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) = nCk * pk * (1-p)n-k
- n: number of trials
- k: number of successes
- p: probability of success on a given trial
- nCk: 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
- 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.