HMMT 十一月 2014 · 冲刺赛 · 第 16 题

英文解析

16. [ 10 ] A particular coin has a chance of landing on heads (H), chance of landing on tails (T), and
    3 3
    chance of landing vertically in the middle (M). When continuously flipping this coin, what is the 1
    probability of observing the continuous sequence HMMT before HMT?3
    Answer: For a string of coin flips S , let P denote the probability of flipping HM M T before 1
    4S
    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