PUMaC 2012 · 组合（A 组） · 第 3 题

PUMaC 2012 — Combinatorics (Division A) — Problem 3

[ 4 ] Jim has two fair 6-sided dice, one whose faces are labelled from 1 to 6, and the second whose faces are labelled from 3 to 8. Twice, he randomly picks one of the dice (each die equally likely) m and rolls it. Given the sum of the resulting two rolls is 9, if is the probability he rolled the n same die twice where m, n are relatively prime positive integers, then what is m + n ?

解析

Note that there is some symmetry: the probability of rolling a 9 is the same for both two rolls of the first die and two rolls of the second dice (consider the correspondence of a face f of the first die with that of a face 9 − f on the second die), and is given by 4 / 36 = 1 / 9 by listing the possible face combinations. If the first and second dice are rolled, the probability of rolling 9 is 6 / 36 = 1 / 6. By conditional probability, the answer is P (rolled 9 | same dice) 1 / 9 2 P (same dice | rolled 9) = = = , 1 1 P (rolled 9) 5 · (1 / 9) + · (1 / 6) 2 2 so the answer is 2 + 5 = 7 . 2