During play—that is, from the flop and onwards—drawing probabilities come down to a question of outs. All situations which have the same number of outs have the same probability of winning. For example, an inside straight draw (e.g. 34 67 missing the 5 for a straight), and a full house draw (e.g. 66KK drawing for one of the pairs to become three-of-a-kind) are equivalent. Each can be satisfied by four cards—four 5s in the first case, and the other two 6s and other two kings in the second.
The probabilities of drawing these outs are easily calculated. At the flop there remain 47 unseen cards, so the probability is (outs ÷ 47). At the turn there are 46 unseen cards so the probability is (outs ÷ 46). The cumulative probability of making a hand on either the turn or river can be determined as the complement of the odds of not making the hand on the turn and not on the river. The probability of not drawing an out is (47 − outs) ÷ 47 on the turn and (46 − outs) ÷ 46 on the river; taking the complement of these conditional probabilities gives the probability of drawing the out by the river which is calculated by the formula
P = 1 – (((47 – outs)/47) x ((46-outs)/46))
For reference, the probability and odds for some of the more common numbers of outs are given here.
Likely drawing to
Outs
Make on turn
Make on river
Make on turn or river
Prob.
Odds
Prob.
Odds
Prob.
Odds
Inside straight flush; Four of a kind
1
0.0213
46.0 : 1
0.0217
45.0 : 1
0.0426
22.5 : 1
Open-ended straight flush; Three of a kind
2
0.0426
22.5 : 1
0.0435
22.0 : 1
0.0842
10.9 : 1
High pair
3
0.0638
14.7 : 1
0.0652
14.3 : 1
0.1249
7.01 : 1
Inside straight; Full house
4
0.0851
10.8 : 1
0.0870
10.5 : 1
0.1647
5.07 : 1
Three of a kind or two pair
5
0.1064
8.40 : 1
0.1087
8.20 : 1
0.2035
3.91 : 1
Either pair
6
0.1277
6.83 : 1
0.1304
6.67 : 1
0.2414
3.14 : 1
Full house or four of a kind; (see note)
Inside straight or high pair
7
0.1489
5.71 : 1
0.1522
5.57 : 1
0.2784
2.59 : 1
Open-ended straight
8
0.1702
4.88 : 1
0.1739
4.75 : 1
0.3145
2.18 : 1
Flush
9
0.1915
4.22 : 1
0.1957
4.11 : 1
0.3497
1.86 : 1
Inside straight or pair
10
0.2128
3.70 : 1
0.2174
3.60 : 1
0.3839
1.60 : 1
Open-ended straight or high pair
11
0.2340
3.27 : 1
0.2391
3.18 : 1
0.4172
1.40 : 1
Inside straight or flush; Flush or high pair
12
0.2553
2.92 : 1
0.2609
2.83 : 1
0.4496
1.22 : 1
13
0.2766
2.62 : 1
0.2826
2.54 : 1
0.4810
1.08 : 1
Open-ended straight or pair
14
0.2979
2.36 : 1
0.3043
2.29 : 1
0.5116
0.955 : 1
Open-ended straight or flush; Flush or pair;
Inside straight, flush or top pair
15
0.3191
2.13 : 1
0.3261
2.07 : 1
0.5412
0.848 : 1
16
0.3404
1.94 : 1
0.3478
1.88 : 1
0.5698
0.755 : 1
17
0.3617
1.76 : 1
0.3696
1.71 : 1
0.5976
0.673 : 1
Inside straight or flush or pair;
Open-ended straight, flush or high pair
18
0.3830
1.61 : 1
0.3913
1.56 : 1
0.6244
0.601 : 1
19
0.4043
1.47 : 1
0.4130
1.42 : 1
0.6503
0.538 : 1
20
0.4255
1.35 : 1
0.4348
1.30 : 1
0.6753
0.481 : 1
Open-ended straight, flush or pair
21
0.4468
1.24 : 1
0.4565
1.19 : 1
0.6994
0.430 : 1
Note: When drawing to a full house or four of a kind with a pocket pair that has hit trips (three of a kind) on the flop, there are 6 outs to get a full house by pairing the board and one out to make four of a kind. This means that if the turn does not pair the board or make four of a kind, there will be 3 additional outs on the river, for a total of 10, to pair the turn card and make a full house. This makes the probability of drawing to a full house or four of a kind on the turn or river 0.334 and the odds are 1.99 : 1. This makes drawing to a full house or four of a kind by the river about 8½ outs.
