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Math Help - Independence

  1. #1
    Senior Member slevvio's Avatar
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    Independence

    A (possibly biased) coin is tossed twice. The result 'heads' is noted if the coin lands heads up on the first toss and tails up on the second toss (HT). In a similar way, the result 'tails' is noted if the outcome is TH. HH and TT are ignored, the entire experiment being repeated until either HT or TH occurs. Show that the result 'heads' is noted with probability 0.5 (however biased the coin).

    Thanks in advance any help with this would be appreciated.
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by slevvio View Post
    A (possibly biased) coin is tossed twice. The result 'heads' is noted if the coin lands heads up on the first toss and tails up on the second toss (HT). In a similar way, the result 'tails' is noted if the outcome is TH. HH and TT are ignored, the entire experiment being repeated until either HT or TH occurs. Show that the result 'heads' is noted with probability 0.5 (however biased the coin).

    Thanks in advance any help with this would be appreciated.
    Let p be the probability that a coin comes up heads on a single toss. Then 1 - p is the probability that a coin comes up tails on a single toss.

    Pr(Heads) = Pr(HT) = p(1 - p).

    Pr(Tails) = Pr(TH) = (1 - p)p.

    Therefore Pr(Heads) = Pr(Tails).

    But Pr(Heads) + Pr(Tails) = 1 since HH and TT are ignored. Therefore .....
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