Problem:
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is , independently of what has happened before. What is the probability that Larry wins the game?
Answer Choices:
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Solution:
Let be the probability that Larry wins the game. Then . To see this, note that Larry can win by knocking the bottle off the ledge on his first throw; if he and Julius both miss, then it is as if they started the game all over. Thus , so or .
OR
For Larry to win on his th throw, there must be misses by Larry and by Julius - followed by a hit by Larry. Because the probability of each of these independent events is , the probability that Larry wins on his th throw is . Therefore the probability that Larry wins the game is given by a geometric series:
The problems on this page are the property of the MAA's American Mathematics Competitions