Problem of the Week 1068

Penalty Kick Odds

Alice: Did you hear about the recent NCAA Division III soccer game where one team advanced on penalty kicks? It went to 20 rounds of kicks before it was decided 16-15. I think that's a record.*

Bob: Wow. The odds against a shootout** lasting that long must be immense.

Alice: Not when the teams are this good at making them. Since the chance of success in a given kick for the two teams can be taken to be 16/20 and 15/20, the probability of a shootout going 20 or more rounds is surprisingly high, about one in 1400.

Bob: I wonder how many rounds it would have to go before the odds were 1 in a million.

Question: What is the least n for which the probability of a shootout going n or more rounds is one in a million, given the success probabilities above?

* True story. Medaille College vs. Nazareth College, Nov. 11, 2006; see

** Shootout rules: Each team takes 5 shots, alternating. If the score is tied after 5, they continue in rounds of 1 shot each until someone leads at the end of a round.

© Copyright 2006 Stan Wagon. Reproduced with permission.

28 November 2006