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

Answer:

11.42 boxes

Step-by-step 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:

$E_2 = (\frac{4}{5})^{-1} = 1.25$

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

$E_3 = (\frac{3}{5})^{-1} = 1.67$

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

$E_4 = (\frac{2}{5})^{-1} = 2.5$

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

$E_5 = (\frac{1}{5})^{-1} = 5$

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

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