Alice and Bob are playing a game where each player rolls $x$ dice and they score a point for each $5$ or $6$ rolled. They are playing with standard 6-sided, fair dice, but Alice gets to re-roll a die whenever she rolls a $1$. My question is:
How much is Alice expected to win by as $x$ increases?
Clearly, as $x\rightarrow\infty$, they each score infinitely many points. I'm curious about the finite behavior. Bob is expected to score $\frac{x}{3}$ points, but how many is Alice expected to score? In the case where $x=1$, she has a $\frac{2}{5}$ chance of scoring, but what happens when $x>1$?
As long as she is rolling ones she won't be scorning any points. The first time she rolls something else she is equally likely to obtain any of the results: $2$, $3$, $4$, $5$, $6$, two of which give her one point so she gets $\frac{2}{5}$ points on average from a single die. Since there are $x$ dice, by linearity of expectation the total numer of points is just $\frac{2}{5}x$. For Bob the average score is by a simillar reasining $\frac{2}{6}x=\frac{1}{3}x$. So the expected numer of the difference is again by linearity of expectation $\frac{2}{5}x-\frac{1}{3}x=\frac{1}{15}x$.