the highest level of project risk .

Keep in mind that the expected value in decision tree analysis comes from a probability algorithm. It’s up to you and your team to determine how to best evaluate the outcomes of the tree.

## Pros and cons of decision tree analysis

Used properly, decision tree analysis can help you make better decisions, but it also has its drawbacks. As long as you understand the flaws associated with decision trees, you can reap the benefits of this decision-making tool.

When you’re struggling with a complex decision and juggling a lot of data, decision trees can help you visualize the possible consequences or payoffs associated with each choice.

Transparent: The best part about decision trees is that they provide a focused approach to decision making for you and your team. When you parse out each decision and calculate their expected value, you’ll have a clear idea about which decision makes the most sense for you to move forward with.

Efficient: Decision trees are efficient because they require little time and few resources to create. Other decision-making tools like surveys, user testing , or prototypes can take months and a lot of money to complete. A decision tree is a simple and efficient way to decide what to do.

Flexible: If you come up with a new idea once you’ve created your tree, you can add that decision into the tree with little work. You can also add branches for possible outcomes if you gain information during your analysis.

There are drawbacks to a decision tree that make it a less-than-perfect decision-making tool. By understanding these drawbacks, you can use your tree as part of a larger forecasting process.

Complex: While decision trees often come to definite end points, they can become complex if you add too many decisions to your tree. If your tree branches off in many directions, you may have a hard time keeping the tree under wraps and calculating your expected values. The best way to use a decision tree is to keep it simple so it doesn’t cause confusion or lose its benefits. This may mean using other decision-making tools to narrow down your options, then using a decision tree once you only have a few options left.

Unstable: It’s important to keep the values within your decision tree stable so that your equations stay accurate. If you change even a small part of the data, the larger data can fall apart.

Risky: Because the decision tree uses a probability algorithm, the expected value you calculate is an estimation, not an accurate prediction of each outcome. This means you must take these estimations with a grain of salt. If you don’t sufficiently weigh the probability and payoffs of your outcomes, you could take on a lot of risk with the decision you choose.

## Decision tree analysis example

In the decision tree analysis example below, you can see how you would map out your tree diagram if you were choosing between building or upgrading a new software app.

As the tree branches out, your outcomes involve large and small revenues and your project costs are taken out of your expected values.

Decision nodes from this example :

Build new scheduling app: $50K

Upgrade existing scheduling app: $25K

Build team productivity app: $75K

Chance nodes from this example:

Large and small revenue for decision one: 40 and 55%

Large and small revenue for decision two: 60 and 38%

Large and small revenue for decision three: 55 and 45%

End nodes from this example:

Potential profits for decision one: $200K or $150K

Potential profits for decision two: $100K or $80K

Potential profits for decision three: $250K or $200K

Although building a new team productivity app would cost the most money for the team, the decision tree analysis shows that this project would also result in the most expected value for the company.

## Use a decision tree to find the best outcome

You can draw a decision tree by hand, but using decision tree software to map out possible solutions will make it easier to add various elements to your flowchart, make changes when needed, and calculate tree values. With Asana’s Lucidchart integration, you can build a detailed diagram and share it with your team in a centralized project management tool .

Decision tree software will make you feel confident in your decision-making skills so you can successfully lead your team and manage projects.

## Related resources

## How to create a winning marketing plan (with examples)

## Project management software and tools: Your best picks for 2023

## SWOT analysis: What it is and how to use it (with examples)

## SMART Goals: How To Write Them and Why They Matter

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## You may also like

## Master Minding

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

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Which of these games would you play to give yourself the best possible chance of winning a prize?

## How to Use Trees and Fish to Diagram Root Causes

How effective we are at solving problems and keeping them solved depends on our ability to address them at their source. When we put out fires but fail to put in measures to prevent similar ones in the future, we fight the same fires again and again. Instead of increasing our fire-fighting budget, we should increase our investment in prevention at the source.

## How to Identify Root Causes

There are a several common techniques for identifying likely root causes. Either for the sake of simplicity, or because it worked in the past, many organizations zero in only one favored technique. Often this becomes an exercise in compliance. People fill out a root cause analysis template to justify a temporary fix, rather than thinking through how to approach the problem.

No single root cause analysis tool or method is versatile or reliable enough to work in every situation. The starter kata approach is increasingly popular due to its simplicity and versatility. Different situations call for different tools. When in doubt, we can go a long way with a combination of fish, trees, and go see.

## Get a Grip on Reality

Solving real problems is easiest when we have a firm grip on reality. Reality tends to exist outside of our heads. But we need our heads in order to process the objective reality that we perceive into actionable thoughts. In short, we need to go see.

Directly observing the process is often the simplest way to find the prime suspects that cause a problem. But this option is not always available. This may be due to the nature of the process or problem. This requires us to use a number of techniques to visualize our thinking.

## Widening the Net through Brainstorming

One common technique to visualize our thought process is brainstorming. A group of people knowledgeable about the problem get together and list what they think are the causes. On the positive side, this is easy to do. It is just talking. Done in a structured way with affinity diagrams to simplify and organize brainstorming and PICK charts to help to prioritize, brainstorming can support rapid idea generation.

There are also risks. When we brainstorm causes, our thinking can be subjective and unscientific. Our thinking may not be grounded in a thorough understanding of the factors affecting the process. Also, without data to weigh one idea versus the next, the most vocal or highest ranked voices can sway the outcome of brainstorming.

## Narrowing Down Options with Fishbone Diagrams

The fishbone diagram is not a form of brainstorming. We should not be generating creative ideas about what could be causing the problem, i.e. guessing. The fishbone diagram works with a predetermined set of factors known to affect the outcome of a process.

In general these are known as the “5M” for manpower, material, method, machinery and measurement. These are the factors that contribute to the majority of problems in manufacturing, healthcare, construction, landscaping and so forth. For retail businesses or the sales process of an organization, we can use “5P” for price, promotion, place, people, product.

