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Math Help - Find the probability questions

  1. #1
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    Exclamation Find the probability questions

    Hey I was wondering if anyone could help me out with these two questions:

    Find the probability of:

    1) a five-card poker hand dealt from a standard deck of 52 playing cards results in full house (3 of a kind, 2 of a kind)

    2) Two face cards are drawnin a row (not replaced) from a standard deck of 52 playing cards given that the first card drawn is a king.



    Thanks so much in advance.
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  2. #2
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    Hello, jenneedshelp!

    Find the probability of:

    1) a five-card poker hand dealt from a standard deck of cards
    results in full house (3 of a kind, 2 of a kind)

    There are: . \binom{52}{5}\:=\:2,598,960 possible 5-card hands.

    For the Triple, there are 13 choices for its value.
    . . There are \binom{4}{3} = 4 ways to select the Triple.

    For the Pair, there are 12 choices for its value.
    . . There are \binom{4}{2} = 6 ways to select the Pair.


    Hence, there are: . 13\cdot4\cdot12\cdot6\:=\:3744 possible Full Houses.

    Therefore: . P(\text{Full House})\;=\;\frac{3774}{2,598,960} \;=\;\frac{6}{4165}



    2) Two face cards are drawn in a row (not replaced) from a standard deck of cards,
    given that the first card drawn is a king.

    We don't need Bayes' Theorem or anything fancy for this one . . .

    There are 12 Face Cards in a standard deck of cards.

    If the first card drawn is a King,
    . . there are 11 Face Cards left among the remaiing 51 cards.

    Therefore: . P(\text{2nd is Face}|\text{1st is a King}) \:=\:\frac{11}{51}

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  3. #3
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    [quote]Find the probability of:

    1) a five-card poker hand dealt from a standard deck of 52 playing cards results in full house (3 of a kind, 2 of a kind)

    3 out of 4 suits: C(4,3)
    2 out of 4 suits: C(4,2)
    2 out of 13 different face values(order matters): P(13,2)
    5 cards out of 52: C(52,5)

    Therefore, \frac{P(13,2)C(4,3)C(4,2)}{C(52,5)}=\frac{3744}{25  98960}=\frac{6}{4165}\approx{0.00144}

    2) Two face cards are drawnin a row (not replaced) from a standard deck of 52 playing cards given that the first card drawn is a king.
    If the first is a king, then there are 11 face cards left, therefore,
    11/51
    Thanks so much in advance.
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  4. #4
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    Thanks so much!
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