Four fair dice are thrown independently 27 times. Then the expected number of times, at least two di

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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

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(11) : Probability of showing 3 or 5, $p=\frac{2}{6}=\frac{1}{3}$\ $	herefore \quad$ Probability of not showing 3 or 5, $q=1-\frac{1}{3}=\frac{2}{3}$\ Experiment is performed with 4 dice independently.\ $	herefore \quad P($ At least 2 dice shows up 3 or 5)\ $={ }^{4} C_{4} p^{4}+{ }^{4} C_{3} p^{3} q+{ }^{4} C_{2} p^{2} q^{2}$\ $=\left(\frac{1}{3}ight)^{4}+4\left(\frac{1}{3}ight)^{3} \cdot\left(\frac{2}{3}ight)+6\left(\frac{1}{3}ight)^{2}\left(\frac{2}{3}ight)^{2}$\ $=\frac{1}{81}+\frac{8}{81}+\frac{24}{81}=\frac{33}{81}$\ $\because \quad$ Experiment performed 27 times.\ $	herefore \quad E(X)=X P(X)=\frac{33}{81} 	imes 27=11$\