To identify factors affecting problems in knowledge work, it is a simple matter of adjusting these categories to suit. Simply select four to six plausible factors are known or agreed to affect the outcome the process. Assign each one to one of the rib bones connected to the spine and draw the cause-and-effect spurs. If an organization that has only a limited idea of what factors contribute to a good outcome for their process, they needs to first spend time understanding their processes. Otherwise, problem solving will be superficial.

## Fishbones Don’t Have Roots, Trees Do

The fishbone diagram is not a template for 5 why analysis. Although it is possible to show 2, 3, 4 or more links in the cause-and-effect chain, getting to root cause is not its purpose. The fishbone diagram is more of a map of possible sources of variation in the process. Fishbone diagrams point the way for further investigation.

A tree diagram looks like a fishbone diagram rotated 45 degrees. However, it serves a very different purpose. Unlike a fishbone diagram whose purpose is to go deeper down the list of suspects, the tree diagram is used to narrow down and eliminate possible causes. Ideally this leads to one or more addressable root causes.

## Finally, Five Times “Why?”

Many organizations latch onto the 5 why analysis as a go-to root cause analysis tool. It’s true that this approach can help us go past the surface and find causes of many less complex problems. Often, when we start with 5 why without first properly framing the problem, we get stuck.

The tree diagram is an appropriate tool for practicing 5 why analysis. It’]s structure lends itself to pursuing multiple lines of questioning, repeating “why?” as often as needed, and branching out multiple times, as roots tend to do.

Used in proper combination, these tools and methods help us to see reality clearly, identify areas of investigation, structure our ideas logically and guide our investigation toward finding the root cause.

Jon has dedicated his 25+ year career to the field of kaizen, continuous improvement, and lean management. Jon spent the first eighteen years of his life in Japan, then graduated from McGill University with a bachelor’s in linguistics.

## DAVID HASKELL

Excellent article! I was really struck by the statement that the “fishbone diagram is not a form of brainstorming.” How fine a point would you put on that? I’ve long treated fishbone diagrams as just another form of affinity diagrams, where the categories are set up front. – So as I think through this: would you say that the distinction is that with brainstorming/affinity the goal is opening up as many possibilities for consideration of a problem as possible, where as with a Fishbone diagramming exercise the inputs should really be targeted hypotheses, that is issues you can readily confirm at the gemba?

Hello David Good question. Anything that helps generate ideas can be used for brainstorming. However, that is not the intent of the fishbone diagram. Brainstorming is creative, open-ended, and imaginative while the fishbone diagram is closed-end, looking at scientific factors and measurable outcomes. Like affinity diagrams, fishbone diagrams list ideas under a header. However, affinity diagrams headers are the result of finding natural groupings and relationships among ideas generated, while the fishbone begins with a list of common factors (the 5M, etc.). Also, the ideas under the affinity are not often causally linked to each other, while they are be design in the fishbone diagram as the branching bones show. Both are constraining / converging tools, the difference is that brainstorming starts out by looking for a large number of ideas while root cause analysis is looking for ideally just one. The issues on the fishbone should not be targeted hypotheses, rather, likely starting points for investigation via go see, tree diagram, 5 why analysis, design of experiments and so forth. Not sure that answered your question on whether it’s a fine point or otherwise.

## Matt Kellett

In the book “how to be successful with continuous improvement” the author used a fish one with problem statement then had teams think of all the causes that could create the problem. They then affinities those caused by similarity and created custome headings for the fish one legs vs using the 5m or 5p headings. Ultimately they would then zero in on the leg with the most likely drivers (Pareto) and select top 3 potential causes on the leg then begin their RCA – 5 why (drip, drip – hopefully most people get that reference. Lol.) or data pulls or gemba etc. Do you see any weaknesses to this approach of grouping vs using the 5m/5p? I’ve tried both. Teams early in their journey like the affinities approach but I’m not sure if there are trade offs that I cant see to that approach. Thoughts?

The weakness with the brainstorming + affinities approach is that you are guessing. Focusing on 5M or other science-based variables points us to where we need to make observation and further study. As you mentioned. It is less a question of whether a team is early in their CI journey and more one of whether they understand their processes and the factors that contribute to good outcomes. Creating the custom headings is OK if that is a means to understanding the process thoroughly, and as long as we don’t reimagine the wheel when existing categories will do.

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## Tree diagram definition:

Coin flip probability using tree diagram:, dice probability using a tree diagram, coin and dice probability using a tree diagram, probability of sampling using a tree diagram, probability without replacement using a tree diagram, practice questions, tree diagram: explanation & examples .

It might be a good idea to refresh the following topics to help understand this article better.

- Basic probability theory.
- Coin flip probabilities.
- Dice Probabilities.
- Probability with replacement.
- Probability without replacement.
- Bernoulli trials.

After reading this article, you should understand the following concepts:

- What a tree diagram is.
- How to make a tree diagram.
- How to solve coin flip problems using tree diagrams.
- How to find dice probabilities using tree diagrams.
- How to use tree diagrams to represent Bernoulli trials.

## What is a tree diagram?

In mathematics, tree diagrams make it easy to visualize and solve probability problems. They are a significant tool in breaking the problem down in a schematic way. While tree diagrams can convert many complicated problems into simple ones, they are not very useful when the sample space becomes too large.

A probability tree diagram represents all the possible outcomes of an event in an organized manner. It starts with a dot and extends into branches. The probability of each outcome is written on its branch.

## How to make a tree diagram

Let’s consider an example and draw a tree diagram for a single coin flip. We know that a coin flip has one of the two possible outcomes: heads ($H$) and tails ($T$). Each outcome has a probability of $1/2$. So we can represent this in a tree diagram as

Now let’s assume that we flip the same coin one more time. Suppose the outcome of the first flip is head, the outcome of the second event can be either heads or tails, and the corresponding branches are shown in red in the diagram below.

Similarly, if we assume that the outcome of the first event is tails, then the possible outcomes of the second flip are depicted in blue in the tree diagram below:

Finally, we can make a complete tree diagram of the two coin flips, as shown below.

Note that two possible outcomes of two coin flips are depicted as $\{HH, HT, TH, TT\}$. To calculate the probability of any single event, we need to multiply the probabilities along the branches. If we need to evaluate the probability of multiple events or a compound event, such as $\{HH, TT\}$, then we add the final probabilities of the individual events down the column. Let us consider an example to clarify these ideas.

