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Math Help - Basic Probability Help Redux

  1. #1
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    Basic Probability Help Redux

    Ok, I just need help showing the answer to this problem mathematically.

    A child has a 1/2 probability of inheriting a certain disease. In a two child family, what is the probability that EXACTLY one child will inherit the disease?

    Obviously, there are four total outcomes.

    {YY, YN, NY, NN}

    And from this, I can see that there are two instances in which exactly one child inherits the disease, so the answer is 1/2.

    I just need help with figuring out how to write this out mathematically. I know I'm just missing something simple.

    I thought maybe it was just:

    A= child 1 is affected, B = child 2 is affected


    Pr(AUB)

    But

    Pr(AUB) = Pr(A)+P(B) - Pr(AnB) = 1/2 +1/2 -1/4 = 3/4

    Which makes sense because the probability of A or B is:

    The probability that child 1 is affected + the probability that child 2 is affected + the probability that both are affected which totals 3/4

    Is the proper way to mathematically write out what I'm looking for:

    Pr(AUB) - Pr(AnB) = [Pr(A)+P(B)-Pr(AnB)] - Pr(AnB)
    = (1/2 + 1/2 - 1/4) - 1/4 = 1/2

    Doing it this way makes sense to me since I'm looking the probability of A or B but not Both, but it seems funky to have to subtract Pr(AnB) twice.

    Any push in the right direction is appreciated. Thanks
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  2. #2
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    Quote Originally Posted by downthesun01 View Post
    Ok, I just need help showing the answer to this problem mathematically.
    A child has a 1/2 probability of inheriting a certain disease. In a two child family, what is the probability that EXACTLY one child will inherit the disease?
    We assume independence.
    It is A and not B or B and not A.
    P(X=2)=P(AB^c)+P(BA^c)=.25+.25=.5
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