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 $305/1001$.


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.

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.