It is worth noting in the preceding table that if a player doesn’t fold before the river, a hand with at least 14 outs after the flop has a better than 50% chance to catch one of its outs by the river. With 20 or more outs, a hand is a better than 2 : 1 favorite to catch at least one out by the river.
The game begins with each player being dealt two cards face down and one card face up. If played with a bring-in, the player with the lowest-ranking upcard pays the bring-in, and betting proceeds after that in normal clockwise order. The bring-in is considered an open, so the next player in turn may not check. If two players have equally ranked low cards, suit may be used to break the tie and assign the bring-in. If there is no bring-in, then the first betting round begins with the player showing the highest-ranking upcard, who may check. In this case, suit should not be used to break ties. If two players have the same high upcard, the one first in clockwise rotation from the dealer acts first.
After the first betting round, another upcard is dealt to each player (after a burn card, and starting at the dealer’s left as will all subsequent rounds), followed by a second betting round beginning with the player whose upcards make the best poker hand. Since fewer than five cards are face up, this means no straights, flushes, or full houses will count for this purpose. On this and all subsequent betting rounds, the player whose face-up cards make the best poker hand will act first, and may check or bet up to the game’s limit.
The second round is followed by a third upcard and betting round, a fourth upcard and betting round, and finally a downcard, a fifth betting round, and showdown if necessary. Seven-card stud can be summarized therefore as “two down, four up, one down”. Upon showdown, each player makes the best five-card poker hand he can out of the seven cards he was dealt.
You may note that seven cards to eight players plus four burn cards makes 60 cards, and there are only 52 in the deck. In most games this is not a problem because several players will have folded in early betting rounds. But there are certainly low-stakes home games where few if any players fold. If this is the case in your game, you may want to limit the game to seven players. If the deck does become exhausted during play, previously-dealt burn cards can be used when only a few cards are needed to complete the deal. If even those are not sufficient, then on the final round instead of dealing a downcard to each player, a single community card is dealt to the center of the table, and is shared by everyone (that is, each player treats it as his seventh card). Under no circumstances can any discarded card from a folded hand be “recycled” for later use. Unlike draw poker, where no cards are ever seen before showdown, stud poker players use the information they get from face-up cards to make strategic decisions, and so a player who sees a certain card folded is entitled to make decisions knowing that the card will never appear in another opponent’s hand.
The face cards, Jacks, Queens, and Kings, in a deck. In Texas hold’em, a flop can be said to be “all paint” if it consists of only these cards.
passive
A style of play characterized by checking and calling.
pat
Already complete. A hand is a pat hand when, say, a straight comes on the first five cards in Texas hold’em.
pick-up
When the house picks up cash from the dealer after a player buys chips.
pigeon
A bad player. Also “donkey”, “fish”.
play twice
See “deal twice”.
pocket pair
In community card poker or stud poker, when two of a player’s private cards make a pair. Also “wired pair”.
poker face
A blank face that does not reveal anything about the cards being held. Often used metaphorically outside the world of poker.
position bet
A bet that is made more due to the strength of the bettor’s position than the strength of the bettor’s cards.
post
To make the required small or big blind bet in Texas Hold ‘em or other games played with blinds rather than antes
post dead
To post a bet amount equal to the small and the big blind combined (the amount of the large blind playing as a live blind, and the amount of the small blind as dead money). In games played with blinds, a player who steps away from the table and misses his turn for the blinds must either post dead or wait for the big blind to re-enter the game. Compare to “dead blind”.
probe bet
A bet after the flop by a player who did not take the lead in betting before the flop (and when the player that did take the lead in betting before the flop declined to act). Compare to “continuation bet”.
proposition player, prop
A player that gets paid an hourly rate to start poker games or to help them stay active. Prop players play with their own money, which distinguishes them from shills, who play with the casino’s money.
