HMMT 十一月 2010 · 冲刺赛 · 第 14 题

HMMT November 2010 — Guts Round — Problem 14

[ 8 ] The positive integer i is chosen at random such that the probability of a positive integer k being 3 th chosen is times the probability of k + 1 being chosen. What is the probability that the i digit after 2 1 the decimal point of the decimal expansion of is a 2? 7

解析

[ 8 ] The positive integer i is chosen at random such that the probability of a positive integer k being 3 th chosen is times the probability of k + 1 being chosen. What is the probability that the i digit after 2 1 the decimal point of the decimal expansion of is a 2? 7 ( ) n 108 1 2 Answer: First we note that the probability that n is picked is × , because this is the 665 2 3 2 sequence whose terms decrease by a factor of each time and whose sum is 1 (recall that probabilities 3 must sum to 1). 1 Now note that = . 142857142857 . . . , meaning that 2 occurs at digits 3, 9, 15, 21, etc. We can then 7 calculate the probability that we ever pick 2 as Guts Round ( ) ( ) ∞ ∞ 6 k +3 6 k ∑ ∑ 1 2 4 2 · = 2 3 27 3 k =0 k =0 4 1 = · ( ) 6 2 27 1 − 3 4 729 = · 27 729 − 64 4 729 = · 27 665 108 = . 665