Example 1 :

A fair coin is flipped three times. Draw a tree diagram to calculate the probability of the following events:

- Getting three Tails.
- Getting two Heads.
- Getting no Heads.

1) Getting three Tails

From the tree diagram, we can see that only one outcome corresponds to the event of getting all three Tails. To get probabilities out of a tree diagram, we multiply the probabilities along the branches. So, the probability of getting three Tails is

$P(\textrm{Three Tails}) = \frac12 \times \frac12 \times \frac12=\frac18$.

2) Getting two Heads

We can see that there are three events that have two Heads,i.e., $E1=\{HHT\}$, $E2=\{HTH\}$ and $E3=\{THH\}$. So we will add the probabilities of each event down the final column of the tree diagram:

$P(E1)=\frac12 \times \frac12 \times \frac12=\frac18$.

$P(E2)=\frac12 \times \frac12 \times \frac12=\frac18$.

$P(E3)=\frac12 \times \frac12 \times \frac12=\frac18$.

So we can write the probability of getting two tails as

$P(\textrm{Two Tails}) = P(E1)+P(E2)+P(E3) = \frac18+\frac18+\frac18=\frac{3}{8}$.

2) Getting no Heads

From the tree diagram, we can see that the probability of getting no Heads is

$P(\textrm{no Heads}) = \frac12 \times \frac12 \times \frac12=\frac18$.

Dice probabilities play an important role in probability theory. We usually consider multiple rolls of a six-sided fair die. The six possible outcomes of each roll, i.e., $\{1,2,3,4,5,6\}$ are considered to be equally likely, and every single outcome has a probability $\frac16$.

Tree diagrams are particularly useful in solving multiple rolls of a fair die when we are interested in a particular number, e.g., questions like getting a single in 2 in three rolls or not getting a 5 in four rolls, etc. Let us consider a few examples.

We roll a single die three times. Find the probability of the following events using a tree diagram:

- We don’t get a 5 in all three attempts.
- We get only one 5 in three attempts.

Let F represent the five and F’ represents not a five.

The event that no five appears in all three attempts is highlighted in red in the tree diagram. We calculate the probability as follows:

$P(F’F’F’)=\frac56 \times \frac56 \times \frac56=\frac{125}{216}$.

There are three outcomes in the tree diagram (highlighted in blue) that correspond to the event that only one five appears in three appempt. The corresponding probability is calculated as

$P(\textrm{One four in three attempts}) = P(FF’F’) + P(F’FF’) + P(F’F’F)$

$\qquad \qquad \qquad \qquad \qquad \qquad \quad = (\frac56 \times \frac56 \times \frac16)+(\frac56 \times \frac56 \times \frac16)+(\frac56 \times \frac56 \times \frac16)=\frac{125}{216}$.

We can combine both coin flip and roll of dice into a single probabilistic experiment, and tree diagrams help visualize and solve such questions. Let’s consider an example where we flip a coin and roll a die simultaneously.

Example: We roll a dice and flip a coin at random. Find the probability of:

a) getting Tails and an even number. b) getting Tails or Heads and an odd number.

a) From the tree diagram, we see that three possibilities correspond to a Tail and an even number, i.e., $(T,2), (T,4), (T,6)$. The probability of getting Tails is $\frac12$, and the probability of getting any single number is $\frac16$ (We have not shown these probabilities on top of branches to reduce clutter in the diagram). The probability of every single event. i.e., $(T,2)$ or $(T,4)$ or $(T,6)$ is then $\frac12 \times \frac16 =\frac{1}{12}$. Finally, we add these individual probabilities to get the final answer

$P(\textrm{Tails and an even}) = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} = \frac{3}{12} = \frac14$.

b) If we get Heads, then there are three possibilities of getting an odd number, as shown in the tree diagram, i.e., $(H,1), (H,3), (H,5)$. The probability of getting Heads is $\frac12$ and getting any single number is $\frac16$. So, the probability of $(H,1)$ or $(H,3)$ or $(H,5)$ is $\frac12 \times \frac16 = \frac{1}{12}$. Similarly, for Tails, we have three possibilities of getting an odd number, i.e., $(T,1), (T,3), (T,5)$. Each possibility has a probability $\frac{1}{12}$. To get the required probability, we need to add the probabilities of all the required possibilities, i.e.,

$P(\textrm{Heads or Tails and an odd number}) = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} = \frac{6}{12} = \frac12$.

In probability theory, many situations deal with sampling from a given collection. For example, sampling a card from a deck of 52 cards, sampling a ball from a bucket of different colored balls, sampling an item from a set of defective and non-defective items, etc. Sampling may be done with replacement, i.e., the sampled object is replaced in the collection. The sampling may be done without replacement, i.e., the sampled object is not replaced in the collection, and so probabilities of the next sample are dependent on the previous sample. In either case, tree diagrams offer a useful tool to visualize and solve these sampling questions.

## Sampling with replacement

Let’s suppose there are thirteen balls in a box. Five balls are Green(G), and eight balls are Red(R). If we draw two balls, one at a time, with replacement, find the probability of the following events:

- Both Balls are Green.
- Both balls are Red.
- The first ball is Green and the second is Red.
- The first ball is Red and the second is Green.

We can solve this question by drawing a tree diagram as shown below:

A bag contains 10 balls. 3 are blue, and 7 are red. A ball is drawn at random and NOT replaced in the bag. Draw a tree diagram to represent the probabilities of drawing two consecutive balls of the same color.

Notice that the probabilities of drawing a Red or Blue ball are different in the second draw as compared to the first draw. For instance, in the first draw, we have $3$ blue and $7$ red balls, so the probability of drawing a Blue ball is $\frac{3}{10}$. For the second draw, if we assume that a Blue ball was drawn in the first draw, then there would be $2$ Blue and $7$ Red balls left, and hence the probability of drawing another Blue ball is $\frac{2}{9}$, as shown in the top branch of the second draw. We calculate all the second draw probabilities using a similar argument and show them on top of their respective branches. Finally, the probability of drawing two balls of the same color is found by adding the probabilities corresponding to $(B,B)$ and $(R,R)$ outcomes, i.e.,

$P(\textrm{Two balls of same color})=P(R,R)+P(B,B)$

$=\frac{7}{15}+\frac{1}{15}=\frac{8}{15}$.

