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11.6 - negative binomial examples, example 11-2 section  .

oil pump

An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. What is the probability that the first strike comes on the third well drilled?

To find the requested probability, we need to find \(P(X=3\). Note that \(X\)is technically a geometric random variable, since we are only looking for one success. Since a geometric random variable is just a special case of a negative binomial random variable, we'll try finding the probability using the negative binomial p.m.f. In this case, \(p=0.20, 1-p=0.80, r=1, x=3\), and here's what the calculation looks like:

\(P(X=3)=\dbinom{3-1}{1-1}(1-p)^{3-1}p^1=(1-p)^2 p=0.80^2\times 0.20=0.128\)

It is at the second equal sign that you can see how the general negative binomial problem reduces to a geometric random variable problem. In any case, there is about a 13% chance thathe first strike comes on the third well drilled.

What is the probability that the third strike comes on the seventh well drilled?

To find the requested probability, we need to find \(P(X=7\), which can be readily found using the p.m.f. of a negative binomial random variable with \(p=0.20, 1-p=0.80, x=7, r=3\):

\(P(X=7)=\dbinom{7-1}{3-1}(1-p)^{7-3}p^3=\dbinom{6}{2}0.80^4\times 0.20^3=0.049\)

That is, there is about a 5% chance that the third strike comes on the seventh well drilled.

What is the mean and variance of the number of wells that must be drilled if the oil company wants to set up three producing wells?

The mean number of wells is:

\(\mu=E(X)=\dfrac{r}{p}=\dfrac{3}{0.20}=15\)

with a variance of:

\(\sigma^2=Var(x)=\dfrac{r(1-p)}{p^2}=\dfrac{3(0.80)}{0.20^2}=60\)

Calcworkshop

Negative Binomial Distribution w/ 7 Worked Examples!

// Last Updated: September 25, 2020 - Watch Video //

Did you know that the negative binomial distribution is a sneaky combination of both the binomial and geometric random variables?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching when to use negative binomial distribution

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

Like before, we have this probability of success that will be the same, but there’s a slight twist.

It’s best if we look to an example, here we go!

Suppose Colette is practicing penalty kicks in preparation for an upcoming soccer match. What is the likelihood that she will make her fifth goal on her eighth shot?

This what we are going to be able to find using the Negative Binomial Distribution!

In general, the negative binomial distribution finds the probability of the Kth success occurring on the Xth trial. Alternatively, it finds x number of successes before resulting in k failures as noted by Stat Trek .

What are the conditions of the Negative Binomial Distribution?

Just like we’ve seen for both the binomial and geometric distribution, we will look at the BINS!

negative binomial distribution mnemonic

Negative Binomial Distribution Mnemonic

Worked Example

So, let’s see how we use these conditions to determine whether a given scenario has a negative binomial distribution.

For example, suppose we shuffle a standard deck of cards, and we turn over the top card. We put the card back in the deck and reshuffle. We repeat this process until we get a 2 Jacks. Is this a negative binomial distribution?

All we have to do is check the BINS! B – binary – yes, either it’s a Jack or not a Jack I – independent – yes, because we replace the card each time, the trials are independent. N – number of trials until you get the kth success – yes, we are told to repeat until we get 2 Jacks. S – success (probability of success) the same – yes, the likelihood of getting a Jack is 4 out of 52 each time you turn over a card.

Therefore, this is an example of a negative binomial distribution.

Okay, so now that we know the conditions of a Negative Binomial Distribution, sometimes referred to as the Pascal Distribution, let’s look at its properties:

pmf and mean and variance of negative binomial distribution

PMF And Mean And Variance Of Negative Binomial Distribution

Notice that the negative binomial distribution, similar to the binomial distribution, does not have a cumulative distribution function. Therefore, if we are asked to find an interval of values, we will have to sum the pmf the desired number of times.

Now, let’s investigate how to use the properties with an example.

Back to Colette and her penalty kicks.

If we are told that the probability Colette scores a goal is 0.78, what is the likelihood that she will make her fifth goal on her eight shot?

negative binomial probability example

Negative Binomial Probability Example

This means that the likelihood of Colette scoring her 5th penalty kick on her 8th shot is 0.1076.

Now let’s determine the number of shots we would expect Colette to attempt in order to make five goals, as well as the standard deviation.

example of a negative binomial with mean and variance and standard deviation

Example Of A Negative Binomial With Mean And Variance And Standard Deviation

What this shows us is that we would expect Colette to take 6.41 shots to make her 5th goal with a standard error or 1.35

And what is interesting to point out is we could have just as easily changed this question from finding Colette’s number of successes to finding the number of failures as well; recognizing that she missed several attempts in her quest to make five penalty kicks.

Throughout this video, we will utilize our conditions for the negative binomial distribution and apply our properties to find expectancy, variance, and probabilities.

Negative Binomial Distribution – Lesson & Examples (Video)

  • Introduction to Video: Negative Binomial Random Variable
  • 00:00:32 – What is the Negative Binomial Distribution and its properties?
  • Exclusive Content for Members Only
  • 00:09:30 – Given a negative binomial distribution find the probability, expectation, and variance (Example #1)
  • 00:18:45 – Find the probability of winning 4 times in X number of games (Example #2)
  • 00:28:36 – Find the probability for the negative binomial (Examples #3-4)
  • 00:36:08 – Find the probability of failure (Example #5)
  • 00:39:15 – Find mean, standard deviation and probability for the distribution (Example #6)
  • 00:45:42 – Find the probability using the negative binomial and binomial distribution (Example #7)
  • Practice Problems with Step-by-Step Solutions
  • Chapter Tests with Video Solutions

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Negative Binomial Distribution

Negative binomial distribution talks about the final success which can be obtained, after a sequence of successes in the preceding trials. Negative binomial distribution refers to the r th success which has been preceded by n - 1 trials, containing r - 1 success.

Let us learn more about the negative binomial distribution, formula, and properties of negative binomial distribution, with the help of examples, FAQs.

What Is Negative Binomial Distribution?

The negative binomial distribution is the distribution of the number of trialn needed to get r th successes. The negative binomial distribution helps in finding r success in x trials. Here we aim to find the specific success event, in combination with the previous needed successes. In negative binomial distribution, the number of trials and the probability of success in each trial are defined clearly. Here we consider the n + r trials needed to get r successes.

Negative Binomial Distribution

Negative Binomial Distribution: f(x) = \(^{n + r - 1}C_{r - 1}.P^r.q^n\)

A binomial experiment is an experiment consisting of a fixed number of independent Bernoulli trials. Each Bernoulli trial is an independent trial and has two possible outcomes, occurrence or non-occurrence (success or failure), and each trial has the same probability of occurrence of successes. The negative binomial distribution talks about the distribution of the number of trials needed to get the defined number of successes. The negative binomial distribution is almost the same as a binomial distribution with one difference: In a binomial distribution we have a fixed number of trials, but in negative binomial distribution we have a fixed number of successes.

A random variable X is supposed to follow a negative binomial distribution if its probability mass function is given by:

f(x) = (n + r - 1)C(r - 1) P r q x , where x = 0, 1, 2, ....., and p + q = 1.

Here we consider a binomial sequence of trials with the probability of success as p and the probability of failure as q.

Let f(x) be the probability defining the negative binomial distribution, where (n + r) trials are required to produce r successes. Here in (n + r - 1) trials we get (r - 1) successes, and the next (n + r) is a success.

Then f(x) = (n + r - 1)C(r - 1) P r-1 q n-1 .p

f(x) = (n + r - 1)C(r - 1) P r q n

Additional Points of Negative Binomial Distribution

The following are the three important points referring to the negative binomial distribution.

  • The mean of the negative binomial distribution is E(X) = rq/P
  • The variance of the negative binomial distribution is V(X)= rq/p 2
  • Here the mean is always greater than the variance. Mean > Variance.

Negative binomial distribution takes an account of all the successes which happen one step before the actual success event, which is further multiplied by the actual success event. Since it takes an account of all the successes one step before the actual success event, it is referred to as a negative binomial distribution. Negative binomial distribution refers to the r th success which has been preceded by n - 1 trial, containing r - 1 success.

