HMMT 二月 2025 · 冲刺赛 · 第 3 题

HMMT February 2025 — Guts Round — Problem 3

[5] Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of outcomes of all the dice he rolls is even.

解析

[5] Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of outcomes of all the dice he rolls is even. Proposed by: Rishabh Das 5 Answer: 12 1 1 1 Solution: There’s a − = probability that he rolls an even number without getting doubles: 2 6 3 1 whatever the first roll is, there is a chance that the second roll is of opposite parity, and we subtract 2 1 the chance that the second roll is the same. 6 1 1 1 There’s a · = probability that he gets doubles and then rolls an even number. 6 2 12 1 1 5 Summing and gets us . 3 12 12