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Math Help - Conditional Probability

  1. #1
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    Conditional Probability

    What is the chance that the seventh card from the top of the deck will be a heart?

    This question really stumped me. I thought [52_C_6 multiply 13_C_1] and then divided by 52_C_7 would be correct but the probability is greater than 1 which is obviously wrong.


    Thank you!
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  2. #2
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    Hello, Felipe!

    What is the chance that the 7th card from the top of the deck will be a heart?

    The deck can be arranged in 52! orderings.

    If the 7th card is a Heart, there are 13 choices.
    The other 51 cards can be arranged in 51! ways.
    . . Hence, there are: 13\cdot51! orderings with the 7th card a Heart.

    Therefore: . P(\text{7th card is a }\heartsuit) \:=\:\dfrac{13\cdot5!!}{52!} \;=\;\dfrac{1}{4}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    With a little Thought, the answer is obvious.

    Consider any card in the deck,
    . . the top card, the bottom card, the 37th card . . .

    What is the probabiity that it is a Heart?


    Since it could any one of the four suits (all equally likely),

    . . P(\text{a card is a }\heartsuit) \:=\:\dfrac{1}{4}
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