Sanderson M. Smith
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POKER 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 is
Question 1132416: A hand consists of 5 cards from a well-shuffled deck of 52 cards. Find the total number of possible 5-card poker hands. A heart flush is a 5-card hand consisting of all heart cards. Find the number of possible heart flushes. Find the probability of being dealt a heart flush. (Round to six decimal places as needed.). There are more AK hands in a range of AA, KK, AK than there are AA and KK hands combined. So out of 28 possible combinations made up from AA, KK and AK, 16 of them come from AK. This means that when our opponent 3bets, the majority of the time he is holding AK and not a big pocket pair. Any Hand - Total number of 3-card poker hands drawn from a 52 card standard deck. 52 C 3 = 22100 Straight Flush - There are 12 differently ranked straight flushes from A-2-3 up to Q-K-A in each of the 4 different suits.
52C5 = 2,598,960A 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.
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 1Hands Of Poker In Order
List Of Best Poker Hands
| A K Q J 10 | K Q J 10 9 | Q J 10 9 8 | J 10 9 8 7 | 10 9 8 7 6 | 9 8 7 65 | 8 7 6 54 | 7 6 5 4 3 | 6 5 4 3 2 | 5 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.
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 1Hands Of Poker In Order
List Of Best Poker Hands
| A K Q J 10 | K Q J 10 9 | Q J 10 9 8 | J 10 9 8 7 | 10 9 8 7 6 | 9 8 7 65 | 8 7 6 54 | 7 6 5 4 3 | 6 5 4 3 2 | 5 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 2Number Of Possible Poker Hands Held
| HAND | N = NUMBER OF WAYS listed by McGervey | Computations and comments | Probability of HANDN/(2,598,960) and approximate odds. |
Straight flush | 40 | There are four suits (spades, hearts, diamond, clubs). Using TABLE 1,4(10) = 40. | 0.0000151 in 64,974 |
Seven feathers casino resort rv park. Four of a kind | 624 | (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 house | 3,744 | (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 |
Flush | 5,108 | (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 |
Straight | 10,200 | (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 kind | 54,912 | (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. | 0.0006091 in 1,641 |
How Many Possible Poker Hands
'I must complain the cards are ill shuffled 'til Ihave a good hand.'
-Swift, Thoughts on Various Subjects
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