Binomial probability refers to the probability of exactly successes on 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 , then the binomial probability is .
Here indicates the number of different combinations of objects selected from a set of objects. Some textbooks use the notation instead of .
Note that if is the probability of success of a single trial, then is the probability of failure of a single trial.
What is the probability of getting heads, when you toss a coin 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 or .
The number of repeated trials:
The number of success trials:
The probability of success on individual trial:
Use the formula for binomial probability.
If the outcomes of the experiment are more than two, but can be broken into two probabilities and such that , the probability of an event can be expressed as binomial probability.
For example, if a six-sided die is rolled times, the binomial probability formula gives the probability of rolling a three on trials and others on the remaining trials.
The experiment has six outcomes. But the probability of rolling a on a single trial is and rolling other than is . Here, .
The binomial probability is: