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Math Help - Conditional probability question help

  1. #1
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    Conditional probability question help

    Can't seem to figure this question out. Can someone please show me how to approach it?

    At bridge hand (13 cards) contains seven clubs, four hearts and two spades. Two cards are drawn at random from the hand without replacement. Find:

    Pr [One card is a club | at least one card is a heart]


    Thanks.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Use the equation form Baye's Theorem.

    P(\text{One card is a club | at least one card is a heart}) = \dfrac{P(\text{One card is a club} \cap \text{at least one card is a heart})}{P(\text{at least one card is a heart})}
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  3. #3
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    Quote Originally Posted by Unknown008 View Post
    Use the equation form Baye's Theorem.

    P(\text{One card is a club | at least one card is a heart}) = \dfrac{P(\text{One card is a club} \cap \text{at least one card is a heart})}{P(\text{at least one card is a heart})}
    Using that, I need getting 7/12 as my answer which is incorrect. The back of the book shows 2/3. Do you mind telling me which numbers I'm supposed to plug in?

    Thanks
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  4. #4
    MHF Contributor Unknown008's Avatar
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    P(at least one card is a heart) = (P(1st card heart) x P(2nd card not heart)) + (P(1st card not heart) x P(2nd card heart)) + (P(both cards heart)) =

    \left(\dfrac{4}{13} \times \dfrac{9}{12}\right) + \left(\dfrac{9}{13} \times \dfrac{4}{12}\right) + \left(\dfrac{4}{13} \times \dfrac{3}{12}\right) = \dfrac{7}{13}

    For the numerator, you must have one club and a heart, or a heart and a club.

    P(Heart and club) = 4/13 x 7/12 = 7/39

    P(club and heart) = 7/13 x 4/12 = 7/39

    Hence, we get: \dfrac{\frac{14}{39}}{\frac{7}{13}} = \dfrac23

    Another way of looking at it is that you have the following combinatios:

    You want to have:
    HC

    out of:
    HH
    SH
    CH

    (H - heart, C- club, S - spade)

    This gives a total of \dfrac{^4C_1 \times ^7C_1}{^4C_2 + (^2C_1 \times ^4C_1) + (^7C_1 \times ^4C_1)}= \dfrac23
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  5. #5
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    Thanks! I missed a couple steps in my calculations.
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