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Geometry

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A fair coin is flipped 3 times. The probability of getting exactly two heads, given that at least one flip results in a head, can be written as ab, where a and b are coprime positive integers. What is the value of a+b?

  • Geometry -

    It is a conditional probability problem where the distribution is binomial.

    The general expression for conditional probability for probability of event A given event B is
    P(A|B)=P(A∩B)/P(B)
    where A = P(exactly 2 heads)
    B=P(at least one head)
    P(A∩B)=P(A) since A is a subset of B.

    Assuming a fair coin, p=1/2, q=1-1/2=1/2
    P(A∩B)
    =P(A)=P(exactly 2 heads)
    =3C2 p^2 q^1
    =3*(1/2)^2 (1/2)
    =3/8

    P(B)=P(at least one heads)
    =1-P(no heads)
    =1- 3C0 (1/2)^0 (1/2)^3
    =1- 1/8
    =7/8

    So
    P(A|B)
    =P(exactly two heads | at least one heads)
    =(3/8) / (7/8)
    =3/7

  • Geometry -

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