2015 AMC 12B Problem 9

Question

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?

\dfrac12

\dfrac35

\dfrac23

\dfrac34

\dfrac45

Answer: C

Concepts:recursive probability geometric sequence

Difficulty rating: 1540

Small hint:

Larry wins immediately with probability

\dfrac12

Big hint:

If both miss (probability

\dfrac14

), the game resets; solve

x = \dfrac12 + \dfrac14 x

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.