A negative binomial distribution is also called a pascal distribution.

Examples Of Negative Binomial Distribution

The following quick examples help in a better understanding of the concept of the negative binomial distribution.

  • If we flip a coin a fixed number of times and count the number of times the coin turns out heads is a binomial distribution. If we continue flipping the coin until it has turned a particular number of heads say the third head-on flipping 5 times, then this is a case of the negative binomial distribution.
  • For a situation involving three glasses to be hit with 7 balls, the probability of hitting the third glass successfully with the seventh ball can be obtained with the help of negative binomial distribution.
  • In a class, if there is a rumor that there is a math test, and the fifth is the second person to believe the rumor, then the probability of this fifth person to be the second person to believe the rumor can be computed using the negative binomial distribution.

Properties Of Negative Binomial Distribution

A negative binomial distribution is a distribution that has the following properties.

  • The negative binomial distribution has a total of n number of trials.
  • Each trial has two outcomes, and one of them is referred to as success and the other as a failure.
  • The probability of success or failure is the same across each of these trials.
  • The probability of success is denoted by p, and the probability of failure is defined as q, and each of these is the same in every trial.
  • The sum of the probability of success and failure is equal to 1. p + q = 1.
  • Each of these trials is independent. The outcome of one trial does not affect the outcome of other trials.
  • The experiment is continued until r success is obtained, and r is defined in advance.
  • The experiment consists of x + r repeated trials, where r is the required number of successes.

Related Topics

The following topics help in a better understanding of negative binomial distribution.

  • Binomial Theorem
  • Theorem Of Total Probability
  • Probability Density Function
  • Multiplication Rule Of Probability
  • Probability of an Impossible Event

Examples on Negative Binomial Distribution

Example 1: Jim is writing an exam with multiple-choice questions, and his probability of attempting the question with the right answer is 60%. What is the probability that Jim gives the third correct answer for the fifth attempted question?

Probability of success P(s) = 60% = 0.6, Probability of failure P(f) = 40% = 0.4.

It is given that Jim gives the third correct answer for the fifth attempted question.

Here we can use the concept of the negative binomial distribution to find the third correct answer for the fifth attempted question.

Here we have x = 5, r = 3, P = 0.6, q = 0.4

The formula for negative binomial distribution is B(x, r, P) = (x - 1)C(r - 1)P r .Q x - r

= (5 - 1)C(3 - 1). (0.6) 3 .(0.4) 5 - 3

= 4C2.(0.6) 3 .(0.4) 2

= 6.(0.216)(0.16)

= (0.1296)(0.16)

Therefore the probability of Jim giving the third correct answer for his fifth attempted question is 0.02.

Example 2: The probability of Ron going on time to school is 80%. What is the probability that Ron goes on time for the eighth time for the first ten days of school?

Probability of Success P(S) = 80% = 0.8, Probability of Failure P(f) = 20% = 0.2

It is given that Ron goes on time for the eight-time for the first ten days of school.

Here also we can use the negative binomial distribution to find the eighth day when he goes on time to school, for the first ten days of school.

Here we have x = 10, r = 8, P = 0.8, q = 0.2

= (10 - 1)C(8 - 1). (0.8) 8 .(0.42 10 - 8

= 9C7.(0.8) 8 .(0.2) 2

= 36.(0.281)(0.04)

= 36(0.01124)

Therefore the probability of Ron going on time for the first ten days is 0.4.

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negative binomial distribution example problems and solutions

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Practice Questions on Negative Binomial Distribution

Faqs on negative binomial distribution.

The negative binomial distribution is the distribution of the number of trials needed to get r th successes. The negative binomial distribution helps in finding r success in x trials. Here we aim to find the specific success event, in combination with the previous needed successes. The formula for negative binomial distribution is f(x) = \(^{n + r - 1}C_{r - 1}.P^r.q^x\).

What Is The Formula For Negative Binomial Distribution?

The formula for negative binomial distribution is f(x) = \(^{n + r - 1}C_{r - 1}.P^r.q^x\). Here n + r is the total number of trials, and r refers to the r th success. Also, p refers to the probability of success, and q refers to the probability of failure, and p + q = 1.

Why Is This Called Negative Binomial Distribution?

Teach yourself statistics

Negative Binomial Distribution

In this lesson, we cover the negative binomial distribution and the geometric distribution. As we will see, the geometric distribution is a special case of the negative binomial distribution.

Negative Binomial Experiment

A negative binomial experiment is a statistical experiment that has the following properties:

  • The experiment consists of x repeated trials.
  • Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
  • The probability of success, denoted by P , is the same on every trial.
  • The trials are independent ; that is, the outcome on one trial does not affect the outcome on other trials.
  • The experiment continues until r successes are observed, where r is specified in advance.

Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:

  • The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads.
  • Each trial can result in just two possible outcomes - heads or tails.
  • The probability of success is constant - 0.5 on every trial.
  • The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
  • The experiment continues until a fixed number of successes have occurred; in this case, 5 heads.

The following notation is helpful, when we talk about negative binomial probability.

  • x : The number of trials required to produce r successes in a negative binomial experiment.
  • r : The number of successes in the negative binomial experiment.
  • P : The probability of success on an individual trial.
  • Q : The probability of failure on an individual trial. (This is equal to 1 - P .)
  • b*( x ; r, P ): Negative binomial probability - the probability that an x -trial negative binomial experiment results in the rth success on the xth trial, when the probability of success on an individual trial is P .
  • n C r : The number of combinations of n things, taken r at a time.
  • n!: The factorial of n (also known as n factorial).

A negative binomial random variable is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution . The negative binomial distribution is also known as the Pascal distribution .

Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a negative binomial experiment. The negative binomial random variable is the number of coin flips required to achieve 2 heads. In this example, the number of coin flips is a random variable that can take on any integer value between 2 and plus infinity. The negative binomial probability distribution for this example is presented below.

Negative Binomial Probability

The negative binomial probability refers to the probability that a negative binomial experiment results in r - 1 successes after trial x - 1 and r successes after trial x . For example, in the above table, we see that the negative binomial probability of getting the second head on the sixth flip of the coin is 0.078125.

Given x , r , and P , we can compute the negative binomial probability based on the following formula:

Negative Binomial Formula . Suppose a negative binomial experiment consists of x trials and results in r successes. If the probability of success on an individual trial is P , then the negative binomial probability is:

b*( x ; r, P ) = x-1 C r-1 * P r * (1 - P) x - r

b*( x ; r, P ) = { (x-1)! / [ (r-1)!(x-r)!] } * P r * (1 - P) x - r

The Mean of the Negative Binomial Distribution

If we define the mean of the negative binomial distribution as the average number of trials required to produce r successes, then the mean is equal to:

where μ is the mean number of trials, r is the number of successes, and P is the probability of a success on any given trial.

Geometric Distribution

The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes ( r ) is equal to 1.

An example of a geometric distribution would be tossing a coin until it lands on heads. We might ask: What is the probability that the first head occurs on the third flip? That probability is referred to as a geometric probability and is denoted by g( x ; P ). The formula for geometric probability is given below.

Geometric Probability Formula . Suppose a negative binomial experiment consists of x trials and results in one success. If the probability of success on an individual trial is P , then the geometric probability is:

g( x ; P ) = P * Q x - 1

Test Your Understanding

The problems below show how to apply your new-found knowledge of the negative binomial distribution (see Example 1) and the geometric distribution (see Example 2).

Negative Binomial Calculator

As you may have noticed, the negative binomial formula requires many time-consuming computations. The Negative Binomial Calculator can do this work for you - quickly, easily, and error-free. Use the Negative Binomial Calculator to compute negative binomial probabilities. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

Example 1 Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. During the season, what is the probability that Bob makes his third free throw on his fifth shot?

Solution: This is an example of a negative binomial experiment. The probability of success ( P ) is 0.70, the number of trials ( x ) is 5, and the number of successes ( r ) is 3.

