Probability · Quick Reference

Binomial Distribution Cheat Sheet

Probability · Lesson 7/8

In one line: the binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success — like counting heads in repeated coin flips.

Key Ideas

1Binomial Conditions. There must be: a fixed number of trials (n), each trial independent, only two outcomes per trial (success/failure), and a constant probability of success (p) each time.

2Binomial Formula. P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) is the number of ways to choose k successes from n trials.

3Combinations, C(n,k). C(n,k) = n! / [k!(n-k)!] — counts how many different orders give exactly k successes.

4Mean and Variance. For a binomial distribution, mean = n*p, and variance = n*p*(1-p).

5When to Use It. Use the binomial distribution whenever you're counting successes across repeated, independent yes/no trials, like defective items or correct guesses.

Worked Examples

A fair coin is flipped 4 times. Find the probability of getting exactly 2 heads.

P(X=2) = 0.375

A test has 5 true/false questions; a student guesses randomly. Find the probability of getting exactly 3 correct.

P(X=3) = 0.3125

For n=10 trials with p=0.3, find the mean and variance of the binomial distribution.

Mean = 3, Variance = 2.1

Formulas

P(X=k) = C(n,k) * p^k * (1-p)^(n-k)

C(n,k) = n! / [k!(n-k)!]

Mean = n*p

Variance = n*p*(1-p)

Practice Yourself

A coin is flipped 3 times. Find P(exactly 2 heads).

C(3,2)*(0.5)^2*(0.5)^1 = 3*0.125 = 0.375

For n=6, p=0.5, find the mean.

For n=8, p=0.25, find the variance.

1.5