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Math Help - In a card game of euchre

  1. #1
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    In a card game of euchre

    there are 24 cards and you get 5 cards what is the probability of having more than 3 black cards
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  2. #2
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    Hello, nafiro!

    I will assume that there are 12 black cards and 12 red cards.


    There are 24 cards and you get 5 cards.
    What is the probability of having more than 3 black cards
    There are: . {24\choose5} \,=\,42,\!504 possible 5-card hands.


    "More than 3 black cards" means "4 blacks cards or 5 black cards".

    . . To get 4 black cards and 1 red, there are: . {12\choose4}{12\choose1} \,=\,495\cdot12 \,=\,5940 ways.
    . . To get 5 black cards, there are: . {12\choose5} \,=\,792 ways.

    Hence, there are: . 5940 + 792 \:=\:6732 ways to get more than 3 black cards.


    Therefore: . P(\text{more than 3 black cards}) \;=\;\frac{6732}{42,\!504} \;=\;\frac{51}{322}

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