PUMaC 2019 · 组合（B 组） · 第 3 题

PUMaC 2019 — Combinatorics (Division B) — Problem 3

Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from 1 to 50 uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability 3/5. If he flips heads, he adds 1 to his score. A player wins the game if their score is higher than the other player’s score. The probability Dukejukem defeats the Trollner to win WASS equals m/n where m, n are coprime positive integers. Compute m + n . i

解析

Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from 1 to 50 uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability 3/5. If he flips heads, he adds 1 to his score. A player wins the game if their score is higher than the other player’s score. The probability Dukejukem defeats the Trollner to win WASS equals m/n where m, n are coprime positive integers. Compute m + n . Proposed by Kapil Chandran. Answer: 751 . Solution: If the coin has probability q of landing heads, the probability of Dukejukem winning is (1 − P (tie)) / 2 + q P (tie), where P (tie) = 1 / 50 is the probability that both players spin the same number on the wheel. This is 251 / 500, so the answer is 751. i