QA - Poker Probabilities

 

 

 

 

 

 

 

Log............................................................................................................................................................................................................ 3

Poker Probabilities..................................................................................................................................................................... 4

Poker (Encyclopædia Britannica).................................................................................................................................... 4

Poker probability in 52-card deck poker..................................................................................................................... 6

Probability..................................................................................................................................................................................... 6

Q: What is the probability getting Straight Flush, Four of a Kind, House etc. when given five Cards?...................... 6

Conditional probabilities.................................................................................................................................................. 11

Q: What is the probability to get a House or Four of a kind if you have Three of a kind, and choose to change two or one of the remaining cards?......................................................................................................................................................................... 11

Q: What is the probability to get a house if you have Two Pair, and choose to change the remaining card?.......... 12

Q: What is the probability to get a house, four of a kind, three of a kind or Two Pair if you have a Pair, and choose to change 2 or 3 of the remaining Cards?............................................................................................................................................................. 13

Q: What is the probability that the other players get at least one pair, two pair etc. when the 5 cards are dealt?. 17

Q: What is the probability that the other players get at least one pair, two pair etc. when all have changed cards? 17

Poker probability in 32-card deck poker................................................................................................................... 18

Introduction............................................................................................................................................................................... 18

Probabilities................................................................................................................................................................................ 18

Q: What is the probability getting Straight Flush, Four of a Kind, House etc. when given five Cards?.................... 18

References........................................................................................................................................................................................ 20

 

Log

Date	Description
2000-01-27	First version published at http://www.pvv.org/~nsaa/poker.html
2000-10-08	Corrected errors and added description in the following calculations : P(Two pair | one pair AND change 3) = 2718/16215 = 2592/16215 P(Three of a kind | one pair AND change 3) = 1917/16215 = 1854/16215 P(House | one pair AND change 3) = 102/16215  = 165/16215   Thanks to jerome.sheahan@nuigalway.ie for these corrections.
2000-11-08	Added explanaition to P(3 of a kind) and P(House)
2004-08-03	Corrected error in the following conditional calculation P(House | A Pair AND Change 2) = (Combin(2 3;2) + Combin(2;1)*Combin(3;1))/Combin(47;2) = (1 3+6)/1081 =7 9/1081 Thanks to pabloposada@puebla.megared.net.mx for this correction   Changed the decimal separator to decimal sign(,) instead of  decimal point(.) since this is in accordance with ISO (ISO 31-0:1992 (E), Note 17).
2005-08-24	Added links to Wikipedia.org and made some further comments to quoted text from Encyclopædia Britannica.

Poker Probabilities

This document was originally created after some hefty discussion between me and some of my friends. What is the best suit: Flush or Strait? If you play Standard five-card draw and you have a pair and an Ace. Should you keep the pair or should you also keep the Ace? What about playing with 32-card deck? This document does these calculations, and many more. For the moment this document has not dealt with the Texas Hold’em poker variant.

 

I’ve also done some quotations from other sources where I have corrected miscalculations. This was done before the first version of this document was published at 2000-01-17 when good poker calculations were not commonly available. Today setup like the Wikipedia article Poker probability explains and state the standard probabilities very well.  

 

Poker (Encyclopædia Britannica)

I will in this section give some commets from an online article about poker published at   http://www.britannica.com/bcom/eb/article/6/0,5716,62116+1,00.html, but now removed.

 

The text in italic is from this article in Encyclopædia Britannica.

 

 Poker

 

 a family of card games, almost invariably played as gambling games. Although played

 internationally, Poker is most popular in North America.

 

Today Poker is played all over the world and is extremely popular everywhere, mainly because of internet, and the possibility to play online outside state control which’s very restrictive in most countries.

 

 A Poker hand usually consists of five cards. Players try for combinations of two or more cards

 of a kind, five-card sequences, or five cards of the same suit. (See below Rank of Hands.)

 

Today Texas Hold’em is the most popular mainstream variant, mainly because it’s played on the television aired World Series of Poker (abbreviated WSOP), widely recognized as the world championship of the game. Most online casinos also offer variants of Texas Hold’em as the standard game in online poker. Texas Hold’em consists of two cards down and five cards up, i.e. seven cards to use for each player at the final round.

 

 Poker is played with a standard 52-card deck in which all suits are of equal value, the cards

 ranking from the ace high, downward through king, queen, jack, and the numbered cards 10 to

 the deuce. The ace may also be considered low to form a straight (sequence) ace through five

 as well as high with king-queen-jack-10.

 

I’ve played poker with 32-card deck in Germany, and as you can see below the ranking of the hand changes!

 

Rank of hands.

