(Optional—more difficult) Consider the Survivor game tree illustrated in Figure 3.11. We might not have guessed exactly the values Rich estimated for the various probabilities, so let’s generalize this tree by considering other possible values. In particular, suppose that the probability of winning the immunity challenge when Rich chooses Continue is x for Rich, y for Kelly, and 1 2 x 2 y for Rudy; similarly, the probability of winning when Rich gives up is z for Kelly and 1 2 z for Rudy. Further, suppose that Rich’s chance of being picked by the jury is p if he has won immunity and has voted Rudy off the island; his chance of being picked is q if Kelly has won immunity and has voted Rudy off the island. Continue to assume that if Rudy wins immunity, he keeps Rich with probability 1, and that Rudy wins the game with probability 1 if he ends up in the final two. Note that in the example of Figure 3.11, we had x 5 0.45, y 5 0.5, z 5 0.9, p 5 0.4, and q 5 0.6. (In general, the variables p and q need not sum to 1, though this happened to be true in Figure 3.11.)
(a) Find an algebraic formula, in terms of x, y, z, p, and q, for the probability that Rich wins the million dollars if he chooses Continue. (Note: Your formula might not contain all of these variables.)
(b) Find a similar algebraic formula for the probability that Rich wins the million dollars if he chooses Give Up. (Again, your formula might not contain all of the variables.)
(c) Use these results to find an algebraic inequality telling us under what circumstances Rich should choose Give Up.
(d) Suppose all the values are the same as in Figure 3.11 except for z. How high or low could z be so that Rich would still prefer to Give Up? Explain intuitively why there are some values of z for which Rich is better off choosing Continue.
(e) Suppose all the values are the same as in Figure 3.11 except for p and q. Assume that since the jury is more likely to choose a “nice” person who doesn’t vote Rudy off, we should have p. 0.5. q. For what values of the ratio (pq)should Rich choose Give Up? Explain intuitively why there are some values of p and q for which Rich is better off choosing Continue.