Binomial Probability

Below is notation frequently used with binomial experiments:

n = the number of trials

p = probability of success on each trial

q = 1 - p = probability of failure on each trial

x = the number of successes in n trials (x is an integer between 0 and n)

P(x) = the probability of x successes in n trials

This notation is used below in the formula for finding binomial probability .

Example 1

Find the probability of rolling a three 4 times in 10 rolls of a fair die.

Solution .

This is a binomial experiment that we studied earlier with n = 10, x = 4, p = 1/6, and q = 1-1/6 = 5/6. Using the formula above, we have

Simplifying we get

If sampling is done without replacement, the events are not independent and as a result the binomial probability formula cannot be used. However, if the sample is relatively small in comparison to the population (i.e. the sample size n is no more than 5% of the population N), then the events can be considered independent.

Example 2

A recent study indicates that half of adult United States males under the age of 40 will develop heart disease at some point in their lives. In a sample of eight U.S. males under the age of 40, find the probability that five of them will develop heart disease assuming no life style changes.

Solution .

With n = 8, x = 5, p = .5 and q = 1 - p = 1 - .5 = .5, we have

Example 3

An urn contains 2 red marbles and 1 blue marble. A marble is selected at random, its color is noted, and the marble is put back into the urn. This is repeated 3 times. Find the probability of getting at most one red marble.

Solution .

Since P(at most 1 red marble) = P( 0 or 1 red) = P(0 red) + P(1 red), we must find P(0 red) and P(1 red) and add them together.

Here n = 3, p = 2/3 (since the marbles are replaced each time), and q = 1 -2/3 = 1/3.

The probability of getting at most one red marble is =