A binomial distribution describes a variable X if 1) there is a fixed number n observations of the variable; 2) all observations are independent of each other; 3) the probability of success p is the same for each observation; and 4) each observation represents one of exactly two possible outcomes (hence the word "binomial" – think "binary"). This last qualification distinguishes binomial distributions from Poisson distributions, which vary continuously rather than discretely.
Such a distribution can be written B(n, p).
Calculating the Probability of a Given Observation
Say a value k lies somewhere along the graph of the binomial distribution, which is symmetrical about the mean np. To calculate the probability that an observation will have this value, this equation must be solved:
P(X = k) = (n : k) pk(1-p)(n-k)
where (n : k) = (n!) ÷ (k!)(n - k)!
The "!" signifies a factorial function, e.g., 27! = 27 x 26 x 25 x ... x 3 x 2 x 1.
Say a basketball player takes 24 free throws and has an established success rate of 75 percent (p = 0.75). What are the chances she will hit exactly 20 of her 24 shots?
First calculate (n : k) as follows:
(n!) ÷ (k!)(n - k)! = 24! ÷ (20!)(4!) = 10,626
pk = (0.75)20 = 0.00317
(1-p) (n-k) = (0.25)4 = 0.00390
Thus P(20) = (10,626)(0.00317)(0.00390) = 0.1314.
This player therefore has a 13.1 percent chance of making exactly 20 out of 24 free throws, in line with what intuition might suggest about a player who would usually hit 18 out of 24 free throws (because of her established success rate of 75 percent).