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 _________.

4 dice are independently thrown. Each die has probability to show 3 or 5 is <br><br>$P = {2 \over 6} = {1 \over 3}$<br><br>$\therefore$ $q = 1 - {1 \over 3} = {2 \over 3}$ (not showing 3 or 5)<br><br>Experiment is performed with 4 dices independently<br><br>$\therefore$ Their binomial distribution is <br><br>$${(q + p)^4} = {(q)^4} + {}^4{C_1}{q^3}p + {}^4{C_2}{q^2}{p^2} + {}^4{C_3}q{p^3} + {}^4{C_4}{P^4}$$<br><br>$\therefore$ In one throw of each dice probability of showing 3 or 5 at least twice is<br><br>= ${p^4} + {}^4{C_3}q{p^3} + {}^4{C_2}{q^2}{p^2}$<br><br>$= {{33} \over {81}}$<br><br>Given such experiment performed 27 times<br><br>$\therefore$ So expected outcomes = np<br><br>= ${{33} \over {81}} \times 27$<br><br>= 11