Skip to main content
OlympiadHQ

Browse · harp

Print

smc

counting and probability senior

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?
(A)
(B)
(C)
(D)
Solution
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 .
Final answer
C