HMMT 十一月 2025 · 冲刺赛 · 第 23 题

HMMT November 2025 — Guts Round — Problem 23

[12] Jacopo and Srinivas are playing a game with a bag of marbles. The bag starts with 6 red marbles and 6 blue marbles. Jacopo begins by drawing a marble from the bag, uniformly at random. When either player draws a marble, if it is red, the same player draws the next marble; otherwise, the other player draws the next marble (uniformly at random). All marbles are drawn without replacement. This process continues until all 12 marbles have been drawn. Compute the expected number of marbles that Jacopo draws. ◦

解析

[12] Jacopo and Srinivas are playing a game with a bag of marbles. The bag starts with 6 red marbles and 6 blue marbles. Jacopo begins by drawing a marble from the bag, uniformly at random. When either player draws a marble, if it is red, the same player draws the next marble; otherwise, the other player draws the next marble (uniformly at random). All marbles are drawn without replacement. This process continues until all 12 marbles have been drawn. Compute the expected number of marbles that Jacopo draws. Proposed by: Derek Liu 45 Answer: 7 Solution: The players must take turns drawing the blue marbles, so Jacopo will always draw 3 blue marbles. Any given red marble will be drawn after either 0, 1, . . . , or 6 blue marbles, all with equal probability; if this quantity is even, Jacopo draws it, and otherwise Srinivas draws it. Hence, there is 4 a probability Jacopo draws any given red marble, so by linearity of expectation, as there are 6 red 7 marbles, Jacopo draws an expected 4 24 6 · = 7 7 red marbles. By linearity of expectation, the total expected number of marbles Jacopo draws is equal to the sum of the expected numbers of blue and red marbles he draws, for an answer of 24 45 3 + = . 7 7 © 2025 HMMT ◦