2015 AMC 12B Problem 9

Problem 9 of 25EasierAlgebra Counting & Probability

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

\dfrac12,

independently of what has happened before. What is the probability that Larry wins the game?

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Solution

Let

x

be the probability Larry wins. He wins right away with probability

\dfrac12,

or both players miss (probability

\dfrac14

) and the game restarts. So

x = \dfrac12 + \dfrac14 x,

giving

\dfrac34 x = \dfrac12

and

x = \dfrac23.

Thus, the correct answer is C.

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