## Bernoulli Trials and tree diagrams

One of the most useful applications of tree diagrams is in visualizing and solving questions related to Bernoulli Trials.

Bernoulli Trials refer to probabilistic events with only two possible outcomes, success and failure. If the probability of success is assumed to be $p$, then the probability of failure is $1-p$. In Bernoulli trials, we assume that the probability of success and failure remains the same for each trial.

There are two important questions that we are usually interested in Bernoulli Trials problems.

- The probability of $k$ successes in $n$ trials.
- The probability of first success in $k$ trials.

Both these questions can be solved using tree diagrams, as shown in the examples.

Example: Suppose a factory is producing light bulbs. The probability that any light bulb is defective is $p = 0.01$. A tester is testing light bulbs at random. What is the probability of the following events:

- Finding 2 defective light bulbs in 3 tests.
- Finding no defective light bulbs in 3 tests.
- The first defective light bulb is found at the third attempt.
- The first defective light bulb is found within the first two attempts.

Let D represents a ”defective light bulb” and D’ represents a ”not defective light bulb”.

The probability of a defective light bulb is given to be $P(D)=0.01$. From basic probability theory, we know that:

$P(D’)=1-P(D)=1-(0.01)=0.99$.

1. Finding 2 defective light bulbs:

$P(\textrm{finding 2 defective light bulbs})=P(D’, D, D)+P(D, D’, D)+P(D, D, D’)$

$ =(0.99\times 0.01 \times 0.01)+(0.01\times 0.99 \times 0.01)+(0.01\times 0.01 \times 0.99)$.

$ =0.000099+0.000099+0.000099=0.000297$.

2. Finding no defective light bulbs:

$P(\textrm{finding no defective light bulbs})=P(D’, D’, D’)$.

$=(0.99 \times 0.99 \times 0.99) = 0.9703$.

3. The first defective light bulb is found at the third attempt:

$P(\textrm{1st defective light bulb at 3rd attempt})=P(D’, D’, D)$.

$=(0.99 \times 0.99 \times 0.01) = 0.009801$.

4. The first defective light bulb is found within the first two attempts:

$P(\textrm{1st defective light bulb at first 2 attempts})=P(D, D, D’)$.

$=(0.01 \times 0.01 \times 0.99) = 0.000099$.

- The letters of the word ‘SUCCESS’ are printed on 7 cards. Jacob chooses a card at random, replaces it, then chooses a card again. Calculate the probability using a tree diagram that only one of the cards he chooses has the letter C printed on it.
- Getting an even number in all three attempts.
- Getting at least two even numbers in three attempts.

3.Three fair coins are tossed simultaneously. Use a tree diagram to determine the probability of getting:

- At least 2 Tails.
- At most, two Heads.
- No Tails at all.

4. Two cards are drawn from a deck of 52 cards without replacement. What is the probability

- Both cards are Kings.
- Atleast one of the cards is a King
- C’ represents Not the letter C.

We can see from the tree diagram that the probability for one of the card he chooses has ‘ C ‘ printed on it is:

$P(\textrm{One of the card is C})=P(C,C’)+P(C’,C)$

$= (\frac27 \times \frac57)+(\frac57 \times \frac27) = \frac{20}{49}$.

$P(\textrm{All even}) = P(E,E,E) = \frac{1}{216}$.

$P(\textrm{Two evens}) = P(E,E,E’) + P(E,E’,E) + P(E’,E,E) = \frac{15}{216}$.

$P(\textrm{atleast two Tails}) = P(T,T,H) + P(T,H,T) + P(H,T,T) + P(T,T,T) = \frac12$.

$P(\textrm{at most two Heads}) = 1 – P(H,H,H) = \frac78$.

$P(\textrm{No tails}) = P(H,H,H) = \frac18$.

$P(\textrm{Both Kings}) = P(K,K) = \frac{1}{221}$.

$P(\textrm{Atleast one King}) = P(K,K’) + P(K’,K) + P(K,K) = \frac{33}{221}$.

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## 7 Powerful Problem-Solving Root Cause Analysis Tools

The first step to solving a problem is to define the problem precisely. It is the heart of problem-solving.

Root cause analysis is the second important element of problem-solving in quality management. The reason is if you don't know what the problem is, you can never solve the exact problem that is hurting the quality.

Manufacturers have a variety of problem-solving tools at hand. However, they need to know when to use which tool in a manner that is appropriate for the situation. In this article, we discuss 7 tools including:

- The Ishikawa Fishbone Diagram (IFD)
- Pareto Chart
- Failure Mode and Effects Analysis (FMEA)
- Scatter Diagram
- Affinity Diagram
- Fault Tree Analysis (FTA)

## 1. The Ishikawa Fishbone Diagram IFD

The model introduced by Ishikawa (also known as the fishbone diagram) is considered one of the most robust methods for conducting root cause analysis. This model uses the assessment of the 6Ms as a methodology for identifying the true or most probable root cause to determine corrective and preventive actions. The 6Ms include:

- Measurement,
- Mother Nature- i.e., Environment

Related Training: Fishbone Diagramming

## 2. Pareto Chart

The Pareto Chart is a series of bars whose heights reflect the frequency or impact of problems. On the Chart, bars are arranged in descending order of height from left to right, which means the categories represented by the tall bars on the left are relatively more frequent than those on the right.

Related Training: EFFECTIVE INVESTIGATIONS AND CORRECTIVE ACTIONS (CAPA) Establishing and resolving the root causes of deviations, problems and failures

This model uses the 5 Why by asking why 5 times to find the root cause of the problem. It generally takes five iterations of the questioning process to arrive at the root cause of the problem and that's why this model got its name as 5 Whys. But it is perfectly fine for a facilitator to ask less or more questions depending on the needs.

