How Many Different Poker Hands Are There
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*How Many Different Poker Hands Are There Bad
*Combinations In Poker
*How Many Different 5-card Poker Hands Are There That Contain No AcesSanderson M. Smith
How many different poker hands are there? (You may give your answer symbolically, you don’t have to write out this large number.) 2-9. A standard deck of playing cards, the cards have 13 different ranks and 4 different suits, each card having one rank and one suit. There are C(4,4) combinations of 4 queens and C(48,1) combinations for the remaining card. So: 1x48 = 48 different 5 card hands containing 4 queens. Remember, the order of the 4 queens is.Home | About Sanderson Smith | Writings and Reflections | Algebra 2 | AP Statistics | Statistics/Finance | ForumPOKER PROBABILITIES (FIVECARD HANDS)
In many forms of poker, one is dealt 5 cards from astandard deck of 52 cards. The number of different 5 -card pokerhands is52C5 = 2,598,960
A wonderful exercise involves having students verify probabilitiesthat appear in books relating to gambling. For instance, inProbabilities in Everyday Life, by John D. McGervey, one findsmany interesting tables containing probabilities for poker and othergames of chance.
This article and the tables below assume the reader is familiarwith the names for various poker hands. In the NUMBER OF WAYS columnof TABLE 2 are the numbers as they appear on page 132 in McGervey’sbook. I have done computations to verify McGervey’s figures. Thiscould be an excellent exercise for students who are studyingprobability.How Many Different Poker Hands Are There Bad
There are 13 denominations (A,K,Q,J,10,9,8,7,6,5,4,3,2) in thedeck. One can think of J as 11, Q as 12, and K as 13. Since an acecan be ’high’ or ’low’, it can be thought of as 14 or 1. With this inmind, there are 10 five-card sequences of consecutive dominations.These are displayed in TABLE 1.TABLE 1A K Q J 10K Q J 10 9Q J 10 9 8J 10 9 8 710 9 8 7 69 8 7 658 7 6 547 6 5 4 36 5 4 3 25 4 3 2 A
The following table displays computations to verify McGervey’snumbers. There are, of course , many other possible poker handcombinations. Those in the table are specifically listed inMcGervey’s book. The computations I have indicated in the table doyield values that are in agreement with those that appear in thebook.TABLE 2HAND
N = NUMBER OF WAYS listed by McGerveyComputations and commentsProbability of HANDN/(2,598,960) and approximate odds.
Straight flush40
There are four suits (spades, hearts, diamond, clubs). Using TABLE 1,4(10) = 40.0.0000151 in 64,974
Four of a kind624
(13C1)(48C1) = 624.
Choose 1 of 13 denominations to get four cards and combine with 1 card from the remaining 48.0.000241 in 4,165
Full house3,744
(13C1)(4C3)(12C1)(4C2) = 3,744.
Choose 1 denominaiton, pick 3 of 4 from it, choose a second denomination, pick 2 of 4 from it.0.001441 in 694
Flush5,108
(4C1)(13C5) = 5,148.
Choose 1 suit, then choose 5 of the 13 cards in the suit. This figure includes all flushes. McGervey’s figure does not include straight flushes (listed above). Note that 5,148 - 40 = 5,108.0.0019651 in 509
Straight10,200
(4C1)5(10) = 45(10) = 10,240
Using TABLE 1, there are 10 possible sequences. Each denomination card can be 1 of 4 in the denomination. This figure includes all straights. McGervey’s figure does not include straight flushes (listed above). Note that 10,240 - 40 = 10,200.0.003921 in 255
Three of a kind54,912
(13C1)(4C3)(48C2) = 58,656.
Choose 1 of 13 denominations, pick 3 of the four cards from it, then combine with 2 of the remaining 48 cards. This figure includes all full houses. McGervey’s figure does not include full houses (listed above). Note that 54,912 - 3,744 = 54,912.0.02111 in 47
Exactly one pair, with the pair being aces.84,480
(4C2)(48C1)(44C1)(40C1)/3! = 84,480.
Choose 2 of the four aces, pick 1 card from remaining 48 (and remove from consider other cards in that denomination), choose 1 card from remaining 44 (and remove other cards from that denomination), then chose 1 card from the remaining 40. The division by 3! = 6 is necessary to remove duplication in the choice of the last 3 cards. For instance, the process would allow for KQJ, but also KJQ, QKJ, QJK, JQK, and JKQ. These are the same sets of three cards, just chosen in a different order.0.03251 in 31
Two pairs, with the pairs being 3’s and 2’s.1,584
McGervey’s figure excludes a full house with 3’s and 2’s.
(4C2)(4C1)(44C1) = 1,584.
Choose 2 of the 4 threes, 2 of the 4 twos, and one card from the 44 cards that are not 2’s or 3’s.
