2015 AMC 12B Problems/Problem 9
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?
Solution
Solution 1
If Larry wins, he either wins on the first move, or the third move, or the fifth move, etc. Let represent "player wins", and represent "player loses". Then the events corresponding to Larry winning are
Thus the probability of Larry winning is
This is a geometric series with ratio , hence the answer is .
Solution 2
Break the problem up into two separate cases: (a) Larry wins on the first throw or (b) Larry wins after the first throw.
a: The probability that Larry wins on the first throw is .
b: The probability that Larry wins after the first throw is half the probability that Julius wins because it only occurs half the time. This probability is , where is the probability that Larry wins.
Therefore, . This equation can be solved for to find that the probability that Larry wins is .
See Also
2015 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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