question about the" roll a die" Tim and Rich each roll a die. Whoever gets the higher number wins; if they both roll the same number, neither wins. Could you please check my answers to following questions and give hints about $d$ and $e$? **a.** What is the probability that Tim wins? **b.** If Rich rolls a $3$, what is the probability that he wins? **c.** If Rich rolls a $3$, what is the probability that Tim wins? **d.** If Rich wins, what is the probability that Tim rolled a $2$? **e.** If Rich wins, what is the probability that Tim rolled a $3$? _Answers_ **a.** I think it is $15/36$. **b.** It is $2/6$. Because Tim could only choose $1$ or $2$. **c.** It is $3/6$. Because Tim could choose $4$, $5$ or $6$.

Notice that the problem is symmetric: both players win with equal probability. The chance of a tie is simply $\frac16$, so for (a) the chance that Tim wins is $\frac12 \cdot Pr(\
eg tie) = \frac12 \cdot (1 - \frac16) = \frac5{12}$.

For (b) Rich wins if Tim rolls below a 3, so $\frac13$ is correct. Likewise, (c) is $\frac12$.

For (d) and (e) we use Bayes' theorem: $P(A|B) = \frac{P(A\cap B)}{P(B)}$:

$$Pr(Tim=2 | Rich wins) = \frac{Pr(Tim=2 \cap Rich wins)}{Pr(Rich wins)} = \frac{\frac16\cdot\frac23}{\frac5{12}} = \frac4{15}$$

(e) is basically the same problem.