Poker site sign up offers. About Poker Sites Bonuses. Not all poker bonuses are created equal. Some sites may offer a lower welcome or poker sign up bonus, but compensate by providing monthly reload or loyalty bonuses (or even both). Choosing a poker site with the best sign up bonus can be quite a difficult task without the right information at hand.0.0006091 in 1,641
’I must complain the cards are ill shuffled ’til Ihave a good hand.’-Swift, Thoughts on Various Subjects
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Copyright © 2003-2009 Sanderson Smith
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*How Many Different Poker Hands Are There Bad
*Combinations In Poker
*How Many Different 5-card Poker Hands Are There That Contain No AcesSanderson M. Smith
How many different poker hands are there? (You may give your answer symbolically, you don’t have to write out this large number.) 2-9. A standard deck of playing cards, the cards have 13 different ranks and 4 different suits, each card having one rank and one suit. There are C(4,4) combinations of 4 queens and C(48,1) combinations for the remaining card. So: 1x48 = 48 different 5 card hands containing 4 queens. Remember, the order of the 4 queens is.Home | About Sanderson Smith | Writings and Reflections | Algebra 2 | AP Statistics | Statistics/Finance | ForumPOKER PROBABILITIES (FIVECARD HANDS)
In many forms of poker, one is dealt 5 cards from astandard deck of 52 cards. The number of different 5 -card pokerhands is52C5 = 2,598,960
A wonderful exercise involves having students verify probabilitiesthat appear in books relating to gambling. For instance, inProbabilities in Everyday Life, by John D. McGervey, one findsmany interesting tables containing probabilities for poker and othergames of chance.
This article and the tables below assume the reader is familiarwith the names for various poker hands. In the NUMBER OF WAYS columnof TABLE 2 are the numbers as they appear on page 132 in McGervey’sbook. I have done computations to verify McGervey’s figures. Thiscould be an excellent exercise for students who are studyingprobability.How Many Different Poker Hands Are There Bad
There are 13 denominations (A,K,Q,J,10,9,8,7,6,5,4,3,2) in thedeck. One can think of J as 11, Q as 12, and K as 13. Since an acecan be ’high’ or ’low’, it can be thought of as 14 or 1. With this inmind, there are 10 five-card sequences of consecutive dominations.These are displayed in TABLE 1.TABLE 1A K Q J 10K Q J 10 9Q J 10 9 8J 10 9 8 710 9 8 7 69 8 7 658 7 6 547 6 5 4 36 5 4 3 25 4 3 2 A
The following table displays computations to verify McGervey’snumbers. There are, of course , many other possible poker handcombinations. Those in the table are specifically listed inMcGervey’s book. The computations I have indicated in the table doyield values that are in agreement with those that appear in thebook.TABLE 2HAND
N = NUMBER OF WAYS listed by McGerveyComputations and commentsProbability of HANDN/(2,598,960) and approximate odds.
Straight flush40
There are four suits (spades, hearts, diamond, clubs). Using TABLE 1,4(10) = 40.0.0000151 in 64,974
Four of a kind624
(13C1)(48C1) = 624.
Choose 1 of 13 denominations to get four cards and combine with 1 card from the remaining 48.0.000241 in 4,165
Full house3,744
(13C1)(4C3)(12C1)(4C2) = 3,744.
Choose 1 denominaiton, pick 3 of 4 from it, choose a second denomination, pick 2 of 4 from it.0.001441 in 694
Flush5,108
(4C1)(13C5) = 5,148.
Choose 1 suit, then choose 5 of the 13 cards in the suit. This figure includes all flushes. McGervey’s figure does not include straight flushes (listed above). Note that 5,148 - 40 = 5,108.0.0019651 in 509
Straight10,200
(4C1)5(10) = 45(10) = 10,240
Using TABLE 1, there are 10 possible sequences. Each denomination card can be 1 of 4 in the denomination. This figure includes all straights. McGervey’s figure does not include straight flushes (listed above). Note that 10,240 - 40 = 10,200.0.003921 in 255
Three of a kind54,912
(13C1)(4C3)(48C2) = 58,656.
Choose 1 of 13 denominations, pick 3 of the four cards from it, then combine with 2 of the remaining 48 cards. This figure includes all full houses. McGervey’s figure does not include full houses (listed above). Note that 54,912 - 3,744 = 54,912.0.02111 in 47
Exactly one pair, with the pair being aces.84,480
(4C2)(48C1)(44C1)(40C1)/3! = 84,480.
Choose 2 of the four aces, pick 1 card from remaining 48 (and remove from consider other cards in that denomination), choose 1 card from remaining 44 (and remove other cards from that denomination), then chose 1 card from the remaining 40. The division by 3! = 6 is necessary to remove duplication in the choice of the last 3 cards. For instance, the process would allow for KQJ, but also KJQ, QKJ, QJK, JQK, and JKQ. These are the same sets of three cards, just chosen in a different order.0.03251 in 31
Two pairs, with the pairs being 3’s and 2’s.1,584
McGervey’s figure excludes a full house with 3’s and 2’s.
(4C2)(4C1)(44C1) = 1,584.
Choose 2 of the 4 threes, 2 of the 4 twos, and one card from the 44 cards that are not 2’s or 3’s.
Poker site sign up offers. About Poker Sites Bonuses. Not all poker bonuses are created equal. Some sites may offer a lower welcome or poker sign up bonus, but compensate by providing monthly reload or loyalty bonuses (or even both). Choosing a poker site with the best sign up bonus can be quite a difficult task without the right information at hand.0.0006091 in 1,641
’I must complain the cards are ill shuffled ’til Ihave a good hand.’-Swift, Thoughts on Various Subjects
Home | About Sanderson Smith | Writings and Reflections | Algebra 2 | AP Statistics | Statistics/Finance | ForumCombinations In Poker
Previous Page | Print This PageHow Many Different 5-card Poker Hands Are There That Contain No Aces
Copyright © 2003-2009 Sanderson Smith
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