problem: coin toss game

Discussion in 'Probability' started by Hamwise Tarot, Jul 22, 2004.

  1. My thanks to anyone who can help with the following problem. It's an
    analog to a behavioral assessment (of basic discrimination ability)
    with a "pass" criterion equivalent to the "win" criterion described
    below. We have used the assessment for many years, but to my knowledge
    no one has articulated the precise probability of passing by chance.

    # The Game #
    Toss a fair coin repeatedly, recording the result on each toss. The
    game can end in two ways:
    1. You land eight heads consecutively, thereby winning.
    2. You land eight tails cumulatively, thereby losing.

    The shortest possible game is 8 tosses: TTTT TTTT or HHHH HHHH.
    The longest possible game is 64 tosses:

    HHHH HHHT HHHH HHHT
    HHHH HHHT HHHH HHHT
    HHHH HHHT HHHH HHHT
    HHHH HHHT HHHH HHH(T/H)

    What the probability (relative frequency) of winning?
     
    Hamwise Tarot, Jul 22, 2004
    #1
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  2. Hamwise Tarot

    Henry Guest

    What's the probability of 8Hs in a row?
    What's the probability of a T without 8Hs in a row?
    What's the probability of 8Ts without 8Hs in a row?
    What's the probability of 8Hs in a row before 8Ts?

    I make the last one 1-(1-0.5^8)^8 = 0.030826
     
    Henry, Jul 23, 2004
    #2
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  3. The game can be viewed as a sequence of tossing sequences each of
    which has one of these 9 forms

    T
    HT
    HHT
    HHHT
    HHHHT
    HHHHHT
    HHHHHHT
    HHHHHHHT
    HHHHHHHH p = 1/256


    You lose if you toss eight times in a row one of sequeces #1..#8, This
    probability is (255/256)^8. Therefore the probability of winning is
    1-(255/256)^8
     
    Horst Kraemer, Jul 23, 2004
    #3
  4. Hamwise Tarot

    Steve Tilley Guest

    Sorry I'm being thick how come the longest game is 64 tosses? Tou could
    have a sequence such as HHTHHTTHHTTTHHTH etc going on for a long time.
     
    Steve Tilley, Jul 23, 2004
    #4
  5. According to the rules the game is over after the 8th tail or after
    the 8th head in a row.
     
    Horst Kraemer, Jul 23, 2004
    #5
  6. That's excellent, thanks you guys.

    (And for your explanation, Horst!)
     
    Hamwise Tarot, Jul 26, 2004
    #6
  7. Hamwise Tarot

    jeffc Guest

    Look at the wording of rule 1 vs. rule 2. It's "consecutive" heads, but
    "cumulative" (total) tails. So the longest run is HHHHHHHT repeated 7 more
    times.
     
    jeffc, Aug 2, 2004
    #7
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