Math Help! Find the probability (as a percentage) of drawing any of the following poker hands in 5-card stud poker:?

Based on the ideas presented there, try to find the probability (as a percentage) of drawing any of the following poker hands in 5-card stud poker:

1) a three-of-a-kind

2) a four-of-a-kind

3 Answers

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  • 3 years ago
    Favorite Answer

     

    Total number of 5-card hands = C(52,5) = 2,598,960

    1) a three-of-a-kind

    13 ways to choose rank (2, 3, .., Aces) that shows up 3 times in hand

    For this rank, there are C(4,3) = 4 ways to choose 3 cards

    C(12,2) = 66 ways to choose 2 remaining (and different) ranks

    For each of these 2 ranks there are 4 ways to choose a card

    Total number of three-of-a-kind hands = 13 * 4 * 66 * 4^2 = 54,912

    Prob = 54,912/2,598,960 = 0.021128 = 2.1128%

    Check:

    http://www.wolframalpha.com/input/?i=poker+three+o...

    2) a four-of-a-kind

    13 ways to choose ranks (2, 3, .., Aces) that shows up 4 times in hand

    For this ranks, there is 1 way to choose all 4 cards

    48 cards remain. Last card must be one of these

    Total number of four-of-a-kind hands = 13 * 1 * 48 = 624

    Prob = 624/2,598,960 = 0.000240096 = 0.0240096%

    Check:

    http://www.wolframalpha.com/input/?i=poker+four+of...

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  • 3 years ago

    The web site at the source link shows the computation.

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  • nbsale
    Lv 6
    3 years ago

    These are all done the same way:

    1. Find how many ways there are to get the specified cards.

    2. Find how many ways there are to fill out the hand with other cards.

    Multiply those together to get the number of hands described.

    3. Find how may ways there are to get any 5 cards.

    Divide that into the prior result to get the probability.

    For 3 of a kind:

    How many ranks of cards are there to chose from? 13, obviously.

    How many ways are there to get 3 of the 4 cards of that rank? 4C3

    How many ways to fill out the hand?

    Once the rank is selected, there are 48 other cards. there are 48 choices for the first, then 44 for the second (it can't be the same card or else you'll get a full house, not just 3 of a kind).

    How many hands are there? 52C5.

    Now just work out the math.

    4 of a kind is similar.

    • Sqdancefan
      Lv 7
      3 years agoReport

      For 3 of a kind, 52*48*44 counts each permutation of the fill cards. The correct number is 52*48*44/2.

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