To solve this problem, we enter these values into the negative binomial formula.

b*( x ; r, P ) = x-1 C r-1 * P r * Q x - r b*( 5 ; 3, 0.7 ) = 4 C 2 * 0.7 3 * 0.3 2 b*( 5 ; 3, 0.7 ) = 6 * 0.343 * 0.09 = 0.18522

Thus, the probability that Bob will make his third successful free throw on his fifth shot is 0.18522.

Let's reconsider the above problem from Example 1. This time, we'll ask a slightly different question: What is the probability that Bob makes his first free throw on his fifth shot?

Solution: This is an example of a geometric distribution, which is a special case of a negative binomial distribution. Therefore, this problem can be solved using the negative binomial formula or the geometric formula. We demonstrate each approach below, beginning with the negative binomial formula.

The probability of success ( P ) is 0.70, the number of trials ( x ) is 5, and the number of successes ( r ) is 1. We enter these values into the negative binomial formula.

b*( x ; r, P ) = x-1 C r-1 * P r * Q x - r b*( 5 ; 1, 0.7 ) = 4 C 0 * 0.7 1 * 0.3 4 b*( 5 ; 3, 0.7 ) = 0.00567

Now, we demonstate a solution based on the geometric formula.

g( x ; P ) = P * Q x - 1 g(5; 0.7) = 0.7 * 0.3 4 = 0.00567

Notice that each approach yields the same answer.

Statology

Statistics Made Easy

An Introduction to the Negative Binomial Distribution

The negative binomial distribution  describes the probability of experiencing a certain amount of failures before experiencing a certain amount of successes in a series of Bernoulli trials.

A Bernoulli trial is an experiment with only two possible outcomes – “success” or “failure” – and the probability of success is the same each time the experiment is conducted.   An example of a Bernoulli trial is a coin flip. The coin can only land on two sides (we could call heads a “success” and tails a “failure”) and the probability of success on each flip is 0.5, assuming the coin is fair.

If a  random variable   X  follows a negative binomial distribution, then the probability of experiencing k  failures before experiencing a total of  r  successes can be found by the following formula:

P(X=k) =  k+r-1 C k  * (1-p) r *p k

  • k:  number of failures
  • r:  number of successes
  • p:  probability of success on a given trial
  • k+r-1 C k :  number of combinations of (k+r-1) things taken k at a time

For example, suppose we flip a coin and define a “successful” event as landing on heads. What is the probability of experiencing 6 failures before experiencing a total of 4 successes?

To answer this, we can use the negative binomial distribution with the following parameters:

  • k:  number of failures = 6
  • r:  number of successes = 4
  • p:  probability of success on a given trial = 0.5

Plugging these numbers in the formula, we find the probability to be:

P(X=6 failures)  = 6+4-1 C 6  * (1-.5) 4 *(.5) 6  = (84)*(.0625)*(.015625) =  0.08203 .

Properties of the Negative Binomial Distribution

The negative binomial distribution has the following properties:

The mean number of failures we expect before achieving r  successes is  pr / (1-p) .

The variance in the number of failures we expect before achieving r  successes is pr  / (1-p) 2 .

For example, suppose we flip a coin and define a “successful” event as landing on heads. 

The mean number of failures (e.g. landing on tails) we expect before achieving 4 successes would be pr/(1-p)  = (.5*4) / (1-.5) = 4 .

The variance in the number of failures we expect before achieving 4 successes would be  pr / (1-p) 2   = (.5*4) / (1-.5) 2 = 8 .

Negative Binomial Distribution Practice Problems

Use the following practice problems to test your knowledge of the negative binomial distribution.

Note:  We will use the Negative Binomial Distribution Calculator to calculate the answers to these questions.

Problem 1

Question:  Suppose we flip a coin and define a “successful” event as landing on heads. What is the probability of experiencing 3 failures before experiencing a total of 4 successes?

Answer: Using the Negative Binomial Distribution Calculator with k = 3 failures, r = 4 successes, and p = 0.5, we find that P(X=3) =  0.15625 .

Problem 2

Question:  Suppose we go door to door selling candy. We consider it a “success” if someone buys a candy bar. The probability that any given person will buy a candy bar is 0.4. What is the probability of experiencing 8 failures before we experience a total of 5 successes?

Answer: Using the Negative Binomial Distribution Calculator with k = 8 failures, r = 5 successes, and p = 0.4, we find that P(X=8) =  0.08514 .

Problem 3

Question:  Suppose we roll a die and define a “successful” roll as landing on the number 5. The probability that the die lands on a 5 on any given roll is 1/6 = 0.167. What is the probability of experiencing 4 failures before we experience a total of 3 successes?

Answer: Using the Negative Binomial Distribution Calculator with k = 4 failures, r = 3 successes, and p = 0.167, we find that P(X=4) =  0.03364 .

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Statistics LibreTexts

3.4: Hypergeometric, Geometric, and Negative Binomial Distributions

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  • Page ID 3263

  • Kristin Kuter
  • Saint Mary's College

In this section, we consider three more families of discrete probability distributions. There are some similarities between the three, which can make them hard to distinguish at times. So throughout this section we will compare the three to each other and the binomial distribution, and point out their differences.

Hypergeometric Distribution

Consider the following example.

Example \(\PageIndex{1}\)

An urn contains a total of \(N\) balls, where some number \(m\) of the balls are orange  and the remaining \(N-m\) are grey . Suppose we draw \(n\) balls from the urn without replacement, meaning once we select a ball we do not place it back in the urn before drawing out the next one. Then some of the balls in our selection may be orange  and some may be grey . We can define the discrete random variable \(X\) to give the number of orange  balls in our selection. The probability distribution of \(X\) is referred to as the hypergeometric distribution , which we define next.

Definition \(\PageIndex{1}\)

Suppose in a collection of \(N\) objects, \(m\) are of type 1 and \(N-m\) are of another type 2. Furthermore, suppose that \(n\) objects are randomly selected from the collection without replacement. Define the discrete random variable \(X\) to give the number of selected objects that are of type 1. Then \(X\) has a hypergeometric distribution with parameters \(N, m, n\). The probability mass function of \(X\) is given by \begin{align*} p(x) &= P(X=x) = P(x\ \text{type 1 objects &}\ n-x\ \text{type 2}) \notag \\ &= \frac{(\text{# of ways to select}\ x\ \text{type 1 objects from}\ m) \times (\text{# of ways to select}\ n-x\ \text{type 2 objects from}\ N-m)}{\text{total # of ways to select}\ n\ \text{objects of any type from}\ N} \notag \\ &= \frac{\displaystyle{\binom{m}{x}\binom{N-m}{n-x}}}{\displaystyle{\binom{N}{n}}} \label{hyperpmf} \end{align*}

In some sense, the hypergeometric distribution is similar to the binomial, except that the method of sampling is crucially different. In each case, we are interested in the number of times a specific outcome occurs in a set number of repeated trials, where we could consider each selection of an object in the hypergeometric case as a trial. In the binomial case we are interested in the number of "successes" in the trials, and in the hypergeometric case we are interested in the number of a certain type of object being selected, which could be considered a "success". However, the trials in a binomial distribution are independent, while the trials in a hypergeometric distribution are not because the objects are selected without replacement. If, in Example 3.4.1 , the balls were drawn with replacement, then each draw would be an independent Bernoulli trial and the distribution of \(X\) would be binomial, since the same number of balls in the urn would be the same each time another ball is drawn. However, when the balls are drawn without replacement, each draw is not independent, since the number of balls in the urn decreases after each draw as well as the number of balls of a given type.

Exercise \(\PageIndex{1}\)

Suppose your friend has 10 cookies, 3 of which are chocolate chip. Your friend randomly divides the cookies equally between herself and you. What is the probability that you get all the chocolate chip cookies?

