A friend bets you $100 on a game involving two six-sided dice, one red and one green. You choose the number of times the pair of dice will be rolled. You win if the number of times a red 6 is rolled is at most 2 and the number of times a green 6 is rolled is at least 2.
a) How many times should the dice be rolled to maximise your chance of winning?
b) With that number of rolls, what are your expected winnings?
Let $R$ and $G$ let the number of $6$ in $n$ rolls on the red and green dice, resp. Both are binomial r.v. with $p=1/6$.
We want to find $n$ which maximises $$P(R\leq2\text{ and }G\geq2).$$ As the events are independent that probability is equal to $$P(R\leq2)\cdot P(G\geq2)=P(R\leq2)\cdot\bigl(1-P(G\leq1)\bigr).$$ Assumably there's no analytic way to solve that problem, so simply try out some values of $n$; here $n=12$ gives the maximum $0.4191$.
It seems that in general the maximum is near $n=2/p$ (that is, the expected value is $2$); it's value is approximately $0.4$, independent of $p$.