# Binomial Distribution- Probability Formula

A set formula that is perfect for probability question - the Binomial Distribution

Let p represent the probability of a success for one event (ie. flipping heads or rolling a 6)
Let n represent the total number of observed events (ie. coin flips or dice rolls)
Let k represent the desired number of successes (ie. number of heads or number of rolled 6's)

Using the binomial, we can quickly calculate the probability of k successes out of n events...

P(k) = nCk * p^k * (1-p)^(n-k)

Example 1: What is the probability of flipping exactly 6 heads out of 10 coin flips?

P(6 heads) = 10C6 * (1/2)^6 * (1/2)^4 = 210/1024 = 105/512

Example 2: What is the probability of rolling exactly 2 sixes with three dice?

P(2 sixes) = 3C2 * (1/6)^2 * (5/6)^1 = 15/216 = 5/72

The numerator of each answer (before simplification) will always represent the number of WAYS to perform the k successes. Obviously you can perform the binomial for any number of successes and then add them up as needed. Also, notice that for k = 0, the binomial simplifies to (1-p)^n, which means that for k >= 1, we use 1 - (1-p)^n