PUMaC 2013 · 团队赛 · 第 14 题

PUMaC 2013 — Team Round — Problem 14

Shuffle a deck of 71 playing cards which contains 6 aces. Then turn up cards from the top until you see an ace. What is the average number of cards required to be turned up to find the first ace?

解析

Shuffle a deck of 71 playing cards which contains 6 aces. Then turn up cards from the top until you see an ace. What is the average number of cards required to be turned up to find the first ace? SOLUTION: Let X be the random variable for which X = 1 if no ace is drawn before the i i 65 ( ) i − 1 i -th step, and X = 0 if not. Clearly, E [ X ] = . Then i i 71 ( ) i − 1 E [ X + · · · + X ] = E [ X ] + · · · + E [ X ] 1 71 1 71 ( ) 71 65 ∑ i − 1 = ( ) 71 i − 1 i =1 71 ∑ (72 − i ) · · · (67 − i ) = 71 · · · 66 i =1 ( ) 71 ∑ 6! 72 − i = 71 · · · 66 6 i =1 ( ) 6! 72 = 71 · · · 66 7 72 = . 7 72 ANSWER: 7