# (Optional—more difficult) Consider the Survivor game tree illustrated in Figure 3.11. We might not..

(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.