Question

Suppose a box of Cracker Jacks contains one of 5 toy prizes: a small rubber ball, a whistle, a Captain America decoder ring, a race car, or a magnifying glass. Each prize is equally likely to be in a box. Question 1. How many boxes of Cracker Jacks would you expect to buy until you obtain a complete set of prizes

Answer 1

11.42 boxes

Explanation:

For the first box bought, there is a 100% chance of getting a unique toy (since you still don't have any). E₁ = 1.

After that, there is a 4 in 5 chance of getting a unique toy from the next box, the expected number of boxes required is:

`E_2 = ((4)/(5))^(-1) = 1.25`

For the next unique toy, there is now a 3 in 5 chance of getting it:

`E_3 = ((3)/(5))^(-1) = 1.67`

Following that logic, there is a 2 in 5 chance of getting the 4th unique toy:

`E_4 = ((2)/(5))^(-1) = 2.5`

Finally, there is a 1 in 5 chance to get the last unique toy:

`E_5 = ((1)/(5))^(-1) = 5`

The expected number of boxes to obtain a full set is:

`E=E_1+E_2+E_3+E_4+E_5\\E=1+1.25+1.67+2.5+5\\E=11.42\ boxes`