It is useful to look at the chances different starting hands have of either improving on the flop, or of weakening on the flop. One interesting circumstance concerns pocket pairs. When holding a pocket pair, overcards (cards of higher rank than the pair) weaken the hand because of the potential that an overcard has paired a card in an opponent’s hand. The hand gets worse the more overcards there are on the board and the more opponents that are in the hand because the probability that one of the overcards has paired a hole card increases. To calculate the probability of no overcard, take the total number of outcomes without an overcard divided by the total number of outcomes.
Where x is the rank 3–K of the pocket pair (assigning values from 3–10 and J–K = 11–13), then the number of overcards is (14 – x) x 4 and the number of cards of rank x of less is 0 – (14-x) x 4 = 4x – 6. The number of outcomes without an overcard is the number of combinations that can be formed with the remaining cards, so the probability P of an overcard on the flop is
P = ((4x-6)/3)/(50/3),
and on the turn and river are
P = ((4x-6)/4)/(50/4) and P = ((4x-6)/5)/(50/5), respectively.
The following table gives the probability that no overcards will come on the flop, turn and river, for each of the pocket pairs from to K.
Holding pocket pair
Overcard on flop
Overcard by turn
Overcard by river
Prob.
Odds
Prob.
Odds
Prob.
Odds
KK
0.7745
0.29 : 1
0.7086
0.41 : 1
0.6470
0.55 : 1
QQ
0.5857
0.71 : 1
0.4860
1.06 : 1
0.4015
1.49 : 1
JJ
0.4304
1.32 : 1
0.3205
2.12 : 1
0.2369
3.22 : 1
TT
0.3053
2.28 : 1
0.2014
3.97 : 1
0.1313
6.61 : 1
99
0.2071
3.83 : 1
0.1190
7.40 : 1
0.0673
13.87 : 1
88
0.1327
6.54 : 1
0.0649
14.40 : 1
0.0310
31.21 : 1
77
0.0786
11.73 : 1
0.0318
30.48 : 1
0.0124
79.46 : 1
66
0.0416
23.02 : 1
0.0133
74.26 : 1
0.0040
246.29 : 1
55
0.0186
52.85 : 1
0.0043
229.07 : 1
0.0009
1057.32 : 1
44
0.0061
162.33 : 1
0.0009
1095.67 : 1
0.0001
8406.78 : 1
33
0.0010
979.00 : 1
0.0001
15352.33 : 1
0.0000
353125.67 : 1
Notice that there is a better than 35% probability that an ace will come by the river if holding pocket kings, and with pocket queens, the odds are slightly in favor of an ace or a king coming by the turn, and a full 60% in favor of an overcard to the queen by the river. With pocket jacks, there’s only a 43% chance that an overcard won’t come on the flop and it’s better than 3:1 that an overcard will come by the river.
Seven-card stud is a variant of stud poker. Until the recent increase in popularity of Texas hold ‘em, seven-card stud was the most popular poker variant in home games across the United States, and in casinos in the eastern part of the country. Seven-card stud is also played in western American casinos, but Texas hold ‘em is far more popular there. Two to eight players can play.
The descriptions below assume that you are familiar with the general game play of poker, and with hand values. They also make no assumptions about what poker betting structure is used. In casino play, it is common to use a small ante and bring-in. In home games, it is typical to use an ante only.
Quick play overview
Play proceeds as follows (“player” refers only to those who have not folded and are still in the game), with betting rounds in-between.
Betting is clockwise, the player with the highest poker hand showing starts (ie 2-2 beats K-Q).
2 cards dealt face down to each player, 1 card dealt face up
Cards that are not of the same suit. The ace of clubs and the king of spades are called ace-king offsuit
open
To bet first.
open ended, open ended straight draw
An outside straight draw. Also “two-way straight draw”.
openers
The cards held by a player in a game of “jackpots” entitling him to open the pot. “Splitting openers” refers to holding onto one of your openers after discarding it to prove you had the necessary cards to open should you win the pot.
option
An optional bet or draw, such as getting an extra card facedown for 50 cents or raising on the big blind when checked all the way around.