Related training: Accident/Incident Investigation and Root Cause Analysis

## 4. Failure Mode and Effects Analysis (FMEA)

FMEA is a technique used to identify process and product problems before they occur. It focuses on how and when a system will fail, not if it will fail. In this model, each failure mode is assessed for:

- Severity (S)
- Occurrence (O)
- Detection (D)

A combination of the three scores produces a risk priority number (RPN). The RPN is then provided a ranking system to prioritize which problem must gain more attention first.

Related Training: Failure Mode Effects Analysis

## 5. Scatter Diagram

A scatter diagram also known as a scatter plot is a graph in which the values of two variables are plotted along two axes, the pattern of the resulting points revealing any correlation present.

To use scatter plots in root cause analysis, an independent variable or suspected cause is plotted on the x-axis and the dependent variable (the effect) is plotted on the y-axis. If the pattern reflects a clear curve or line, it means they are correlated. If required, more sophisticated correlation analyses can be continued.

Related Training: Excel Charting Basics - Produce Professional-Looking Excel Charts

## 6. Affinity Diagram

Also known as KJ Diagram, this model is used to represent the structure of big and complex factors that impact a problem or a situation. It divides these factors into small classifications according to their similarity to assist in identifying the major causes of the problem.

## 7. Fault Tree Analysis (FTA)

The Fault Tree Analysis uses Boolean logic to arrive at the cause of a problem. It begins with a defined problem and works backward to identify what factors contributed to the problem using a graphical representation called the Fault Tree. It takes a top-down approach starting with the problem and evaluating the factors that caused the problem.

Related Training: Fault Tree Analysis: A Risk Management Tool

Finding the root cause isn't an easy because there is not always one root cause. You may have to repeat your experiment several times to arrive at it to eliminate the encountered problem. Using a scientific approach to solving problem works. So, its important to learn the several problem-solving tools and techniques at your fingertips so you can use the ones appropriate for different situations.

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## Probability Tree Diagrams

Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ... tree diagrams to the rescue!

Here is a tree diagram for the toss of a coin:

We can extend the tree diagram to two tosses of a coin:

How do we calculate the overall probabilities?

- We multiply probabilities along the branches
- We add probabilities down columns

Now we can see such things as:

- The probability of "Head, Head" is 0.5×0.5 = 0.25
- All probabilities add to 1.0 (which is always a good check)
- The probability of getting at least one Head from two tosses is 0.25+0.25+0.25 = 0.75
- ... and more

That was a simple example using independent events (each toss of a coin is independent of the previous toss), but tree diagrams are really wonderful for figuring out dependent events (where an event depends on what happens in the previous event) like this example:

## Example: Soccer Game

You are off to soccer, and love being the Goalkeeper, but that depends who is the Coach today:

- with Coach Sam the probability of being Goalkeeper is 0.5
- with Coach Alex the probability of being Goalkeeper is 0.3

Sam is Coach more often ... about 6 out of every 10 games (a probability of 0.6 ).

So, what is the probability you will be a Goalkeeper today?

Let's build the tree diagram. First we show the two possible coaches: Sam or Alex:

The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)

Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):

If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):

The tree diagram is complete, now let's calculate the overall probabilities. This is done by multiplying each probability along the "branches" of the tree.

Here is how to do it for the "Sam, Yes" branch:

(When we take the 0.6 chance of Sam being coach and include the 0.5 chance that Sam will let you be Goalkeeper we end up with an 0.3 chance.)

But we are not done yet! We haven't included Alex as Coach:

An 0.4 chance of Alex as Coach, followed by an 0.3 chance gives 0.12.

Now we add the column:

0.3 + 0.12 = 0.42 probability of being a Goalkeeper today

(That is a 42% chance)

One final step: complete the calculations and make sure they add to 1:

0.3 + 0.3 + 0.12 + 0.28 = 1

Yes, it all adds up.

You can see more uses of tree diagrams on Conditional Probability .

So there you go, when in doubt draw a tree diagram, multiply along the branches and add the columns. Make sure all probabilities add to 1 and you are good to go.

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## Decision Tree Analysis Examples and How to Use Them

By Letícia Fonseca , May 05, 2022

The purpose of a decision tree analysis is to show how various alternatives can create different possible solutions to solve problems. A decision tree, in contrast to traditional problem-solving methods, gives a “visual” means of recognizing uncertain outcomes that could result from certain choices or decisions.

For those who have never worked with decision trees before, this article will explain how they function and it will also provide some examples to illustrate the ideas.

## Click to jump ahead:

What is a decision tree analysis, what is the importance of using a decision tree analysis, when do you use or apply a decision tree analysis, easy 5 step process of a decision node analysis, decision tree analysis examples, decision tree symbols you need to know, how to create a decision node diagram with venngage, faqs on decision tree analysis.

A decision tree is a diagram that depicts the many options for solving an issue. Given particular criteria, decision trees usually provide the best beneficial option, or a combination of alternatives, for many cases. By employing easy-to-understand axes and graphics, a decision tree makes difficult situations more manageable. An event, action, decision, or attribute linked with the problem under investigation is represented by each box or node.

For risk assessment, asset values, manufacturing costs, marketing strategies, investment plans, failure mode effects analyses (FMEA), and scenario-building, a decision tree is used in business planning. Data from a decision tree can also build predictive models.

There are four basic forms of decision tree analysis , each with its own set of benefits and scenarios for which it is most useful. These subtypes include decision under certainty, decision under risk, decision-making, and decision under uncertainty. In terms of how they are addressed and applied to diverse situations, each type has its unique impact.

Return to Table of Contents

Business owners and other decision-makers can use a decision tree to help them consider their alternatives and the potential repercussions of each one. The examination of a decision tree can be used to:

- Determine the level of risk that each option entails. Before making a final decision, you can see how changing one component impacts others, so you can identify where more research or information is needed. Data from decision trees can also be utilized to build predictive models or to analyze an expected value.
- Demonstrate how particular acts or occurrences may unfold in the context of other events. It’s easy to see how different decisions and possible outcomes interact when you’re looking at decision trees.
- Concentrate your efforts. The most effective ways for reaching the desired and final outcome are shown in decision trees. They can be utilized in a multitude of industries, including goal setting, project management, manufacturing, marketing, and more.

