HMMT 二月 2005 · COMB 赛 · 第 8 题

HMMT February 2005 — COMB Round — Problem 8

Every second, Andrea writes down a random digit uniformly chosen from the set { 1 , 2 , 3 , 4 } . She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1?

解析

Every second, Andrea writes down a random digit uniformly chosen from the set { 1 , 2 , 3 , 4 } . She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1? Solution: 15 / 44 Let p be the probability that the last number she writes down is 1 when the first n number she writes down is n . Suppose she starts by writing 2 or 4. Then she can continue writing either 2 or 4, but the first time she writes 1 or 3, she stops. Therefore 1 p = p = . Suppose she starts by writing 1. Then she stops if she writes 1, 2 4 2 1 2, or 4, but continues if she writes 3. Therefore p = (1 + p ). If she starts by 1 3 4 writing 3, then she stops if she writes 2 or 4 and otherwise continues. Therefore 1 1 1 3 p = ( p + p ) = (1 + 5 p ). Solving gives p = and p = . The probability we 3 1 3 3 3 1 4 16 11 11 1 15 want to find is therefore ( p + p + p + p ) = . 1 2 3 4 4 44