Let random variable \(X=\) number of chocolate chip cookies you get. Then \(X\) is hypergeometric with \(N=10\) total cookies, \(m=3\) chocolate chip cookes, and \(n=5\) cookies selected by your friend to give to you. We want the probability that you get all the chocolate chip cookies, i.e., \(P(X=3)\), which is $$P(X=3) = \frac{\displaystyle{\binom{3}{3}\binom{7}{2}}}{\displaystyle{\binom{10}{5}}} = 0.083\notag$$

Note that \(X\) has a hypergeometric distribution and not binomial because the cookies are being selected (or divided) without replacement.

Geometric Distribution & Negative Binomial Distribution

The geometric and negative binomial distributions are related to the binomial distribution in that the underlying probability experiment is the same, i.e., independent trials with two possible outcomes. However, the random variable defined in the geometric and negative binomial case highlights a different aspect of the experiment, namely the number of trials needed to obtain a specific number of "successes". We start with the geometric distribution.

Definition \(\PageIndex{2}\)

Suppose that a sequence of independent Bernoulli trials is performed, with \(p = P(\text{"success"})\) for each trial. Define the random variable \(X\) to give the number of trial at which the first success occurs. Then \(X\) has a geometric distribution with parameter \(p\). The probability mass function of \(X\) is given by \begin{align} p(x) = P(X=x) &= P(1^{st}\ \text{success on}\ x^{th}\ \text{trial}) \notag \\ &= P(1^{st}\ (x-1)\ \text{trials are failures &}\ x^{th}\ \text{trial is success}) \notag \\ &= (1-p)^{x-1}p, \quad\text{for}\ x = 1, 2, 3, \ldots \label{geompmf} \end{align}

Exercise \(\PageIndex{2}\)

Verify that the pmf for a geometric distribution (Equation \ref{geompmf}) satisfies the two properties for pmf's , i.e.,

  • \(p(x) \geq 0\), for \(x=1, 2, 3, \ldots\)
  • \(\displaystyle{\sum^{\infty}_{x=1} p(x) = 1}\)   Hint: It's called "geometric" for a reason!
  • Note that \(0\leq p\leq 1\), so that we also have \(0\leq (1-p) \leq 1\) and \(0\leq (1-p)^{x-1} \leq 1\), for \(x=1, 2, \ldots\). Thus, it follows that \(p(x) = (1-p)^{x-1}p \geq 0\).
  • Recall the formula for the sum of a geometric series: $$\sum^{\infty}_{x=1} ar^{x-1} = \frac{a}{1-r}, \quad\text{if}\ |r|<1.\notag$$ Note that the sum of the geometric pmf is a geometric series with \(a=p\) and \(r = 1-p < 1\). Thus, we have $$\sum_{x=1}^{\infty} p(x) = \sum^{\infty}_{x=1} (1-p)^{x-1}p = \frac{p}{1 - (1-p)} = \frac{p}{p} = 1\ \checkmark\notag$$

Example \(\PageIndex{2}\)

Each of the following is an example of a random variable with the geometric distribution.

  • Toss a fair coin until the first heads occurs. In this case, a "success" is getting a heads ("failure" is getting tails) and so the parameter \(p = P(h) = 0.5\).
  • Buy lottery tickets until getting the first win. In this case, a "success" is getting a lottery ticket that wins money, and a "failure" is not winning. The parameter \(p\) will depend on the odds of wining for a specific lottery.
  • Roll a pair of fair dice until getting the first double 1's. In this case, a "success" is getting double 1's, and a "failure" is simply not getting double 1's (so anything else). To find the parameter \(p\), note that the underlying sample space consists of all possible rolls of a pair of fair dice, of which there are \(6\times6 = 36\) because each die has 6 possible sides. Each of these rolls is equally likely, so $$p = P(\text{double 1's}) = \frac{\text{# of ways to roll double 1's}}{36} = \frac{1}{36}.\notag$$

The negative binomial distribution generalizes the geometric distribution by considering any number of successes.

Definition \(\PageIndex{3}\)

Suppose that a sequence of independent Bernoulli trials is performed, with \(p = P(\text{"success"})\) for each trial. Fix an integer \(r\) to be greater than or equal to 2 and define the random variable \(X\) to give the number of trial at which the \(r^{th}\) success occurs. Then \(X\) has a negative binomial distribution with parameters \(r\) and \(p\). The probability mass function of \(X\) is given by \begin{align*} p(x) = P(X=x) &= P(r^{th}\ \text{success is on}\ x^{th}\ \text{trial}) \\ &= \underbrace{P(1^{st}\ (r-1)\ \text{successes in}\ 1^{st}\ (x-1)\ \text{trials})}_{\text{bniomial with}\ n=x-1} \times P(r^{th}\ \text{success on}\ x^{th}\ \text{trial}) \\ &= \binom{x-1}{r-1}p^{r-1}(1-p)^{(x-1)-(r-1)}\times p \\ &= \binom{x-1}{r-1}p^r(1-p)^{x-r}, \quad\text{for}\ x=r, r+1, r+2, \ldots \end{align*}

Example \(\PageIndex{3}\)

For examples of the negative binomial distribution, we can alter the geometric examples given in Example 3.4.2 .

  • Toss a fair coin until get 8 heads. In this case, the parameter \(p\) is still given by \(p = P(h) = 0.5\), but now we also have the parameter \(r = 8\), the number of desired "successes", i.e., heads.
  • Buy lottery tickets until win 5 times. In this case, the parameter \(p\) is still given by the odds of winning the lottery, but now we also have the parameter \(r = 5\), the number of desired wins.
  • Roll a pair of fair dice until get 100 double 1's. In this case, the parameter \(p\) is still given by \(p = P(\text{double 1's}) = \frac{1}{36}\), but now we also have the parameter \(r = 100\), the number of desired "successes".

In general, note that a geometric distribution can be thought of a negative binomial distribution with parameter \(r=1\).

Note that for both the geometric and negative binomial distributions the number of possible values the random variable can take is infinite. These are still discrete distributions though, since we can "list" the values. In other words, the possible values are countable . This is in contrast to the Bernoulli, binomial, and hypergeometric distributions, where the number of possible values is finite.

We again note the distinction between the binomial distribution and the geometric and negative binomial distributions. In the binomial distribution, the number of  trials  is fixed, and we count the number of "successes". Whereas, in the geometric and negative binomial distributions, the number of  "successes"  is fixed, and we count the number of trials needed to obtain the desired number of "successes".

Introduction to Probability

Lesson 15 negative binomial distribution, motivating example.

On a (American) roulette wheel, there are 38 spaces: 18 black, 18 red, and 2 green. You’ve been at the casino for a while now and decide to leave after you have won 3 bets on red. What is the probability that you leave the casino after placing exactly 5 bets on red?

To answer the question posed at the beginning of the lesson, we need a distribution like the geometric, except that stops after \(3\) \(\fbox{1}\) s have been drawn (instead of after the first \(\fbox{1}\) ). The negative binomial is that distribution.

Theorem 15.1 (Negative Binomial Distribution) If a random variable can be described as the number of draws, with replacement , from the box \[ \overbrace{\underbrace{\fbox{0}\ \ldots \fbox{0}}_{N_0}\ \underbrace{\fbox{1}\ \ldots \fbox{1}}_{N_1}}^N \] until \(r\) \(\fbox{1}\) s have been drawn, then its p.m.f. is given by \[\begin{align} f(x) &= \dfrac{\binom{x-1}{r-1} N_0^{x-r} \cdot N_1^r}{N^x}, & x&=r, r+1, r+2, \ldots \tag{15.1} \end{align}\] where \(N = N_1 + N_0\) is the number of tickets in the box.

We say that the random variable has a \(\text{NegativeBinomial}(r, N_1, N_0)\) distribution, and \(r\) , \(N_1\) , \(N_0\) are called parameters of the distribution.

Like the geometric distribution, there is no upper bound on the possible values of a negative binomial random variable. We might have to wait arbitrarily long to collect \(r\) \(\fbox{1}\) s. Also, note that the minimum possible value of a negative binomial random variable is \(r\) . This makes sense because you need to have drawn at least \(r\) times before you can have \(r\) \(\fbox{1}\) s.