The right to raise possessed by the big blind if there have been no raises.
out of pocket
A game which gives the players the ability to add more money to their stack in the middle of a hand.
outside straight, outside straight draw
“Two-way straight draw”.
overcall
To call a bet after others have called, esp. big bets. Jim bet, Alice called, then Ted overcalled.
overcard
A community card with a higher rank than a player’s pocket pair.
A higher card. Ted held two overcards to Jill’s pair with two cards to come.
overpair
In community card games such as Texas Hold’em and Omaha, a pocket pair with a higher rank than any community card.
over the top
To reraise. Ted raised $20, then Alice came over the top for $60 more.
The value of a starting hand can change dramatically after the flop. Regardless of initial strength, any hand can flop the nuts—for example, if the flop comes with three 2s, any hand holding the fourth 2 has the nuts. Conversely, the flop can undermine the perceived strength of any hand—A♣A♥ would not be happy to see
♠ 9♠ 10♠ on the flop because of the straight and flush possibilities.
There are
(50/3) = 19,600
possible flops for any given starting hand. By the turn the total number of combinations has increased to
(50/4) = 230,300
and on the river there are
(50/5) = 2,118,760
possible boards to go with the hand.
The following are some general probabilities about what can occur on the board. These assume a “random” starting hand for the player.
Board consisting of
Making on flop
Making by turn
Making by river
Prob.
Odds
Prob.
Odds
Prob.
Odds
Three or more of same suit
0.05177
18.3 : 1
0.13522
6.40 : 1
0.23589
3.24 : 1
Four or more of same suit
0.01056
93.7 : 1
0.03394
28.5 : 1
Rainbow flop (all different suits)
0.39765
1.51 : 1
0.10550
8.48 : 1
Three cards of consecutive rank (but not four consecutive)
0.03475
27.8 : 1
0.11820
7.46 : 1
0.25068
2.99 : 1
Four cards to a straight (but not five)
0.03877
24.8 : 1
0.18991
4.27 : 1
Three or more cards of consecutive rank and same suit
0.00217
459 : 1
0.00869
114 : 1
0.02172
45.0 : 1
Three of a kind (but not a full house or four of a kind)
0.00235
424 : 1
0.00935
106 : 1
0.02128
46 : 1
A pair (but not two pair or three or four of a kind)
0.16941
4.90 : 1
0.30417
2.29 : 1
0.42450
1.36 : 1
Two pair (but not a full house)
0.01037
95.4 : 1
0.04716
20.2 : 1
An interesting fact to note from the table above is that more than 60% of the flops will have at least two of the same suit—you’re likely to either be drawing to a flush or worried about one.
Roll your own is poker jargon used for a particular ruleset in certain poker variants, particularly in stud poker.In traditional stud poker games, cards are simply dealt to each player, either face up or face down according to the rules of the game being played, followed by betting. In roll your own games this is different in one of three possible ways. These are called:
simultaneous choose-after
in turn choose-after
choose-before
In simultaneous choose-after in every round where an upcard is normally dealt, each player is instead dealt a downcard. All players then look at all of their downcards and choose one to turn face up, then all players turn their chosen card at once.
In in turn choose-after the game begins the same way with each player being dealt a downcard, but then the first player to act (determined by the rules of the particular game) turns over his choice of upcard, then the next player can use that information to decide which of his cards to turn up, and then all players follow in turn.
Choose-before is always played in turn. On upcard rounds, before a card is dealt to each player, that player must choose whether he wants to receive it up or down. If he wants to receive it up, he says so. If he wants to receive it down, he must first turn one of his already-dealt downcards face up, so that all players will still have the same number of up and down cards. When using this method cards are not dealt to players starting at the dealer’s left as usual, but start with the high hand showing.
Roll your own should not be confused with rollouts, which, while similar, is fundamentally different.
Some forms of poker, often called lowball, sometimes called “low poker,” reward poor poker hands (in the traditional sense). There are three common variations on this idea, differing in whether aces are treated as high cards or low cards, and whether or not straights and flushes are used. The methods are:
Ace-to-five low: The lowest possible hand is -4-3-2-A, called a wheel. Aces are low and straights and flushes are ignored. This is the most common method.
