# Three gamers and consecutive dice rolling: the first to throw a six drops out the game

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.

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

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 $$a.$$