 

 The traditional ranking is (1) straight flush (five cards of the same suit in sequence, the highest

 ace-king-queen-jack-ten being called a royal flush; (2) four of a kind, plus any fifth card; (3)

 full house; (4) flush; (5) straight; (6) three of a kind; (7) two pair; (8) one pair; (9) no pair,

 highest card determining the winner.

 

Below I’ve calculated all the possibilities, not just give the rank. As I stated above, 32-card deck, changes this rank.

 

 To determine the winner in hands in which there are hands of the same rank, the one

 containing the highest card wins. If the high cards are identical, the second highest wins, and so

 on. With full houses, the higher three of a kind wins; with two pairs, the highest pair wins, or if

 the pairs are identical, the odd high card wins, as in the case of identical pairs. For the occasion

 when none of the above applies, e.g., two flushes with identical cards in different suits, house

 rules may apply (the winning hand being the one in the higher bridge suit, or the two hands

 may split the pot). There is no universally accepted code of Poker rules. A code prepared by

 Oswald Jacoby in 1940 and a set of rules in the United States Playing Card Company's Official

 Rules of Card Games, published from 1945, are the ones usually adopted subject to house rules

 in the United States

.

Many people playing poker start arguing what to do when two people have the same flush with identical cards in different suits.  Make a rule before the game have started or split the pot 50/50 as default, if nothing else has been agreed upon.

 

 Wild cards. Most serious poker players decry the use of wild cards, but among the less serious,

 a wild card can be declared by the dealer (the deuce is most popular, but any rank can be

 used, or a distinguishable face card: the one-eyed jack). When there are wild cards in the

 game, the highest hand becomes five of a kind, though some house rules preserve the sanctity

 of the royal flush

 

Be aware that wild card can seriously change the probability and I will really recommend not using Wild cards in any serious game. You should at least be aware of the probabilities and have the correct rank calculated (if it changes it – I’ve not calculated this so I can’t really say). But to site a passage in Wikipedia’s article Wild card (poker)

Another issue with wild cards is that they distort the hand frequencies. In 5-card stud, the stronger hands are less frequent than the weaker hands; i.e., no pair is most common, followed by one pair, two pair, three of a kind, etc. When you add wild cards, the stronger hands gain frequency while the weaker hands lose frequency. For example, if you have a pair and a wild card, you will always choose three of a kind rather than two pair. This causes three of a kind to be more common than two pair.

Poker probability in 52-card deck poker

 

About the rank.

 

Probability

Q: What is the probability getting Straight Flush, Four of a Kind, House etc. when given five Cards?

 

A: Probability getting different hands when dealt five Cards

 

(Rank of Hands) Probability getting this hand:	Out of comb(52;5) = 2598960 ways to draw five cards you will get the following hands these number of times	Exact probability	Approx. Probability 1/	Approx. Probability num.
Royal Flush	4	1/649740	1/649740	0,00000154
Straight Flush	36	9/649740	1/72193	0,0000139
Four of a Kind	624 [5]	1/4165	1/4165	0,000240
House	3744[11]	6/4165	1/694	0,00144
Flush	5108 [8]	1277/649740	1/509	0,00197
Straight	10200	5/1274	1/255	0,00392
Three of a Kind	54912	88/4165	1/47	0,0211
Two Pair	123552	198/4165	1/21	0,0475
One Pair	1098240[11]	352/833	1/2.4	0,423
None	1302540	1302540/2598960	1/2	0,501

 

 

Probability getting at least this hand:	Out of comb(52;5) = 2598960 ways to draw five cards you will get the following hands these number of times	Exact probability	Approx. Probability 1/	Approx. Probability num.
Royal Flush	4		1/649740	0,00000154
Straight Flush	40		1/64974	0,0000154
Four of a Kind	664		1/3914	0,000256
House	4408		1/590 [1]	0,00170
Flush	9516		1/273	0,00366
Straight	19716		1/132	0,00759
Three of a Kind	74628		1/35	0,0287
Two Pair	198180		1/13	0,0763
One Pair	1296420		1/2 [1]	0,499
None	2598960		1	1

 

1 in 5 for at least a pair of Jacks (11) ^[1]

 

Why:

 

P(House) = 13*12*COMBIN(4;3)*COMBIN(4;2) = 3744

 

 Why?

 13*12 because: 13 Columns horizontal and 12 rows vertical

   1-1-1-2-2        2-2-2-1-1        ...   13-13-13-1-1

   1-1-1-3-3        2-2-2-3-3              13-13-13-2-2

   1-1-1-4-4        2-2-2-4-4              13-13-13-3-3

   ...