We will derive the formulas (15.1) and (15.2) later in this lesson. For now, let’s see how these formulas can be applied to real problems.

Example 15.1 (Three Wins in Roulette) There are 38 equally likely spaces on a roulette wheel, 18 of which are red. So we set up a box model where the \(\fbox{1}\) s represent the red spaces:

\[ \overbrace{\underbrace{\fbox{1}\ \ldots \fbox{1}}_{N_1=18}\ \underbrace{\fbox{0}\ \ldots \fbox{0}}_{N_0=20}}^{N=38} \]

The number of draws until we get \(r=3\) \(\fbox{1}\) s corresponds to the number of bets we make until we have won 3 times. So the number of bets follows a \(\text{NegativeBinomial}(r=3, N_1=18, N_0=20)\) distribution.

Therefore, we know its p.m.f. by (15.2) : \[ f(x) = \binom{x-1}{3-1} \left( \frac{20}{38} \right)^{x-3} \left( \frac{18}{38} \right)^3. \]

Now, let’s derive the p.m.f. of the binomial distribution.

Proof (Theorem 15.1 ). To calculate the p.m.f. at \(x\) , we need to determine the probability that it takes exactly \(x\) draws to get \(r\) \(\fbox{1}\) s.

First, there are \(N^x\) equally likely ways to draw \(x\) tickets from \(N\) with replacement, taking order into account. (We must count ordered outcomes because the unordered outcomes are not all equally likely. See Lesson 4 .)

Next, we count the outcomes where the \(r\) th \(\fbox{1}\) happens on the \(x\) th draw. We proceed in two steps:

Count outcomes that look like \[\begin{equation} \underbrace{\fbox{0}, \ldots, \fbox{0}}_{x-r}, \underbrace{\fbox{1}, \ldots, \fbox{1}}_{r}, \tag{15.3} \end{equation}\] where the \(r\) \(\fbox{1}\) s are all at the end. There are \(N_0\) choices for the first \(\fbox{0}\) , \(N_0\) choices for the second \(\fbox{0}\) , and in fact, \(N_0\) choices for each of the \(x-r\) \(\fbox{0}\) s, since we are drawing with replacement. Likewise, there are \(N_1\) choices for each of the \(r\) \(\fbox{1}\) s. By the multiplication principle of counting (Theorem 1.1 ), there are \[\begin{equation} N_0^{x-r} \cdot N_1^r. \tag{15.4} \end{equation}\] ways to get an outcome like (15.3) , in that exact order.

Account for the possibility that the \(\fbox{1}\) s and \(\fbox{0}\) s were drawn in a different order than (15.3) . Unlike the binomial, we cannot rearrange the \(\fbox{1}\) s and \(\fbox{0}\) s any order we like. With the negative binomial, the \(x\) th draw must be a \(\fbox{1}\) , since we need the \(r\) th \(\fbox{1}\) to come on the \(x\) th draw. Other than this last draw, we have complete freedom to rearrange the remaining \(r-1\) \(\fbox{1}\) s among the first \(x-1\) draws. Therefore, there are \(\binom{x-1}{r-1}\) valid arrangements of the \(\fbox{1}\) s and \(\fbox{0}\) s where that the \(r\) th \(\fbox{1}\) comes on the \(x\) th draw.

So the total number of (ordered) ways to get the \(r\) th \(\fbox{1}\) s on the \(x\) th draw is: \[ \binom{x-1}{r-1} \cdot N_0^{x-r} \cdot N_1^r. \]

Dividing this by the total number of outcomes, \(N^x\) , gives the p.m.f.: \[ f(x) = P(X = x) = \frac{\binom{x-1}{r-1} \cdot N_0^{x-r} \cdot N_1^r}{N^x}. \]

Visualizing the Distribution

Let’s graph the negative binomial distribution for different values of \(n\) , \(N_1\) , and \(N_0\) .

First, we fix the number of \(\fbox{1}\) s at \(r=5\) and vary the composition of the box.

negative binomial distribution example problems and solutions

Not surprisingly, as we increase the number of \(\fbox{1}\) s in the box, the \(\fbox{1}\) s are drawn sooner, so the \(r=5\) th \(\fbox{1}\) comes after fewer draws.

Next, we study the effect of increasing \(r\) on the negative binomial distribution.

negative binomial distribution example problems and solutions

Not surprisingly, when we require more \(\fbox{1}\) s by increasing \(r\) , the distribution shifts to the right, indicating that it takes longer to achieve that goal.

Calculating Negative Binomial Probabilities on the Computer

The negative binomial distribution is built into many software packages. However, we have to check which definition the package is using. Some packages define a negative binomial random variable to be the number of \(\fbox{0}\) s that were drawn, instead of the total number of draws.

Solution. First, we will set up a box model for the number of calls . We have a box with

  • \(N_0 = 85\) tickets labeled \(\fbox{0}\)
  • \(N_1 = 15\) tickets labeled \(\fbox{1}\)

to represent the 15% chance of making a sale. We will draw from this box until we have drawn 10 \(\fbox{1}\) s, representing the 10 successful calls. We will assume that his success on one call is independent of his success on any other call, so we make the draws with replacement .

This shows that the number of calls, which we will call \(X\) , follows a \(\text{NegativeBinomial}(r=10, N_1=15, N_0=85)\) distribution.

Here’s how we would calculate the probability using the Python library Symbulate . We first specify the parameters of the negative binomial distribution. Note that Symbulate requires that the parameters be \(r\) and \(p\) , so we have to convert \(N_1=15, N_0=85\) into \(p = 0.15\) .

Calculating the probability directly involves evaluating the p.m.f. at infinitely many values, so we look at the complement. We can evaluate the p.m.f. at all of these values using the .pmf() method and add the probabilities using sum() .

Alternatively, we could also calculate this using the c.d.f. and the complement rule:

You can play around with the Python code in this Colab notebook .

It is also possible to do this calculation in R, a statistical programming language. R defines the negative binomial distribution a bit differently; it only counts the number of \(\fbox{0}\) s that were drawn, rather than the total number of draws. So we have to remember to subtract the \(r=10\) \(\fbox{1}\) s from the total number of draws before passing the values to dnbinom or pnbinom .

We can also use the c.d.f. function:

You can play around with the R code in this Colab notebook .

Essential Practice

  • Complete the sentence: The geometric distribution is a special case of the negative binomial distribution where \(r=\) _____.

Calculate the following probabilities.

You toss a coin 4 times. The probability that you get (exactly) 2 heads.

You toss a coin until you get 2 heads. The probability that it takes (exactly) 4 tosses.

In Major League Baseball’s Home Run Derby, each contestant is allowed to keep swinging the bat until they have made 10 “outs”. (An “out” is anything that is not a home run.) If Barry Bonds has a 70% chance of hitting a home run on any given swing, what is the probability that he hits at least 10 home runs before his turn is up?

Additional Exercises

  • A medical researcher is recruiting 20 subjects for a study on an experimental drug for COVID-19. Each person that she interviews has a 60% chance of being eligible to participate in the study. What is the probability that she will have to interview more than 40 people?
  • Your coach tells you that you cannot leave basketball practice until you have made at least \(20\) free throws. If you free throw probability is \(80\%\) , find the probability that you are out of practice after taking an even amount of free throws.
  • You have two coins. One coin is a fair coin with a \(.5\) probability of landing on heads. The other coin is a biased coin with a \(.25\) probability of landing on heads. You pick one of these two coins at random, and begin flipping until you get \(5\) heads. It takes you \(12\) flips in order to get your \(5\) heads. What is the probability that the coin you picked was the fair coin? What is the probability you picked the biased coin?

