HMMT 十一月 2012 · 冲刺赛 · 第 20 题

HMMT November 2012 — Guts Round — Problem 20

[ 11 ] An elementary school teacher is taking a class of 20 students on a field trip. To make sure her students don’t get lost, she uses the buddy system—using the complete class roster, she pairs the students into 10 pairs and leaves if no person reports that someone from his or her pair is missing. In particular, the teacher will not notice if both students from a pair go missing. Suppose each student 1 independently gets lost with probability . Given that the teacher leaves, what is the probability that 10 no student got lost?

解析

[ 11 ] 10 81 Answer: The teacher will leave if the students from each pair are either both present or both 10 82 81 not present; the probability that both are present is and the probability that neither are present 100 1 is . If the teacher leaves, then the probability that both students in any given pair did not get lost 100 ( ) 10 81 10 81 81 81 100 is = . Since there are ten pairs, the overall probability is = . 81 1 10 82 82 82

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