Binomial Distribution

	Binomial Distribution

The binomial distribution is used to model situations having only 2 possible outcomes, usually labeled as success or failure. For a random variable to follow a binomial distribution, the number of trials must be fixed, and the probability of success must be equal for all trials. The binomial probability density function is

where p(x,n,p) is the probability of exactly x successes in n trials with a probability of success equal to p on each trial. Note that

This notation is referred to as "n choose x", and is equal to the number of combination of size x made from n possibilities. This function is found on most calculators.

The binomial cumulative distribution function is

where P(x,n,p) is the probability of exactly x or fewer successes in n trials with a probability of success equal to p on each trial.

The mean and variance of the binomial distribution are

m = np

s² = np(1-p)

Example
The probability of a salesman making a successful sales call is 0.2, and 8 sales calls are made in a day. What is the probability of making exactly 3 successful sales calls in a day? What is the probability of making more than 2 successful sales calls in a day?

Solution
The probability of exactly 2 successes in 8 trials is

This solution is found using the Excel function

=BINOMDIST(2,8,0.2,0)

The probability of more than 2 successes is equal to the one minus the probability of 2 or fewer successes. The binomial probability density function can be used to compute the probability of exactly 0 successes and the probability of exactly 1 success, but it is easier to use the Excel function

=BINOMDIST(2,8,0.2,1)

The "1" at the end of the function above returns the cumulative binomial distribution. The probability of 2 or fewer successes is 0.7969. The probability of more than 2 successes is 1- 0.7969 = 0.2031. Click here to download this solution in Microsoft Excel.

Before electronic spreadsheets were common, the poisson distribution was used to approximate the binomial distribution because poisson tables were more accommodating than binomial tables. This approximation is now useless; why approximate a value when you can get an exact answer. The requirement for a valid approximation is p must be small and n must be large. The approximation is done by using np as the mean of the poisson distribution.