Negative Binomial Distribution | Introduction and examples

Negative Binomial Distribution | Introduction and examples

Negative Binomial Experiment

The experiment consists of x repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. The experiment continues until r successes are observed, where r is specified in advance. Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because: The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads. Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials. The experiment continues until a fixed number of successes have occurred; in this case, 5 heads. Cite: Stat Trek [1]

Geometric distribution is a special case of the negative binomial distribution.

$$ b(x; r, P) = C_{x-1}^{r-1} \times P^r \times (1 - P)^{x - r} $$

  • $x$: The number of trials required to produce r successes in a negative binomial experiment.
  • $r$: The number of successes in the negative binomial experiment.
  • $P$: The probability of success on an individual trial.
  • $Q$: The probability of failure on an individual trial. (This is equal to 1 - P.)
  • $b(x; r, P)$: Negative binomial probability - the probability that an x-trial negative binomial experiment results in the rth success on the xth trial, when the probability of success on an individual trial is P.
  • $C_n^r$: The number of combinations of n things, taken r at a time.

The Mean of the Negative Binomial Distribution

$μ = r / P$

  • $μ$ is the mean number of trials
  • $r$ is the number of successes
  • $P$ is the probability of a success on any given trial.

Alternative Views of the Negative Binomial Distribution

$μ_ R = kP/Q$

  • $R$: The negative binomial random variable
  • $k$: the number of successes before the binomial experiment results in k failures

$μK = rQ/P$

  • $K$: The negative binomial random variable
  • $r$: the number of failures before the binomial experiment results in r successes

Geometric Distribution

The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes ® is equal to 1 . Cite: Stat Trek [1:1]

$ b(x; r, P) = C_{x-1}^{r-1} \times P^r \times (1 - P)^{x - r} $ $b(x; 1, P) = C_{x-1}^{0} \times P^1 \times (1 - P)^{x - 1}$ $b(x; 1, P) = P^1 (1 - P)^{x - 1}$ $b(x; 1, P) = P Q^{x - 1}$

Geometric Probability Formula. Suppose a negative binomial experiment consists of x trials and results in one success. If the probability of success on an individual trial is P, then the geometric probability is:

$$ g(x; P) = P Q^ {x - 1} $$

Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. During the season, what is the probability that Bob makes his third free throw on his fifth shot? [1:2]

From the example, we can know: $P = 0.7$ $x = 4$ $r = 3$ So, we can have: $b(5; 3, .7) = C_{4}^{2} \times P^3 \times (1 - P)^{5-3}$ $b(5; 3, .7) = C_{4}^{2} \times 0.7^3 \times 0.3^2$ $b(5; 3, .7) = 6 \times 0.7^3 \times 0.3^2 $ $b(5; 3, .7) = 18.522 % $

Stat Trek: Negative Binomial Distribution ↩︎ ↩︎ ↩︎

https://karobben.github.io/2021/04/11/LearnNotes/negbinomial/

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  • • Deriving negative binomial distribution • Formula for negative binomial distribution • Relation of geometric distribution to the negative binomial distribution
  • Identifying Negative Binomial Distributions Identify which of the following experiments below are negative binomial distributions? i. A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times? ii. Cards are drawn out of a deck until 2 exactly aces are drawn. What is the probability that a total of 10 cards will be drawn? iii. An urn contains 3 red balls and 2 black balls. If 2 balls are drawn with replacement what is the probability that 1 of them will be black? iv. Roll a die until the first six comes up. What is the probability that this will take 3 rolls?
  • Determining the Negative Binomial Distribution A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times?
  • Determining the Cumulative Negative Binomial Distribution A sculptor is making 3 exhibits for an art gallery. There is a probability of 0.75 that every piece of wood she carves into will be good enough to be part of the exhibit. What is the probability that she uses 4 pieces of wood or less?

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Topic Notes

What is a binomial distribution.

  • The experiment has a fixed number of trials. And so, the value of n n n is defined.
  • Each trial has only two possible outcomes, the result is either a success or a failure.
  • The probability of success for each individual trial is equal.
  • There are only two possible outcomes for each trial in the experiment.
  • The probability of success in each trial is constant, in other words, it doesnt matter how many trials are run in the experiment, the value of the probability of success in each is the same.
  • There is a set number of trials to run in the experiment, and each of these trials is independent from the others.

Negative binomial distribution examples

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Negative Binomial Distribution : Properties, Applications and Examples

What is negative binomial distribution.

\theta

It is similar to a binomial distribution but with one key difference, in a binomial distribution, the number of trials is fixed, while in the negative binomial distribution, the number of successes is fixed.

Table of Content

Properties of Negative Binomial Distribution

Probability density function (pdf) of negative binomial distribution, mean and variance of negative binomial distribution.

  • Applications of Negative Binomial Distribution in Business Statistics

Examples of Negative Binomial Distribution

The negative binomial distribution has specific characteristics,

  • It involves a total of ‘n’ trials.
  • Each trial has two possible outcomes, success and failure.
  • The probability of success (denoted as ‘p’) is the same for each trial.
  • The probability of failure (denoted as ‘q’) is also consistent across trials, with p + q equaling 1.
  • Trials are independent; the outcome of one trial doesn’t influence others.
  • The experiment continues until a predetermined number of ‘r’ successes are achieved.
  • In total, there are ‘x + r’ repeated trials to reach the desired number of successes.

The Negative Binomial Distribution ‘s Probability Density Function (PDF) describes the likelihood of getting a certain number of successes before a specific number of failures happen in a series of independent trials. The formula for the PDF looks like this,

The mean and variance of the Negative Binomial Distribution can help us understand the average and spread of the number of trials needed to achieve a certain number of successes.

I. Mean of Negative Binomial Distribution

Ii. variance of negative binomial distribution.

\theta^2

Applications of Negative Binomial Distribution

The Negative Binomial Distribution finds applications in various aspects of business statistics. Here are a few practical scenarios where it is commonly used,

1. Project Management: In project management, the Negative Binomial Distribution can be applied to estimate the number of trials (attempts or tasks) required to complete a project with a certain number of successful outcomes (milestones achieved or tasks completed). This helps in project planning and resource allocation.

2. Quality Control: In manufacturing and quality control, the distribution can be used to model the number of defective items produced before reaching a certain number of acceptable items. This aids in setting quality standards and optimizing production processes.

3. Customer Service and Call Centers: The Negative Binomial Distribution is often employed to model the number of customer service calls a representative needs to handle before resolving a certain number of issues. This is valuable for workforce management and optimizing service efficiency.

4. Marketing Campaigns: Marketers can use the Negative Binomial Distribution to predict the number of attempts (such as advertisement exposures or promotional events) needed to achieve a specific number of desired responses, like customer purchases or sign-ups.

5. Insurance and Risk Management: In the insurance industry, the distribution can be utilized to model the number of claims or losses a company might experience before reaching a certain level of profitability. This aids in risk assessment and setting appropriate insurance premiums.

6. Inventory Management: Businesses dealing with inventory can use the Negative Binomial Distribution to model the number of orders or deliveries needed to restock inventory before selling a certain quantity of products. This helps in maintaining optimal stock levels and minimizing holding costs.

The probability of Mary solving a puzzle correctly is 75%. What is the probability that Mary solves the puzzle correctly for the fifth time in the first eight attempts?

Here, the occurrence of success is at the 5 th time, so k = 5.

Number of Trials, X = 8.

P(X=x)=^{x-1}C_{k-1}\theta^k(1-\theta)^{x-k}

P (X = 8) = 35 × 0.2373046875 × 0.015625

P (X = 8) = 0.12977600097

P (X = 8) = 0.13

∴ the probability that Mary solves the puzzle correctly for the fifth time in the first eight attempts is approximately 0.13.

The probability of Alex hitting the target in archery is 90%. What is the probability that Alex hits the target for the second time in four attempts?

Here, the occurrence of hitting the target is at 2 nd time, so k = 2.

Number of Trials, X = 4.

P(X=4)=^{4-1}C_{2-1}(0.90)^2(1-0.90)^{4-2}

P (X = 4) = 3 x 0.81 x 0.01

P (X = 4) = 0.0243

∴ the probability that Alex hits the target for the second time in the first four attempts is 0.02.

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Negative Binomial Distribution Examples

  • May 11, 2023

Negative Binomial Distribution

In this tutorial, we will provide you step by step solution to some numerical examples on negative binomial distribution to make sure you understand the negative binomial distribution clearly and correctly.

