HMMT 十一月 2010 · GEN2 赛 · 第 10 题

英文解析

10. [ 7 ] Justine has a coin which will come up the same as the last flip of the time and the other side of
    3 3
    the time. She flips it and it comes up heads. She then flips it 2010 more times. What is the probabilitythat the last flip is heads?
    2010
    3 +1
    Answer: Let the “value” of a flip be 1 if the flip is different from the previous flip and let
    2010
    2 · 3
    it be 0 if the flip is the same as the previous flip. The last flip will be heads if the sum of the values of
    ( ) ( ) ( )
    1005
    2 i 2010 − 2 i
    ∑
    2010 1 2
    all 2010 flips is even. The probability that this will happen is .
    2 i 3 3
    i =0
    We know that
    ( ) ( ) ( ) ( ) ( ) ( )
    1005
    2 i 2010 − 2 i 2 i +1 2010 − (2 i +1)
    ∑
    2010 1 2 2010 1 2 + =
    2 i 3 3 2 i + 1 3 3
    i =0
    ( ) ( ) ( ) ( )
    2010
    k 2010 − k 2010
    ∑
    2010 1 2 1 2 = + = 1
    k 3 3 3 3
    k =0
    ( ) ( ) ( ) ( ) ( ) ( )and
    1005
    2 i 2010 − 2 i 2 i +1 2010 − (2 i +1)
    ∑
    2010 1 2 2010 1 2 − =
    2 i 3 3 2 i + 1 3 3
    i =0
    ( ) ( ) ( ) ( ) ( )
    2010
    k 2010 − k 2010 2010
    ∑
    2010 − 1 2 − 1 2 1 = + =
    k 3 3 3 3 3
    k =0
    1 2010
    2010
    1+( )
    3 +1
    Summing these two values and dividing by 2 gives the answer = .3
    2010
    2 2 · 3
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