What is the probability that a fair six-sided die lands on an even number three out of five times it is rolled?

What is the probability that a fair six-sided die lands on an even number three out of five times it is rolled?

For one roll, the outcomes are $1, 2, 3, 4, 5, 6$ of which $2, 4, 6$ are even, so the probability is $3/6=1/2$. But how to deal with "three out of five times" part?

Answer

Because the number of even sides are equal to odd sides, you can simplify the problem to "What is the probability that a coin lands on heads three times out of five?

For which the probability is $5$ pick $3$ out of $32$ or $$\frac{1}{32} \cdot \frac{5 \cdot 4 \cdot 3}{3\cdot 2 \cdot 1}=\frac{10}{32}= \frac{5}{16}.$$