Question

During a single day at radio station WMZH, the probability that a particular song is

played is 76%. What is the probability that this song will be played on exactly 2 days
out of 6 days? Round your answer to the nearest thousandth.

Answer 1

0.029 = 2.9% probability that this song will be played on exactly 2 days out of 6 days.

Explanation:

For each day, there are only two possible outcomes. Either the song is played, or it is not. The probability of the song being played on a day is independent of any other day. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

76% probability the song is played on a given day:

This means that
`p = 0.76`

6 days:

This means that
`n = 6`

What is the probability that this song will be played on exactly 2 days?

This is P(X = 2). So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(6,2).(0.76)^(2).(0.24)^(4) = 0.029`

0.029 = 2.9% probability that this song will be played on exactly 2 days out of 6 days.