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$3$ prize types

Expected valueEasy

Each cereal box contains one of $3$ prize types, equally likely and independently of other boxes. What is the expected number of boxes needed to get all $3$ types?

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Hint

Add the waiting times for each new type.

Solution

The first box always gives a new type, which takes $1$ box.
With $1$ type held, a new one appears with probability $\frac{2}{3}$ , taking $\frac{3}{2}$ boxes on average; with $2$ held the chance is $\frac{1}{3}$ , taking $3$ boxes.
The total is $1 + \frac{3}{2} + 3 = \frac{11}{2}$ .

Answer: $\frac{11}{2}$

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