MathDB
Balls and Bottles

Source: 2015 AMC12B #9

February 26, 2015
probabilityratiogeometric seriesAMC

Problem Statement

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 12\frac{1}{2}, independently of what has happened before. What is the probability that Larry wins the game?
<spanclass=latexbold>(A)</span>12<spanclass=latexbold>(B)</span>35<spanclass=latexbold>(C)</span>23<spanclass=latexbold>(D)</span>34<spanclass=latexbold>(E)</span>45<span class='latex-bold'>(A) </span>\frac{1}{2}\qquad<span class='latex-bold'>(B) </span>\frac{3}{5}\qquad<span class='latex-bold'>(C) </span>\frac{2}{3}\qquad<span class='latex-bold'>(D) </span>\frac{3}{4}\qquad<span class='latex-bold'>(E) </span>\frac{4}{5}