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Distinct faces on dice

Expected valueMedium

$3$ fair $6$ -sided dice are rolled independently. What is the expected number of distinct faces that appear? Give a fraction a/b in lowest terms.

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Hint

Use one indicator per face value: does that value appear at least once?

Solution

For a fixed face value, the chance that a single die does not show it is $\frac{5}{6}$ .
All $3$ dice miss it with probability $(\frac{5}{6})^{3} = \frac{125}{216}$ , so it appears with probability $\frac{91}{216}$ .
There are $6$ face values, so by linearity the expectation is $6 \times \frac{91}{216}$ .
This simplifies to $\frac{91}{36}$ .

Answer: $\frac{91}{36}$