(The game of craps) The game of craps is played by rolling two fair, six-sided dice. On the first roll, if the sum of the two numbers showing equals 2, 3, or 12, then the player immediately loses. If the sum equals 7 or 11, then the player immediately wins. If the sum equals any other value, then this value becomes the player's "point." The player then repeatedly rolls the two dice, until such time as he or she either rolls the point value again (in which case he or she wins) or rolls a 7 (in which case he or she loses).
(a) Suppose the player's point is equal to 4. Conditional on this, what is the conditional probability that he or she will win (i.e., will roll another 4 before rolling a 7)?
(b) For 2 ≤ i ≤ 12, let pi be the conditional probability that the player will win, conditional on having rolled i on the first roll. Compute pi for all i with 2 ≤ i ≤ 12