Question

On a particular game show, there are 8 covered buckets and 2 of them contain a ball.

To win the game, a contestant must select both buckets that contain a ball. Find the
probability that a contestant wins the game if he/she gets to select 4 of the buckets.

Answer 1

0.2143 = 21.43% probability that a contestant wins the game if he/she gets to select 4 of the buckets.

Explanation:

The buckets 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 covered buckets, so N = 8.

4 buckets are selected, so n = 4.

2 contain a ball, which means that k = 2.

Find the probability that a contestant wins the game if he/she gets to select 4 of the buckets.

This is P(X = 2). So

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

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

0.2143 = 21.43% probability that a contestant wins the game if he/she gets to select 4 of the buckets.