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問題:What is the probability of obtaining exactly 2 heads if it is known that at least 1 head appeared?

答案:P(2H/at least 1H occurs) = 28/255

需要算式過程

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1 個解答

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  • chao
    Lv 4
    1 0 年前
    最佳解答

    Conditional probablity: P(A / B) = P (A and B)/P(B)

    P(2H/at least 1H occurs) = P([2H] and [at least 1H occurs]) / P(at least 1H occurs) = X/Y

    X= P([2H] and [at least 1H occurs])= ?

    So we know at least 1 H occurs, and there is another H somewhere.

    How many ways can it be?

    (1 H in 1st) (1 H in 2nd) (1 H in 3rd) (1H in 4th)

    HHTTTTTT / THHTTTTT / TTHHTTTT / TTTHHTTT

    HTHTTTTT / THTHTTTT / TTHTHTTT / TTTHTHTT

    HTTHTTTT / THTTHTTT / TTHTTHTT / TTTHTTHT

    HTTTHTTT / THTTTHTT / TTHTTTHT / TTTHTTTH

    HTTTTHTT / THTTTTHT / TTHTTTTH

    HTTTTTHT / THTTTTTH

    HTTTTTTH

    (1 H in 5th) (1 H in 6th) (1 H in 7th)

    TTTTHHTT / TTTTTHHT / TTTTTTHH

    TTTTHTHT / TTTTTHTH

    TTTTHTTH

    So the total ways is: 7 6 5 4 3 2 1=28

    There are total of 2x2x2x2x2x2x2x2 = 256 ways

    So X= P([2H] and [at least 1H occurs])= 28/256

    ----------------------------------------------------------

    Y= P(at least 1H occurs) = 1- P(no H occurs) = 1- (1/256) = 255/256

    Therefore,

    P(2H/at least 1H occurs) = P([2H] and [at least 1H occurs]) / P(at least 1H occurs) = X/Y= (28/256)/(255/256) = 28/255 Q.E.D.

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