3 players, Achilles, Briseis, and Chryseis, take turns to roll a die in the order $ABC,ABC,\ldots$ . Each player drops out of the game immediately upon throwing a six.
(a) For each player, find the probability that he or she is the first to roll a six.
(b) Let $D_{n}$ be the event that the third player to roll a six does so on the $n$-th roll. Describe the event $E$ given by
$$E = \left(\bigcup_{n=1}^{\infty}D_{n}\right)$$
(c) Show that $\textbf{P}(E) = 0$.
(d) Find the probability that the Achilles rolls a six before Briseis rolls a six.
(e) Show that the probability that Achilles is last to throw a six is $305/1001$.
MY ATTEMPT
Unfortunately, I have no idea how to tackle this problem. But it is worthy emphasizing that it is not a homework. I am really interested in knowing the result. Thanks in advance.
Best Answer
Let $a,b,c$ be the respective probabilities that $A,B,C$ is first to roll a $6.$ The only way that B can be first to roll a $6$ is if Achilles does not roll a $6$ at his first turn. Then Briseis is in the same position as Achilles was at his first turn, so $$b={5a\over6}$$ Similarly, $$c={25a\over36}$$ Clearly, $a+b+c=1,$ so you can solve for $a.$
That should get you started.