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Two distributions that are similar in statistics are the Binomial distribution and the Poisson distribution.

This tutorial provides a brief explanation of each distribution along with the similarities and differences between the two.

**The Binomial Distribution**

The **Binomial distribution** describes the probability of obtaining *k* successes in *n* binomial experiments.

If a random variable *X* follows a binomial distribution, then the probability that *X* = *k* successes can be found by the following formula:

**P(X=k) = _{n}C_{k} * p^{k} * (1-p)^{n-k}**

where:

**n:**number of trials**k:**number of successes**p:**probability of success on a given trialthe number of ways to obtain_{n}C_{k}:*k*successes in*n*trials

For example, suppose we flip a coin 3 times. We can use the formula above to determine the probability of obtaining 0 heads during these 3 flips:

**P(X=0) **= _{3}C_{0} * .5^{0} * (1-.5)^{3-0} = 1 * 1 * (.5)^{3} = **0.125**

**The Poisson Distribution**

The **Poisson distribution** describes the probability of experiencing *k* events during a fixed time interval.

If a random variable *X* follows a Poisson distribution, then the probability that *X* = *k* events can be found by the following formula:

**P(X=k) = λ ^{k} * e^{– λ} / k!**

where:

**λ:**mean number of successes that occur during a specific interval**k:**number of successes**e:**a constant equal to approximately 2.71828

For example, suppose a particular hospital experiences an average of 2 births per hour. We can use the formula above to determine the probability of experiencing 3 births in a given hour:

**P(X=3) **= 2^{3} * e^{– 2} / 3! = **0.18045**

**Similarities & Differences**

The Binomial and Poisson distribution share the following **similarities**:

- Both distributions can be used to model the number of occurrences of some event.
- In both distributions, events are assumed to be independent.

The distributions share the following key **difference**:

- In a Binomial distribution, there is a fixed number of trials (e.g. flip a coin 3 times)
- In a Poisson distribution, there could be any number of events that occur during a certain time interval (e.g. how many customers will arrive at a store in a given hour?)

**Practice Problems: When to Use Each Distribution**

In each of the following practice problems, determine whether the random variable follows a Binomial distribution or Poisson distribution.

**Problem 1: Network Failures**

A tech company wants to model the probability that a certain number of network failures occur in a given week. Suppose it’s known that an average of 4 network failures occur each week. Let *X* be the number of network failures in a given week. What type of distribution does the random variable *X* follow?

Answer: *X* follows a Poisson distribution because we’re interested in modeling the number of network failures in a given week and there is no upper limit on the number of failures that could occur. This is not a Binomial distribution because there is not a fixed number of trials.

**Problem 2: Shooting Free-Throws**

Tyler makes 70% of all free-throws he attempts. Suppose he shoots 10 free-throws. Let *X* be the number of times Tyler makes a basket during the 10 attempts. What type of distribution does the random variable *X* follow?

Answer: *X* follows a Binomial distribution because there is a fixed number of trials (10 attempts), the probability of “success” on each trial is the same, and each trial is independent.

**Additional Resources**

Binomial Distribution Calculator

Poisson Distribution Calculator