Final answer:
The probability that Amy wins in one play of the game is 1/6, and the probability that Bill wins is 5/6.
Step-by-step explanation:
In one play of the game, Amy rolls a fair die. If the die shows a 1, Amy wins. If the die does not show 1, then Bill flips a fair coin. If the coin turns up heads, Bill wins.
a) To find the probability that Amy wins, we need to determine the probability of rolling a 1 on a fair die. Since there is only 1 possible outcome out of 6 total outcomes (1, 2, 3, 4, 5, 6) that results in Amy winning, the probability of Amy winning is 1/6.
b) To find the probability that Bill wins, we need to consider the cases where Amy does not win, which is when the die shows a number other than 1. In these cases, Bill flips a fair coin. The probability of getting heads on a fair coin is 1/2. So the probability that Bill wins if Amy does not win is 1/2. Since the events of Amy winning and Bill winning are mutually exclusive, the probability that Bill wins is 1 - (probability of Amy winning) = 1 - (1/6) = 5/6.