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Math Help - card problem

  1. #1
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    card problem

    Cards are dealt at random and without replacement from a standard 52 ard deck. What is the probability that the second king is dealt on the fifth card?

    The answer in the back of the book is 0.016.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Cards are dealt at random and without replacement from a standard 52 ard deck. What is the probability that the second king is dealt on the fifth card?

    The answer in the back of the book is 0.016.
    Let K denote a king and O any other card.

    The answer is the sum of the probabilities that the first 5 cards are KOOOK,
    OKOOK, OOKOK and OOOKK.

    RonL
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  3. #3
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    Hello, lllll!

    Cards are dealt at random and without replacement from a standard 52 card deck.
    What is the probability that the second king is dealt on the fifth card?

    The answer in the back of the book is 0.016.
    The deck contains 4 Kings and 48 Others.

    Among the first four cards, there must be one King and three Others.

    If the the order was KOOO, the probability is: . \frac{4}{52}\cdot\frac{48}{51}\cdot\frac{47}{50}\c  dot\frac{46}{49} \;=\;\frac{17,296}{270,725}

    Since the first King can be in any of four positions,

    . . the probability is: . 4 \times \frac{17,296}{270,725} \:=\:\frac{69,184}{270,725}


    The fifth card must be one of the three remaining Kings: . \frac{3}{48}


    Therefore, the probability is: . \frac{69,184}{270,725} \times \frac{3}{48} \;=\;\frac{4324}{270,725} \;=\;0.015971927 \;\approx\;\boxed{0.016}

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