# Four fair dice are thrown independently

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______.

Solution:

Probability of getting at least two 3 's or 5 's in one trial

$={ }^{4} C_{2}\left(\frac{2}{6}\right)^{2}\left(\frac{4}{6}\right)^{2}+{ }^{4} C_{3}\left(\frac{2}{6}\right)^{3}\left(\frac{4}{6}\right)+{ }^{4} C_{4}\left(\frac{2}{6}\right)^{4}$

$=\frac{33}{3^{4}}=\frac{11}{27}$

$E(x)=n p=27\left(\frac{11}{27}\right)=11 .$