HMMT 十一月 2010 · 冲刺赛 · 第 14 题

英文解析

14. [ 8 ] The positive integer i is chosen at random such that the probability of a positive integer k being
    3 thchosen is times the probability of k + 1 being chosen. What is the probability that the i digit afterthe decimal point of the decimal expansion of is a 2?12
    ( )7
    108 1 2 n
    Answer: First we note that the probability that n is picked is × , because this is the
    665 2 3
    sequence whose terms decrease by a factor of each time and whose sum is 1 (recall that probabilities 2
    must sum to 1).3
    Now note that = . 142857142857 . . . , meaning that 2 occurs at digits 3, 9, 15, 21, etc. We can then 1
    calculate the probability that we ever pick 2 as 7
    Guts Round
    ( ) ( )
    ∞ ∞
    6 k +3 6 k
    ∑ ∑
    1 2 4 2
    · =
    2 3 27 3
    k =0 k =0
    4 1 = ·
    ( )
    2726
    1 −
    4 7293 = ·
    27 729 − 64
    4 729 = ·
    27 665 = .108
    665