To compute a binomial probability, you need to calculate and multiply three separate factors:
- the number of ways to select exactly k successes,
- the probability of success (p) raised to the k power, and
- the probability of failure (q) raised to the (n-k) power.
Since success and failure are the only two possibilities, q = 1 - p. The probability of k successes in a binomial(n,p) distribution is:
The exclamation point is mathematical notation for factorial. You compute a factorial by multiplying all the integers in sequence from one up to the number. For example,
The value of 1! is, intuitively, equal to 1. The value of 0! is, not so intuitively, equal to 1 also. Here are some examples:
X=number of successful transplants is binomial(4,0.3). What is the probability of exactly one successful transplant?
X=number of girls is binomial(3,0.5). What is the probability of exactly two girls?
X=# of vaccinated volunteers with flu resistance is binomial(20,0.94). What is the probability that more than 17 of the volunteers will develop resistance?
The calculations in the last example are a bit tedious, which is why most statisticians rely on computers.