Ace-to-six low: Also called 6-4 low, since the lowest possible hand is
-4-3-2-A. Aces are low and straights and flushes count as high hands.
Deuce-to-six low: The other, mostly unused, possibility would be 6-5 low. Aces are high, straights and flushes are ignored.
Some games are played high-low split, where the player with the best traditional poker hand (called the “high hand”) splits the pot with the best low hand. The low hand is decided by one of the methods above. According to Official Rules of Card Games by Albert Morehead[1], the low hand in high-low is generally the deuce-to-seven low, although many on-line casinos use ace-to-five low, with a qualifier, e.g., no card higher than an 8. Low hands tie more frequently than high hands, especially in community card games, so it is not uncommon for such a hand to win a small fraction of a poker pot. For example, if one player has the high hand on showdown, and two other players tie for the best low hand, the high hand wins half of the pot and each low hand wins only a quarter of the pot. Playing ace-to-five high-low greatly increases the chances of the “scoop”–winning both hands–because a low flush or straight may count for both high and low.
References
^ Albert H. Morehead (1996). Official Rules of Card Games. Ballantine Books. ISBN 0449911586.
A card that isn’t wild or otherwise modified by the game rules. In some houses, a natural hand beats an equivalent hand that uses wild cards, though this is not generally the case.
When evaluating a hand before the flop, it’s useful to have some idea of how likely the hand is dominated. A dominated hand is a hand that is beaten by another hand (the dominant hand) and is extremely unlikely to win against it. Often the dominated hand has only a single card rank that can improve the dominated hand to beat the dominant hand (not counting straights and flushes.) For example, KJ is dominated by KQ—both hands share the king and the queen kicker is beating the jack kicker. Barring a straight or flush, the KJ will need a jack on the board to improve against the KQ (and will still be losing if a queen comes on the board also.) A pocket pair is dominated by a pocket pair of higher rank.
Pocket pairs
Barring a miracle straight or flush, a pocket pair needs to make three of a kind to beat a higher pocket pair.
To calculate the probability that another player has a higher pocket pair, first consider the case against a single opponent. The probability that a single opponent has a higher pair can be stated as the probability that the first card dealt to the opponent is a higher rank than the pocket pair and the second card is the same rank as the first. Where r is the rank of the pocket pair (assigning values from 2–10 and J–A = 11–14), there are (14 − r) × 4 cards of higher rank. Subtracting the two cards for the pocket pair leaves 50 cards in the deck. After the first card is dealt to the player there are 49 cards left, 3 of which are the same rank as the first. So the probability of a single opponent being dealt a higher pocket pair is
P = (((14-r)x4)/50) x (3/49)
The following approach extends this equation to calculate the probability that one or more other players has a higher pocket pair.
Multiply the base probability for a single player for a given rank of pocket pairs by the number of opponents in the hand;
Subtract the adjusted probability that more than one opponent has a higher pocket pair. (This is necessary because this probability effectively gets added to the calculation multiple times when multiplying the single player result.)
Where n is the number of other players still in the hand and Pma is the adjusted probability that multiple opponents have higher pocket pairs, then the probability that at least one of them has a higher pocket pair is
P = ((84-6r)/1225) x n – Pma.
The calculation for Pma depends on the rank of the player’s pocket pair, but can be generalized as
Pma = P
+ 2P
+ … + (n-1)Pn,
where P
is the probability that exactly two players have a higher pair, P
is the probability that exactly three players have a higher pair, etc. As a practical matter, even with pocket 2s against 9 opponents, P
< 0.0015 and P
< 0.00009, so just calculating P
and P
gives an adequately precise result.
The following table shows the probability that before the flop another player has a larger pocket pair when there are one to nine other players in the hand.
