HMMT 十一月 2018 · 冲刺赛 · 第 12 题

HMMT November 2018 — Guts Round — Problem 12

[ 8 ] Consider an unusual biased coin, with probability p of landing heads, probability q ≤ p of landing 1 tails, and probability of landing on its side (i.e. on neither face). It is known that if this coin is 6 1 flipped twice, the likelihood that both flips will have the same result is . Find p . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2018, November 10, 2018 — GUTS ROUND Organization Team Team ID#

解析

[ 8 ] Consider an unusual biased coin, with probability p of landing heads, probability q  p of landing 1 tails, and probability of landing on its side (i.e. on neither face). It is known that if this coin is 6 1 flipped twice, the likelihood that both flips will have the same result is . Find p . 2 Proposed by: Brian Reinhart 2 Answer: 3 2 2 1 1 The probability that both flips are the same is p + q + . For this to be , we must have 36 2 ✓ ◆ 2 1 5 1 1 2 2 2 p + q + = p + p + = 36 6 36 2 2 1 2 Using the quadratic formula, p = or . Since p q , we have that p = . 3 6 3