A fair die is rolled six times. Find the mathematical expectation of the number of different faces that appear.

#
Let $\xi_{i}$ be a random variable equal to 1 if the face with $i$ points has appeared at least once and 0 if it has never appeared. Then the number of different faces that have appeared is

$\xi=\xi_{1}+\ldots+\xi_{6}$. Transitioning to expectations, we get: $\mathrm{E} \xi=6 \mathrm{E} \xi_{1}$, since all variables $\xi_{i}$ are identically distributed.

The expected value $\mathrm{E} \xi_{1}$ is obviously equal to $\mathrm{P}\left(\xi_{1}=1\right)=1-(5 / 6)^{6}$.

Thus, $\mathrm{E} \xi=6\left(1-\left(\frac{5}{6}\right)^{6}\right)=\frac{6^{6}-5^{6}}{6^{5}}$.

## Answer

$\frac{6^{6}-5^{6}}{6^{5}} \approx 3.99$