## Advantages of using a tree diagram as a decision-making tool

Decision tree analysis can be used to make complex decisions easier. They explain how changing one factor impacts the other and how it affects other factors by simplifying concepts. A summary of data can also be included in a decision tree as a reference or as part of a report. They show which methods are most effective in reaching the outcome, but they don’t say what those strategies should be.

Even if new information arises later that contradicts previous assumptions and hypotheses, decision-makers may find it difficult to change their minds once they have made and implemented an initial choice. Decision-makers can use decision-making tools like tree analysis to experiment with different options before reaching a final decision; this can help them gain expertise in making difficult decisions.

When presented with a well-reasoned argument based on facts rather than simply articulating their own opinion, decision-makers may find it easier to persuade others of their preferred solution. A decision tree is very useful when there is any uncertainty regarding which course of action will be most advantageous or when prior data is inadequate or partial.

Before implementing possible solutions, a decision tree analysis can assist business owners and other decision-makers in considering the potential ramifications of different solutions.

## Disadvantages of using a tree diagram as a decision-making tool

Rather than displaying real outcomes, decision trees only show patterns connected with decisions. Because decision trees don’t provide information on aspects like implementation, timeliness, and prices, more research may be needed to figure out if a particular plan is viable.

This type of model does not provide insight into why certain events are likely while others are not, but it can be used to develop prediction models that illustrate the chance of an event occurring in certain situations.

A decision tree diagram employs symbols to represent the problem’s events, actions, decisions, or qualities. Given particular criteria, decision trees usually provide the best beneficial option, or a combination of alternatives, for many cases.

By employing easy-to-understand axes and drawings, as well as breaking down the critical components involved with each choice or course of action, decision trees help make difficult situations more manageable. This type of analysis seeks to help you make better decisions about your business operations by identifying potential risks and expected consequences.

In this case, the tree can be seen as a metaphor for problem-solving: it has numerous roots that descend into diverse soil types and reflect one’s varied options or courses of action, while each branch represents the possible and uncertain outcomes. The act of creating a “tree” based on specified criteria or initial possible solutions has to be implemented.

You may start with a query like, “What is the best approach for my company to grow sales?” After that, you’d make a list of feasible actions to take, as well as the probable results of each one. The goal of a decision tree analysis is to help you understand the potential repercussions of your decisions before you make them so that you have the best chance of making a good decision.

Regardless of the level of risk involved, decision tree analysis can be a beneficial tool for both people and groups who want to make educated decisions.

This style of problem-solving helps people make better decisions by allowing them to better comprehend what they’re entering into before they commit too much money or resources. The five-step decision tree analysis procedure is as follows:

- Determine your options

Which can help deal with an issue or answer a question. A problem to be addressed, a goal to be achieved, and additional criteria that will influence the outcome are all required for decision tree analysis to be successful, especially when there are multiple options for resolving a problem or a topic.

- Examine the most effective course of action

Taking into account the potential rewards as well as the risks and expenses that each alternative may entail. If you’re starting a new firm, for example, you’ll need to decide what kind of business model or service to offer, how many employees to hire, where to situate your company, and so on.

- Determine how a specific course will affect your company’s long-term success.

Depending on the data being studied, several criteria are defined for decision tree analysis. For instance, by comparing the cost of a drug or therapy to the effects of other potential therapies, decision tree analysis can be used to determine how effective a drug or treatment will be. When making decisions, a decision tree analysis can also assist in prioritizing the expected values of various factors.

- Use each alternative course of action to examine multiple possible outcomes

This way you can decide which decision you believe is the best and what criteria it meets (the “branches” of your decision tree). Concentrate on determining which solutions are most likely to bring you closer to attaining your goal of resolving your problem while still meeting any of the earlier specified important requirements or additional considerations.

- To evaluate which choice will be most effective

Compare the potential outcomes of each branch. Implement and track the effects of decision tree analysis to ensure that you appropriately assess the benefits and drawbacks of several options so that you can concentrate on the ones that offer the best return on investment while minimizing the risks and drawbacks.

Many businesses employ decision tree analysis to establish an effective business, marketing, and advertising strategies. Based on the probable consequences of each given course of action, decision trees assist marketers to evaluate which of their target audiences may respond most favorably to different sorts of advertisements or campaigns.

A decision tree example is that a marketer might wonder which style of advertising strategy will yield the best results. The decision tree analysis would assist them in determining the best way to create an ad campaign, whether print or online, considering how each option could affect sales in specific markets, and then deciding which option would deliver the best results while staying within their budget.

Another decision tree diagram example is when a corporation that wishes to grow sales might start by determining their course of action, which includes the many marketing methods that they can use to create leads. Before making a decision, they may use a decision tree analysis to explore each alternative and assess the probable repercussions.

If a company chooses TV ads as their proposed solution, decision tree analysis might help them figure out what aspects of their TV adverts (e.g. tone of voice and visual style) make consumers more inclined to buy, so they can better target new customers or get more out of their advertising dollars.

Related: 15+ Decision Tree Infographics to Visualize Problems and Make Better Decisions

Venngage is an online tool that allows you to quickly design attractive and informative decision trees. You’ll need two key components to make a decision node analysis:

- Decision Nodes

Decision nodes are the building blocks of decision tree analysis, and they represent the various options or courses of action open to people or groups. Typically, decision trees have 4-5 decision nodes. To ensure that you can analyze your data afterward, decision nodes should have the same kind as your data: numerical, categorical, etc.

- Decision Branches

In a decision node, decision branches contain both the results and information connected to each choice or alternative. Decision branches normally appear before and after Decision Nodes, however, they can appear in a variety of numbers and directions.

Venngage has built-in templates that are already arranged according to various data kinds, which can assist in swiftly building decision nodes and decision branches. Here’s how to create one with Venngage:

- Sign up for a free account here .
- From Home or your dashboard, click on Templates.

