HMMT 十一月 2014 · 冲刺赛 · 第 16 题

HMMT November 2014 — Guts Round — Problem 16

[ 10 ] A particular coin has a chance of landing on heads (H), chance of landing on tails (T), and 3 3 1 chance of landing vertically in the middle (M). When continuously flipping this coin, what is the 3 probability of observing the continuous sequence HMMT before HMT?

解析

[ 10 ] A particular coin has a chance of landing on heads (H), chance of landing on tails (T), and 3 3 1 chance of landing vertically in the middle (M). When continuously flipping this coin, what is the 3 probability of observing the continuous sequence HMMT before HMT? 1 Answer: For a string of coin flips S , let P denote the probability of flipping HM M T before S 4 1 1 1 HM T if S is the starting sequence of flips. We know that the desired probability, p , is P + P + P . H M T 3 3 3 Now, using conditional probability, we find that 1 1 1 P = P + P + P H HH HM HT 3 3 3 1 1 1 = P + P + P . H HM T 3 3 3 We similarly find that 1 1 1 P = P = P + P + P M T H M T 3 3 3 1 1 P = P + P HM HM M H 3 3 1 1 1 P = + P + P . HM M M H 3 3 3 1 1 Solving gives P = P = P = . Thus, p = . H M T 4 4