Definition of Negative Binomial Distribution

A discrete random variable $X$ is said to have negative binomial distribution if its p.m.f. is given by $$ \begin{aligned} P(X=x)&= \binom{x+r-1}{r-1} p^{r} q^{x},\\ & \quad \quad x=0,1,2,\ldots; r=1,2,\ldots\\ & \quad\quad \qquad 0<p, q<1, p+q=1. \end{aligned} $$

Mean of Negative Binomial Distribution

The mean of negative binomial distribution is $E(X)=\dfrac{rq}{p}$.

Variance of Negative Binomial Distribution

The variance of negative binomial distribution is $V(X)=\dfrac{rq}{p^2}$.

A large lot of tires contains 5% defectives. 4 tires are to be chosen for a car.

a. Find the probability that you find 2 defective tires before 4 good ones.

b. Find the probability that you find at most 2 defective tires before 4 good ones.

c. Find the mean and variance of the number of defective tires you find before finding 4 good tires.

Let $X$ denote the number of defective tires you find before you find 4 good tires. A large lot of tires contains 5% defectives. So the probability of good tire is $p=0.95$.

The random variable $X\sim NB(4, 0.95)$.

The probability mass function of $X$ is $$ \begin{aligned} P(X=x)&= \binom{x+4-1}{x} (0.95)^{4} (0.05)^{x},\quad x=0,1,2,\ldots\\ &= \binom{x+3}{x} (0.95)^{4} (0.05)^{x},\quad x=0,1,2,\ldots \end{aligned} $$

negative binomial distribution example problems and solutions

a. The probability that you find 2 defective tires before 4 good tires is $$ \begin{aligned} P(X=2)&= \binom{2+3}{2} (0.95)^{4} (0.05)^{2}\\ &= \binom{5}{2} (0.8145)\times (0.0025)\\ &= 10*(0.00204)\\ &= 0.0204 \end{aligned} $$ b. The probability that you at most 2 defective tires before 4 good tires is $$ \begin{aligned} P(X\leq 2)&=\sum_{x=0}^{2}P(X=x)\\ &= P(X=0)+P(X=1)+P(X=2)\\ &= \binom{0+3}{0} (0.95)^{4} (0.05)^{0}+\binom{1+3}{1} (0.95)^{4} (0.05)^{1}\\ &\quad +\binom{2+3}{2} (0.95)^{4} (0.05)^{2}\\ &= 1*(0.8145)+4*(0.04073)+10*(0.00204)\\ &= 0.8145+0.1629+0.0204\\ &=0.9978 \end{aligned} $$ c. The mean of the number of defective tires you find before finding 4 good tires is $$ \begin{aligned} E(X) &= \frac{rq}{p}\\ &= \frac{4*0.05}{0.95}\\ &= 0.2105. \end{aligned} $$ The variance of the number of defective tires you find before finding 4 good tires is

$$ \begin{aligned} V(X) &= \frac{rq}{p^2}\\ &= \frac{4*0.05}{0.95^2}\\ &= 0.2216. \end{aligned} $$

The probability of male birth is 0.5. A couple wishes to have children until they have exactly two female children in their family.

a. Write the probability distribution of $X$, the number of male children before two female children.

b. What is the probability that the family has four children?

c. What is the probability that the family has at most four children?

d. What is the expected number of male children this family have?

e. What is the expected number of children this family have?

A couple wishes to have children until they have exactly two female children in their family.

Birth of female child is consider as success and birth of male child is consider as failure. The probability of male birth is $q=0.5$. So the probability of female birth is $p=1-q=0.5$. The number of female children (successes) $r=2$.

Here $X$ denote the number of male children before two female children. Then the random variable $X$ follows a negative binomial distribution $NB(2,0.5)$.

a. The probability distribution of $X$ (number of male children before two female children) is $$ \begin{aligned} P(X=x)&= \binom{x+2-1}{x} (0.5)^{2} (0.5)^{x},\quad x=0,1,2,\ldots\\ &= \binom{x+1}{x} (0.5)^{2} (0.5)^{x},\quad x=0,1,2,\ldots \end{aligned} $$

negative binomial distribution example problems and solutions

b. The family has four children means 2 male and 2 female. Thus probability that a family has four children is same as probability that a family has 2 male children before 2 female children. $$ \begin{aligned} P(X=2) & = \frac{(2+1)!}{1!2!}(0.5)^{2}(0.5)^{2} \\ & = 0.1875 \end{aligned} $$ c. The family has at the most four children means $X$ is less than or equal to 2. Thus, the probability that a family has at the most four children is $$ \begin{aligned} P(X\leq 2) & = \sum_{x=0}^{2} P(X=x)\\ & = p(0) + p(1) + p(2)\\ & = 0.25+ 0.25+0.1875\\ &= 0.6875 \end{aligned} $$ where $$ \begin{aligned} p(0) & = \frac{(0+1)!}{1!0!}(0.5)^{2}(0.5)^{0}\\ & = 0.25. \end{aligned} $$

$$ \begin{aligned} p(1) & = \frac{(1+1)!}{1!1!}(0.5)^{2}(0.5)^{1}\\ & = 0.25. \end{aligned} $$

$$ \begin{aligned} p(2) & = \frac{(2+1)!}{1!2!}(0.5)^{2}(0.5)^{2}\\ & = 0.1875. \end{aligned} $$

d. The expected number of male children is $$ \begin{aligned} E(X)& = \frac{rq}{p}\\ & = \frac{2\times0.5}{0.5}\\ & = 2 \end{aligned} $$ e. The expected number of children is $$ \begin{aligned} E(X+2)& = E(X) + 2\\ &= 2+2. \end{aligned} $$

Related Resources

negative binomial distribution example problems and solutions

  • May 9, 2023

Examples of Binomial Distribution Problems and Solutions

On this page you will learn:

  • Binomial distribution definition and formula.
  • Conditions for using the formula.
  • 3 examples of the binomial distribution problems and solutions.

Many real life and business situations are a pass-fail type. For example, if you flip a coin, you either get heads or tails. You either will win or lose a backgammon game. There are only two possible outcomes – success and failure , win and lose.

And the key element here also is that likelihood of the two outcomes may or may not be the same.

So, what is binomial distribution?

Let’s define it:

In simple words, a binomial distribution is the probability of a success or failure results in an experiment that is repeated a few or many times.

The prefix “bi” means two. We have only 2 possible incomes.

Binomial probability distributions are very useful in a wide range of problems, experiments, and surveys. However, how to know when to use them?

Let’s see the necessary conditions and criteria to use binomial distributions:

  • Rule 1: Situation where there are only two possible mutually exclusive outcomes (for example, yes/no survey questions).
  • Rule2: A fixed number of repeated experiments and trials are conducted (the process must have a clearly defined number of trials).
  • Rule 3: All trials are identical and independent (identical means every trial must be performed the same way as the others; independent means that the result of one trial does not affect the results of the other subsequent trials).
  • Rule: 4: The probability of success is the same in every one of the trials.

Notations for Binomial Distribution and the Mass Formula:

  • P is the probability of success on any trail.
  • q = 1- P – the probability of failure
  • n  – the number of trails/experiments
  • x  – the number of successes, it can take the values 0, 1, 2, 3, . . . n.
  • n C x  = n!/x!(n-x)  and denotes the number of combinations of n elements taken x at a time.

Assuming what the  n C x  means, we can write the above formula in this way:

Just to remind that the ! symbol after a number means it’s a factorial. The factorial of a non-negative integer x is denoted by x!. And x! is the product of all positive integers less than or equal to x. For example, 4! = 4 x 3 x 2 x 1 = 24.

Examples of binomial distribution problems:

  • The number of defective/non-defective products in a production run.
  • Yes/No Survey (such as asking 150 people if they watch ABC news).
  • Vote counts for a candidate in an election.
  • The number of successful sales calls.
  • The number of male/female workers in a company

So, as we have the basis let’s see some binominal distribution examples, problems, and solutions from real life.

