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?

$\dfrac12$

$\dfrac35$

$\dfrac23$

$\dfrac34$

$\dfrac45$

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.

2015 AMC 12B Problem 9

Solution:

Problem 9 in Other Years