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

*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.*

**p**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:

where

The "!" signifies a factorial function, e.g., 27! = 27 × 26 × 25 × ... × 3 × 2 × 1.

## Example

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:

Thus

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).