Let’s say that 80% of all business startups in the IT industry report that they generate a profit in their first year. If a sample of 10 new IT business startups is selected, find the probability that exactly seven will generate a profit in their first year.

First, do we satisfy the conditions of the binomial distribution model?

  • There are only two possible mutually exclusive outcomes – to generate a profit in the first year or not (yes or no).
  • There are a fixed number of trails (startups) – 10.
  • The IT startups are independent and it is reasonable to assume that this is true.
  • The probability of success for each startup is 0.8.

We know that:

n = 10, p=0.80, q=0.20, x=7

The probability of 7 IT startups to generate a profit in their first year is:

This is equivalent to:

Interpretation/solution: There is a 20.13% probability that exactly 7 of 10 IT startups will generate a profit in their first year when the probability of profit in the first year for each startup is 80%.

The important points here are to know when to use the binomial formula and to know what are the values of p, q, n, and x.

Also, binomial probabilities can be computed in an Excel spreadsheet using the = BINOMDIST function .

Your basketball team is playing a series of 5 games against your opponent. The winner is those who wins more games (out of 5).

Let assume that your team is much more skilled and has 75% chances of winning. It means there is a 25% chance of losing.

What is the probability of your team get 3 wins?

We need to find out.

In this example:

n = 5, p=0.75, q=0.25, x=3

Let’s replace in the formula to get the answer:

Interpretation: the probability that you win 3 games is 0.264.

A box of candies has many different colors in it. There is a 15% chance of getting a pink candy. What is the probability that exactly 4 candies in a box are pink out of 10?

We have that:

n = 10, p=0.15, q=0.85, x=4

When we replace in the formula:

Interpretation: The probability that exactly 4 candies in a box are pink is 0.04.

The above binomial distribution examples aim to help you understand better the whole idea of binomial probability.

If you need more examples in statistics and data science area, our posts descriptive statistics examples and categorical data examples might be useful for you.

About The Author

negative binomial distribution example problems and solutions

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Silvia Valcheva is a digital marketer with over a decade of experience creating content for the tech industry. She has a strong passion for writing about emerging software and technologies such as big data, AI (Artificial Intelligence), IoT (Internet of Things), process automation, etc.

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Negative Binomial Distribution Discrete Random Variable problems

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Solution to this Negative Binomial Discrete Random Variable Distribution Probability practice  problem is given in the video below!

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  1. Negative Binomial Distribution

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  4. Solved The negative binomial distribution is a

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  1. Chapter 2 5 2 Binomial distribution example (MAU)

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  3. 5. PRACTICE PROBLEM ON BINOMIAL DISTRIBUTION

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COMMENTS

  1. 11.6

    11.6 11.6 - Negative Binomial Examples Example 11-2 An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. What is the probability that the first strike comes on the third well drilled? Solution To find the requested probability, we need to find P ( X = 3.

  2. Negative Binomial Distribution (w/ 7 Worked Examples!)

    S - success (probability of success) the same - yes, the likelihood of getting a Jack is 4 out of 52 each time you turn over a card. Therefore, this is an example of a negative binomial distribution.

  3. 11.4: The Negative Binomial Distribution

    The distribution defined by the density function in (1) is known as the negative binomial distribution; it has two parameters, the stopping parameter \ (k\) and the success probability \ (p\). In the negative binomial experiment, vary \ (k\) and \ (p\) with the scroll bars and note the shape of the density function.

  4. Negative Binomial Distribution: Uses, Calculator & Formula

    For example, the negative binomial distribution can answer the following questions. What is the probability of the following: Rolling the 5 th six on the 20 th roll of a die? Getting the 10 th defective item on the 1000 th item inspected? Selecting the 10 th woman as the 15 th participant?

  5. Negative Binomial Distribution

    The following quick examples help in a better understanding of the concept of the negative binomial distribution. If we flip a coin a fixed number of times and count the number of times the coin turns out heads is a binomial distribution.

  6. Negative Binomial Distribution

    Combinations & permutations Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by , is the same on every trial. ; that is, the outcome on one trial does not affect the outcome on other trials.

  7. An Introduction to the Negative Binomial Distribution

    Problem 1 Question: Suppose we flip a coin and define a "successful" event as landing on heads. What is the probability of experiencing 3 failures before experiencing a total of 4 successes? Answer: Using the Negative Binomial Distribution Calculator with k = 3 failures, r = 4 successes, and p = 0.5, we find that P (X=3) = 0.15625. Problem 2

  8. 3.4: Hypergeometric, Geometric, and Negative Binomial Distributions

    Example 3.4.3. For examples of the negative binomial distribution, we can alter the geometric examples given in Example 3.4.2. Toss a fair coin until get 8 heads. In this case, the parameter p is still given by p = P(h) = 0.5, but now we also have the parameter r = 8, the number of desired "successes", i.e., heads.

  9. Negative Binomial Distribution and Examples| 3-Step Rules

    For books, we may refer to these: https://amzn.to/34YNs3W OR https://amzn.to/3x6ufcEThis video explains how to find the probability using Negative Binomial...

  10. Negative Binomial Experiment / Distribution: Definition, Examples

    μ = r / P where r is the number of trials P=probability of success for any trial

  11. Lesson 15 Negative Binomial Distribution

    Motivating Example On a (American) roulette wheel, there are 38 spaces: 18 black, 18 red, and 2 green. You've been at the casino for a while now and decide to leave after you have won 3 bets on red. What is the probability that you leave the casino after placing exactly 5 bets on red? Theory

  12. Negative binomial distribution

    For example, we could use the negative binomial distribution to model the number of days n (random) a certain machine works (specified by r) before it breaks down. The Pascal distribution (after Blaise Pascal) and Polya distribution (for George Pólya) are special cases of the negative binomial distribution.

  13. Negative Binomial Distribution

    Negative Binomial Distribution | Introduction and examples. The experiment consists of x repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; that is, the outcome on ...

  14. Negative binomial distribution

    Lessons Identifying Negative Binomial Distributions Identify which of the following experiments below are negative binomial distributions? i. A fair coin is flipped until head comes up 4 times. What is the probability that the coin will be flipped exactly 6 times? ii. Cards are drawn out of a deck until 2 exactly aces are drawn.

  15. Introduction to the Negative Binomial Distribution

    An introduction to the negative binomial distribution, a common discrete probability distribution. In this video I define the negative binomial distribution...

  16. Negative Binomial Distribution : Properties, Applications and Examples

    Properties of Negative Binomial Distribution. The negative binomial distribution has specific characteristics, It involves a total of 'n' trials. Each trial has two possible outcomes, success and failure. The probability of success (denoted as 'p') is the same for each trial. The probability of failure (denoted as 'q') is also ...

  17. Negative Binomial Distribution Examples

    Solution Let X denote the number of defective tires you find before you find 4 good tires. A large lot of tires contains 5% defectives. So the probability of good tire is p = 0.95. The random variable X ∼ N B ( 4, 0.95). The probability mass function of X is

  18. Binomial Distribution Examples, Problems and Formula

    Rule 1: Situation where there are only two possible mutually exclusive outcomes (for example, yes/no survey questions). Rule2: A fixed number of repeated experiments and trials are conducted (the process must have a clearly defined number of trials).

  19. Negative Binomial Distribution Discrete Random Variable problems

    Detailed video tutorial on finding solutions to Probability example questions and problems using Negative Binomial Distribution. Math, Science, Test Prep, Music Theory Easy Video Tutorials For Your Class. MathCabin.com; 📕 Perfect Score SAT Math eBook; Live Online SAT Math Practice Test;

  20. Binomial Distribution

    Maths Math Article Binomial Distribution Binomial Distribution In probability theory and statistics, the binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure.

  21. Binomial Distribution Questions [Solved]

    Question 1: Find the binomial distribution of getting a six in three tosses of an unbiased dice. Solution: Let X be the random variable of getting six. Then X can be 0, 1, 2, 3. Here, n = 3 p = Probability of getting a six in a toss = ⅙ q = Probability of not getting a six in a toss = 1 - ⅙ = ⅚