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Smaller of $2$ rolls

Expected valueEasy

$2$ fair $4$ -sided dice, numbered $1$ to $4$ , are rolled. What is the expected value of the smaller result?

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Hint

Use the tail sum: the expectation is the sum of P(min ≥ k).

Solution

The smaller result is at least $k$ when both dice are at least $k$ , which has probability $(\frac{5 - k}{4})^{2}$ .
For $k = 1, 2, 3, 4$ these are $\frac{16}{16}$ , $\frac{9}{16}$ , $\frac{4}{16}$ and $\frac{1}{16}$ .
The expectation is their sum, $\frac{30}{16} = \frac{15}{8}$ .

Answer: $\frac{15}{8}$