Question

A comedy club hosts an amateur comedy night, a good opportunity for aspiring comedians to

perform. Of the 8 amateur comedians who signed up for tonight's show, 6 perform satirical

comedy,

If the host randomly selects 4 comedians from the sign-up list, what is the probability that all

of them practice satirical?

Write your answer as a decimal rounded to four decimal places.

Answer 1

0.2143 probability that all of them practice satirical.

Explanation:

The comedians are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x sucesses is given by the following formula:

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

In which:

x is the number of sucesses.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

`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)!)`

In this question:

8 comedians means that
`N = 8`

6 practice satirical means that
`k = 6`

Sample of 4 means that
`n = 4`

What is the probability that all of them practice satirical?

This is P(X = 4). So

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

`P(X = 4) = h(4,8,4,6) = (C_(6,4)*C_(2,0))/(C_(8,4)) = 0.2143`

0.2143 probability that all of them practice satirical.