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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.

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  • $\begingroup$ Hint for (b): The game ends when the third six comes up. $\endgroup$ – amd Jan 12 at 23:41
  • $\begingroup$ The question in c is incorrect. It should be $\textbf{P}(E) =1$ $\endgroup$ – Ross Millikan Jan 12 at 23:53
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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.

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