Question

Info Details Every cereal box has a gift inside, but you cannot tell from the outside what the gift is. The store manager assures you that 22 of the 63 boxes on the shelf have the secret decoder ring. The other 41 boxes on the shelf have a different gift inside. If you randomly select two boxes of cereal from the shelf to purchase, what is the probability that both of them have the secret decoder ring

Answer 1

0.1183 = 11.83% probability that both of them have the secret decoder ring.

Explanation:

The boxes 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:

63 boxes means that
`N = 63`

22 have the secret decoder ring, which means that
`k = 22`

Two are selected, which means that
`n = 2`

What is the probability that both of them have the secret decoder ring?

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(2,63,2,22) = (C_(22,2)*C_(41,0))/(C_(63,2)) = 0.1183`

0.1183 = 11.83% probability that both of them have the secret decoder ring.