Two of the most commonly used distributions in statistics are the binomial distribution and the geometric 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 Geometric Distribution**

The **geometric distribution** describes the probability of experiencing a certain amount of failures before experiencing the first success in a series of binomial experiments.

If a random variable *X* follows a geometric distribution, then the probability of experiencing *k* failures before experiencing the first success can be found by the following formula:

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

where:

**k:**number of failures before first success**p:**probability of success on each trial

For example, suppose we want to know how many times we’ll have to flip a fair coin until it lands on heads. We can use the formula above to determine the probability of experiencing 3 “failures” before the coin finally lands on heads:

**P(X=3) **= (1-.5)^{3}(.5) = **0.0625**

**Similarities & Differences**

The binomial and geometric distribution share the following **similarities**:

- The outcome of the experiments in both distributions can be classified as “success” or “failure.”
- The probability of success is the same for each trial.
- Each trial is independent.

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

- In a binomial distribution, there is a fixed number of trials (i.e. flip a coin 3 times)
- In a geometric distribution, we’re interested in the number of trials required
*until*we obtain a success (i.e. how many flips will we need to make before we see Tails?)

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

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

**Problem 1: Rolling Dice**

Jessica plays a game of luck in which she keeps rolling a dice until it lands on the number 4. Let *X* be the number of rolls until a 4 appears. What type of distribution does the random variable *X* follow?

Answer: *X* follows a geometric distribution because we’re interested in estimating the number of rolls required until we finally get a 4. This is not a binomial distribution because there is not a fixed number of trials.

**Problem 2: Shooting Free-Throws**

Tyler makes 80% 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

Geometric Distribution Calculator