Suppose there are $n$ types of toys, which you are collecting one by one. Each time you collect a toy, it is equally likely to be any of the $n$ types. What is the expected number of distinct toy types that you have after you have collected $t$ toys? (Assume that you will definitely collect $t$ toys, whether or not you obtain a complete set before then.)
This is the famous coupon-collection problem see e.g. the this book, page 148 or here.
- Label the toys from $1$ to $n$ and use the indicator variable $I_j$ to indicate, if we collected the $j$-th toy. Hence, $I_j=1$ if we collected the $j$-the toy at least once.
- Let $X$ be the random variable counting the number of distinct toys after $t$ trials. Then $X=\sum_{j=1}^n I_j$.
- Now we can calculate the expectation value \begin{align} E[X] &= E \left[\sum_{j=1}^n I_j \right] \\ &= \sum_{j=1}^n E \left[I_j \right] ~~~(\text{by linearity}) \\ &= n \cdot E \left[I_1 \right] ~~~(\text{by symmetry}) \\ &= n \cdot \left( 1 - \left( \frac{n-1}{n} \right)^t \right) ~~~(\text{as pointed out by lulu}) \end{align}