HMMT 十一月 2008 · 冲刺赛 · 第 6 题

HMMT November 2008 — Guts Round — Problem 6

[ 6 ] Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6-sided die. If she gets a 1 or 2, she goes to DC. If she rolls a 3 , 4 , or 5, she goes to Russia. If she rolls a 6, she rolls again. What is the probability that she goes to DC? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND 2 2

解析

[ 6 ] Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6-sided die. If she gets a 1 or 2, she goes to DC. If she rolls a 3 , 4 , or 5, she goes to Russia. If she rolls a 6, she rolls again. What is the probability that she goes to DC? 2 Answer: On each roll, the probability that Sarah decides to go to Russia is 3 / 2 times the 5 probability she decides to go to DC. So, the total probability that she goes to Russia is 3 / 2 times the total probability that she goes to DC. Since these probabilities sum to 1 (they are the only two 2 3 eventual outcomes) Sarah goes to DC with probability and Russia with probability . 5 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUND 2 2