# Binomial Probability

Binomial probability refers to the probability of exactly x successes on n repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment ).

If the probability of success on an individual trial is p , then the binomial probability is .

Here indicates the number of different combinations of x objects selected from a set of n objects. Some textbooks use the notation instead of .

Note that if p is the probability of success of a single trial, then (1 – p ) is the probability of failure of a single trial.

Example:

What is the probability of getting 6 heads, when you toss a coin 10 times?

In a coin-toss experiment, there are two outcomes: heads and tails. Assuming the coin is fair , the probability of getting a head is 1/2 or 0.5.

The number of repeated trials: n = 10

The number of success trials: x = 6

The probability of success on individual trial: p = 0.5

Use the formula for binomial probability.

Simplify.

≈ 0.205

If the outcomes of the experiment are more than two, but can be broken into two probabilities p and q such that p + q = 1, the probability of an event can be expressed as binomial probability.

For example, if a six-sided die is rolled 10 times, the binomial probability formula gives the probability of rolling a three on 4 trials and others on the remaining trials.

The experiment has six outcomes. But the probability of rolling a 3 on a single trial is 1/6 and rolling other than 3 is 5/6. Here, 1/6 + 5/6 = 1.

The binomial probability is:

Simplify.

≈ 0.054