- There are hundreds of templates to pick from, but Venngage’s built-in Search engine makes it simple to find what you’re looking for.

- Once you have chosen the template that’s best for you, click Create to begin editing.

- Venngage allows you to download your project as a PNG, PNG HD, or PDF file with a Premium plan, and an Interactive PDF, PowerPoint, or HTML file with a Business plan.

Venngage also has a business feature called My Brand Kit that enables you to add your company’s logo, color palette, and fonts to all your designs with a single click.

For example, you can make the previous decision tree analysis template reflect your brand design by uploading your brand logo, fonts, and color palette using Venngage’s branding feature.

Not only are Venngage templates free to use and professionally designed, but they are also tailored for various use cases and industries to fit your exact needs and requirements.

A business account also includes the real-time collaboration feature , so you can invite members of your team to work simultaneously on a project.

Venngage allows you to share your decision tree online as well as download it as a PNG or PDF file. That way, your design will always be presentation-ready.

## How important is a decision tree in management?

Project managers can utilize decision tree analysis to produce successful solutions, making it a key element of their success process. They can use a decision tree to think about how each decision will affect the company as a whole and make sure that all factors are taken into account before making a decision.

This decision tree can assist you in making smarter investments as well as identifying any dangers or negative outcomes that may arise as a result of certain choices. You will have more information on what works best if you explore all potential outcomes so that you can make better decisions in the future.

## What is a decision tree in system analysis?

For studying several systems that work together, a decision tree is useful. You can use decision tree analysis to see how each portion of a system interacts with the others, which can help you solve any flaws or restrictions in the system.

## Create a professional decision tree with Venngage

Simply defined, a decision tree analysis is a visual representation of the alternative solutions and expected outcomes you have while making a decision. It can help you quickly see all your potential outcomes and how each option might play out.

Venngage makes the process of creating a decision tree simple and offers a variety of templates to help you. It is the most user-friendly platform for building professional-looking decision trees and other data visualizations. Sign up for a free account and give it a shot right now. You might be amazed at how much easier it is to make judgments when you have all of your options in front of you.

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## Tree Diagrams Of Problem Solving Technique

Powerful drawing solution.

- PROBLEM ANALYSIS. Root Cause Analysis Tree Diagram

## Using Fishbone Diagrams for Problem Solving

## Fishbone Diagram Problem Solving

## Root Cause Analysis

## PROBLEM ANALYSIS. Identify and Structure Factors

## Cause and Effect Analysis

## Decision Making

## Cause and Effect Analysis - Fishbone Diagrams for Problem Solving

## TQM Diagram Tool

## Bar Diagrams for Problem Solving. Create business management bar charts with Bar Graphs Solution

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## Corbettmaths

Videos, worksheets, 5-a-day and much more, tree diagrams practice questions, click here for questions, click here for answers.

probability, trees

## Tree Diagrams

In these lessons, we will look at more examples of probability word problems. We will use tree diagrams to help us solve the problems. We will see that tree diagrams can be used to represent the set of all possible outcomes involving one or more experiments.

Related Pages Using Tree Diagrams In Probability Probability Tree Diagrams More Lessons On Probability Probability Worksheets

Probability tree diagrams are useful for both independent (or unconditional) probability and dependent (or conditional) probability.

The following tree diagram shows the probabilities when a coin is tossed two times. Scroll down the page for more examples and solutions on using probability tree diagrams.

## Tree Diagrams For Independent Events

a) Draw a tree diagram for the experiment. b) What is the probability that the spinners stop at “3” and “4”? c) Find the probability that the spinners do not stop at “3” and “4”. d) What is the probability that the first spinner does not stop at “1”?

Example: Box A contains 3 cards numbered 1, 2 and 3. Box B contains 2 cards numbered 1 and 2. One card is removed at random from each box.

a) Draw a tree diagram to list all the possible outcomes. b) Find the probability that: (i) the sum of the numbers is 4 (ii) the sum of the two numbers is even. (iii) the product of the two numbers is at least 5. (iv) the sum is equal to the product.

Example: A bag contains 4 cards numbered 2, 4, 6, 9. A second bag contains 3 cards numbered 2, 3, 6. One card is drawn at random from each bag.

a) Draw a tree diagram for the experiment. b) With the help of the tree diagram, calculate the probability that the two numbers obtained: (i) have different values. (ii) are both even. (iii) are both prime. (iv) have a sum greater than 5. (v) have a product greater than 16.

How to use probability tree diagrams for independent events (or unconditional probability)?

Example: Jenny has a bag with 7 blue sweets and 3 red sweets. She picks a sweet at random from the bag, replaces it and picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks a) 2 red sweets b) no red sweets c) at least 1 blue sweet d) 1 sweet of each color

How to do a probability tree?

Probability Trees and Independent Events Multiplying and adding probabilities of independent events

Example: Sarah picks a marble from a bag. There are 8 marbles in the bag and 5 of them are green. The rest are red. She looks at the marble and then places it into the bag. She then picks another marble. Complete a probability tree.

How to solve probability problems using tree diagrams?

The following videos gives more examples of solving probability problems using tree diagrams. A tree diagram can help you generate all the outcomes.

Example: Show the possible outcomes of playing Rock, Paper, Scissors. Find the probability of a tie game.

Example: You are ordering a sandwich. Make a list of all the possibilities. Bread - white, sourdough Meat - ham, turkey Cheese - American, Provolone Find the probability of making a sandwich with both white bread and ham.

## Tree Diagrams For Dependent Events

Example: Jimmy has a bag with seven blue sweets and 3 red sweets in it. He picks up a sweet at random from the bag, but does not replaces it and then picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that he picks: (a) two red sweets (b) no red sweets (c) at least one blue sweet (d) one sweet of each color

Probability Diagrams for events that involve with and without replacements.

Example: A bag contains 4 red sweets and 5 blue sweets. Draw a probability tree diagram when a) the sweets are taken with replacement. b) the sweets are taken without replacement.

Example: A bag contains four light bulbs, of which two are defective. We draw bulbs without replacement until a working bulb is selected. Set up the tree diagram for this experiment, find the probability of each outcome, and determine the probability that at most two draws occur.

