Question

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is _________.

Hint:

For one throw of the four dice, the number of dice showing 3 or 5 is binomial with \(n=4,\ p=\frac13\); find \(P(X\ge2)\) and multiply by 27.

Answer 1

Correct Answer: 11

A die shows 3 or 5 with probability \(p=\frac{2}{6}=\frac13\). Let \(X\) = number of such dice among the four, \(X\sim B(4,\frac13)\).
\(P(X\ge2)=1-P(0)-P(1)=1-\left(\frac23\right)^4-4\cdot\frac13\left(\frac23\right)^3=1-\frac{16}{81}-\frac{32}{81}=\frac{33}{81}=\frac{11}{27}\).
Expected number of times in 27 throws \(=27\cdot\frac{11}{27}=11\).
Answer: 11