probability - How do wer find this expected value?

I'm just a little confused on this. I'm pretty sure that I need to use indicators for this but I'm not sure how I find the probability. The question goes like this:

A company puts five different types of prizes into their cereal boxes, one in each box and in equal proportions. If a customer decides to collect all five prizes, what is the expected number of boxes of cereals that he or she should buy?

I have seen something like this before and I feel that I'm close, I'm just stuck on the probability. So far I have said that

$$X_i=\begin{cases}1 & \text{if the $i^{th}$ box contains a new prize}\\ 0 & \text{if no new prize is obtained} \end{cases}$$ I know that the probability of a new prize after the first box is $\frac45$ (because obviously the person would get a new prize with the first box) and then the probability of a new prize after the second prize is obtained is $\frac35$, and so on and so forth until the fifth prize is obtained. What am I doing wrong?! Or "what am I missing?!" would be the more appropriate question.

Answer

As the expected number of tries to obtain a success with probability $p$ is $\frac{1}{p}$, you get the expect number :

$$1+\frac{5}{4}+\frac{5}{3}+\frac{5}{2}+5=\frac{12+15+20+30+60}{12}=\frac{137}{12}\approx 11.41$$