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Problem Solving › Problem Solving Approaches › Tree Diagrams Article • 9 min read Tree Diagrams Simplifying Complexity MTCT By the Mind Tools Content Team Each idea, solution or event has its own "branch." Smileus / © iStockphoto

Also known as the tree method, problem tree technique, situational analysis or problem analysis, this tool allows us to map or diagram the problem. The structure of a problem tree is: At the roots are the causes of the problem The trunk represents the main problem In the leaves and branches are the effects or consequences.

Root Cause Analysis (RCA) can be decomposed into 4 steps: Identify and describe clearly the problem - Write down the specific problem. Writing the issue helps you formalize the problem and describe it completely. It also helps a team focus on the same problem. Identify any issues that contributed to the problem - Ask Why the problem happens ...

An issue tree, also called logic tree, is a graphical breakdown of a question that dissects it into its different components vertically and that progresses into details as it reads to the right.: 47 Issue trees are useful in problem solving to identify the root causes of a problem as well as to identify its potential solutions. They also provide a reference point to see how each piece fits ...

Tree diagrams and conditional probability AP.STATS: VAR‑4 (EU) , VAR‑4.D (LO) , VAR‑4.D.1 (EK) , VAR‑4.D.2 (EK) CCSS.Math: HSS.CP.A.5 , HSS.CP.B.6 , HSS.CP.B.8 , HSS.CP.B

Tags. Problem tree analysis is central to many forms of project planning and is well developed among development agencies. Problem tree analysis (also called Situational analysis or just Problem analysis) helps to find solutions by mapping out the anatomy of cause and effect around an issue in a similar way to a mind map, but with more structure.

A tree diagram is a new management planning tool that depicts the hierarchy of tasks and subtasks needed to complete and objective. The tree diagram starts with one item that branches into two or more, each of which branch into two or more, and so on. The finished diagram bears a resemblance to a tree, with a trunk and multiple branches.

Tree diagrams are a helpful tool for problem-solving, including troubleshooting and root cause analysis. Software developers can use tree diagrams to analyze coding issues and find fixes during website and app development projects. Anticipating potential workflow issues

A tree diagram is a great visual and thinking tool and is used for a variety of reasons - task mapping, decision, probability tree, to show a logical break down of an argument. We will concentrate on the simpler - task mapping usage It is a task mapping tool for implementation.

Root Cause Analysis (RCA) can be decomposed into 4 steps: Identify and describe clearly the problem - Write down the specific problem. Writing the issue helps you formalize the problem and describe it completely. It also helps a team focus on the same problem. Identify any issues that contributed to the problem - Ask Why the problem happens ...

View Templates Summary Decision tree analysis involves visually outlining the potential outcomes, costs, and consequences of a complex decision. These trees are particularly helpful for analyzing quantitative data and making a decision based on numbers.

So P (H or 4) is 712. Again, we can work this out from the tree diagram, by selecting every branch which includes a Head or a 4: Each of the ticked branches shows a way of achieving the desired outcome. So P (H or 4) is the sum of these probabilities: P(H or 4) = P(H, 4) + P(H, not 4) + P(T, 4) = 112 + 5 12 + 112 = 7 12.

SECTION 9.4 PROBLEM SET: PROBABILITY USING TREE DIAGRAM. Use a tree diagram to solve the following problems. Suppose you have five keys and only one key fits to the lock of a door. What is the probability that you can open the door in at most three tries? A coin is tossed until a head appears.

A tree diagram looks like a fishbone diagram rotated 45 degrees. However, it serves a very different purpose. Unlike a fishbone diagram whose purpose is to go deeper down the list of suspects, the tree diagram is used to narrow down and eliminate possible causes. Ideally this leads to one or more addressable root causes.

A tree diagram represents the hierarchy of the events that need to be completed when solving a problem. The tree diagram starts with one node, and each node has its branches that further extend into more branches, and a tree-like structure is formed. It might be a good idea to refresh the following topics to help understand this article better.

1. The Ishikawa Fishbone Diagram IFD. The model introduced by Ishikawa (also known as the fishbone diagram) is considered one of the most robust methods for conducting root cause analysis. This model uses the assessment of the 6Ms as a methodology for identifying the true or most probable root cause to determine corrective and preventive actions.

Probability Tree Diagrams Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ... tree diagrams to the rescue! Here is a tree diagram for the toss of a coin: There are two "branches" (Heads and Tails) The probability of each branch is written on the branch

By Letícia Fonseca, May 05, 2022. The purpose of a decision tree analysis is to show how various alternatives can create different possible solutions to solve problems. A decision tree, in contrast to traditional problem-solving methods, gives a "visual" means of recognizing uncertain outcomes that could result from certain choices or ...

As we have already seen, tree diagrams play an important role in solving probability problems. A tree diagram helps us not only visualize, but also list all possible outcomes in a systematic fashion. Furthermore, when we list various outcomes of an experiment and their corresponding probabilities on a tree diagram, we gain a better ...

Root Cause Analysis Tree Diagram. Use the Root Cause Diagram to perform visual root cause analysis. Root Cause Analysis Tree Diagram is constructed separately for each highly prioritized factor. The goal of this is to find the root causes for the factor and list possible corrective action. <br>ConceptDraw Office suite is a software for problem analysis. Tree Problem Solving

Root Cause Analysis Tree Diagram. Use the Root Cause Diagram to perform visual root cause analysis. Root Cause Analysis Tree Diagram is constructed separately for each highly prioritized factor. The goal of this is to find the root causes for the factor and list possible corrective action. <br>ConceptDraw Office suite is a software for problem analysis. Tree Diagrams Of Problem Solving Technique

The Corbettmaths Practice Questions on Tree Diagrams. Videos, worksheets, 5-a-day and much more

Show Solution Example: A bag contains 4 cards numbered 2, 4, 6, 9. A second bag contains 3 cards numbered 2, 3, 6. One card is drawn at random from each bag. a) Draw a tree diagram for the experiment. b) With the help of the tree diagram, calculate the probability that the two numbers obtained: (i) have different values.