Binomial vs. Geometric Distribution: Similarities & Differences

Two 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) = _nC_k * p^k * (1-p)^n-k

where:

n: number of trials
k: number of successes
p: probability of success on a given trial
_nC_k: the number of ways to obtain 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) = ₃C₀ * .5⁰ * (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)^kp

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.