   1-1-1-13-13-13   2-2-2-13-13-13         13-13-13-12-12

 

Lets look at one of these 1-1-1-2-2

 You have Combin(4;3) to draw 1-1-1 and Combin(4;2) to draw 2-2

 

 

P(3 of a kind) = 13*COMBIN(48;2)*(COMBIN(4;3) - P(House)

               = 13*1128*4 - 3744

               = 54912

 

 Why?

 13*COMBIN(48;2) 13 Columns horizontal and Combin(48;2) rows vertical as  above

   1-1-1-2-2        2-2-2-1-1        ...   13-13-13-1-1

   1-1-1-2-3        2-2-2-1-3              13-13-13-1-2

   1-1-1-2-4

   1-1-1-2-5

   ...

   1-1-1-3-3        2-2-2-3-3              13-13-13-2-2

   1-1-1-3-4        2-2-2-3-4              13-13-13-2-3

   1-1-1-3-5

   ...

   1-1-1-4-4        2-2-2-4-4              13-13-13-3-3

   ...

   1-1-1-13-13-13   2-2-2-13-13-13         13-13-13-12-12

 

 Lets look at 1-1-1-2-3. You have Combin(4;3) ways to draw 1-1-1.

 Since we now have Counted all the 1-1-1-2-2,1-1-1-3-3 etc. we must

 substract the house calculation.

 

-Another way to look at it is like this

 You have 13 Columns.

 Lets look at 1-1-1-y-x. You have Combin(4;3) ways to draw 1-1-1 and you

 have Combin(48;2) to draw y-x

 Since we now have Counted all the 1-1-1-2-2,1-1-1-3-3 etc. we must

 substract the house calculation.

 

From [14] : http://www.thewizardofodds.com/game/pokerodd.html

Royal Flush

The number of different royal flushes are four (one for each suit).

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.

Flush

There are 4 suits to choose from and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From 1,287 subtract 10 for the ten high cards that can lead a straight, resulting in a straight flush, leaving 1,277. Then multiply for 4 for the four suits, resulting in 5,108 ways to form a flush.

Straight

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. The number of ways to arrange five cards of four different suits is 4⁵ = 1024. Next subtract 4 from 1024 for the four ways to form a flush, resulting in a straight flush, leaving 1020. The total number of ways to form a straight is 10*1020=10,200.

Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 4² = 54,912.

Two Pair

There are (13:2) = 78 ways to arrange the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 6² * 44 = 123,552 ways to arrange a two pair.

One Pair

There are 13 ranks to choose from for the pair and combin(4,2) = 6 ways to arrange the two cards in the pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 4³ = 1,098,240 ways to arrange a pair.

Nothing

First find the number of ways to choose five different ranks out of 13 which is combin(13,5) = 1287. Then subtract 10 for the 10 different high cards that can lead a straight, to be left with 1277. Each card can be of 1 of 4 suits so there are 4⁵=1024 different ways to arrange the suits in each of the 1277 combinations. However we must subtract 4 from the 1024 for the four ways to form a flush, leaving 1020. So the final number of ways to arrange a high card hand is 1277*1020=1,302,540.

 

Specific High Card Lets find the probability of drawing a jack high, for example. There must be four different cards in the hand all less than a jack, of which there are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the 9-8-7-6-5 combination which would form a straight, leaving 125. From above we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of ways to form a jack high hand. For ace high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A are both valid straights.

 

 

 

From [11] http://www.sscnet.ucla.edu:80/soc/faculty/campbell/210a_Fall1997/210a_notes_10_14_97.htm

 

The first thing we need to know is how many elementary events there are that can occur. We just calculated it, it's 52!/(47!*5!)=2598960. Now all we have to do is work out how many hands correspond to each of the above three situations, and divide by this number.

 

PAIR:

To dealing with the probability of the pairs first, the first thing is to work out how many possible pairs there are. Well for any given value, there are (4 2) pairs that can be drawn, and there are 13 possible values, so there are 13 (4 2) ways of having a pair. How many combinations of the remaining 12 values are there that do not result in a pair among the remaining three cards? (12 3) Thus given 12 remaining values, there are (12 3) of picking three distinct ones from them, for example, 2 3 4, 2 3 5, 2 3 6, .... K Q A. Of course, for each of the three cards any suit is OK, we can have any combination of the 4 suits, so we have to multiply by 4^3. Thus the probability of having one pair, and three distinct remaining cards, is 13 (4 2) (12 3) 4^3 / (52 5). If we work it out, it's about 0.40.

 

FULL HOUSE:

A similar approach can be taken for the full house. There are 13 (4 3) 12 (4 2) ways of having a full house, so the total probability

of