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# 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

- MathmomLv 73 years agoFavorite 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:

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- SqdancefanLv 73 years ago
The web site at the source link shows the computation.

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- nbsaleLv 63 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.

- SqdancefanLv 73 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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