In an exciting scene1 from the 1980 space-opera film “Flash Gordon”, two protagonists take turns in putting their arms into different holes of a large tree stump, where the wood beast lives, in order to see who is subjected to its fatal sting. In a twist on this competition, the sting may take two days to kill, so the game continues even if a sting is made.

Suppose that there are nine holes in this stump, all of which are identical looking except that three of them lead to a fatal sting. You are to take on the role of one of the protagonists.

1.     (a)  Suppose you go first. What is the probability that you do not get stung on your first turn?

2.     (b) Suppose you went first and did not get stung. What is the probability that your competitor also does not get stung on their first turn, given that you did not get stung on your first turn?

3.     (c) Suppose you went first and got stung. What is the probability that your competitor does not get stung on their first turn, given that you got stung on your first turn?

4.     (d)  Suppose that we stop the game after each player has taken exactly one turn. Combining your results from parts (a)–(c), or otherwise, show that the probability that you do not get stung is the same regardless of whether you go first or second.

5.     (e)  Now suppose we keep playing until every hole has been tried exactly once. Would it be better to go first or second? Justify your answer using probability calculations.