Probability of facing a
larger pair when holding
Against 1
Against 2
Against 3
Against 4
Against 5
Against 6
Against 7
Against 8
Against 9
KK
0.0049
0.0098
0.0147
0.0196
0.0244
0.0293
0.0342
0.0391
0.0439
QQ
0.0098
0.0195
0.0292
0.0388
0.0484
0.0579
0.0673
0.0766
0.0859
JJ
0.0147
0.0292
0.0436
0.0577
0.0717
0.0856
0.0992
0.1127
0.1259
TT
0.0196
0.0389
0.0578
0.0764
0.0946
0.1124
0.1299
0.1470
0.1637
99
0.0245
0.0484
0.0718
0.0946
0.1168
0.1384
0.1593
0.1795
0.1990
88
0.0294
0.0580
0.0857
0.1125
0.1384
0.1634
0.1873
0.2101
0.2318
77
0.0343
0.0674
0.0994
0.1301
0.1595
0.1874
0.2138
0.2387
0.2619
66
0.0392
0.0769
0.1130
0.1473
0.1799
0.2104
0.2389
0.2651
0.2890
55
0.0441
0.0862
0.1263
0.1642
0.1996
0.2324
0.2623
0.2892
0.3129
44
0.0490
0.0956
0.1395
0.1806
0.2186
0.2532
0.2841
0.3109
0.3334
33
0.0539
0.1048
0.1526
0.1967
0.2370
0.2729
0.3040
0.3300
0.3503
22
0.0588
0.1141
0.1654
0.2124
0.2546
0.2914
0.3222
0.3464
0.3633
The following table gives the probability that a hand is facing two or more larger pairs before the flop. From the previous equations, the probability Pm is computed as
Pm = P
+ P
+ … + Pn.
Probability of facing multiple
larger pairs when holding
Against 2
Against 3
Against 4
Against 5
Against 6
Against 7
Against 8
Against 9
KK
< 0.00001
0.00001
0.00003
0.00004
0.00007
0.00009
0.00012
0.00016
QQ
0.00006
0.00018
0.00037
0.00061
0.00091
0.00128
0.00171
0.00220
JJ
0.00017
0.00051
0.00102
0.00171
0.00257
0.00360
0.00482
0.00621
TT
0.00033
0.00099
0.00200
0.00335
0.00504
0.00709
0.00950
0.01226
99
0.00054
0.00164
0.00330
0.00553
0.00836
0.01177
0.01580
0.02045
88
0.00081
0.00244
0.00493
0.00828
0.01253
0.01769
0.02378
0.03084
77
0.00112
0.00341
0.00689
0.01160
0.01758
0.02487
0.03351
0.04353
66
0.00149
0.00454
0.00918
0.01550
0.02353
0.03335
0.04503
0.05861
55
0.00191
0.00583
0.01182
0.01998
0.03040
0.04318
0.05840
0.07619
44
0.00239
0.00728
0.01480
0.02506
0.03821
0.05438
0.07371
0.09635
33
0.00291
0.00890
0.01812
0.03075
0.04698
0.06699
0.09099
0.11919
22
0.00349
0.01068
0.02180
0.03706
0.05673
0.08107
0.11034
0.14484
From a practical perspective, however, the odds of out drawing a single pocket pair or multiple pocket pairs are not much different. In both cases the large majority of winning hands require one of the remaining two cards needed to make three of a kind.
Hands with one ace
When holding a single ace (referred to as Ax), it is useful to know how likely it is that another player has a better ace—an ace with a higher second card. The weaker ace is dominated by the better ace. The probability that a single opponent has a better ace is the probability that they have either AA or Ax where x is a rank other than ace that is higher than the player’s second card. When holding Ax, the probability that the other player has AA is (3/50) x (2/49) ~ 0.00245. Where x is the rank 2–K of the second card (assigning values from 2–10 and J–K = 11–13) the probability that a single opponent has a better ace is calculated by the formula
P = ((3/50) x (2/49)) + ((3/50) x (((13-x) x 4)/49) x 2) = (3/1225) + (12 x (13 – x))/1225 = (159 – 12x)/1225.
The probability (3/50) x (((13-x) x 4)/49) of a player having Ay, where y is a rank such that x < y <= K, is multiplied by the two ways to order the cards A and y in the hand.
When facing two opponents, for any given starting hand the number of possible combinations of hands the opponents can have is
(50/2)(48/2) = 1,381800
hands. For calculating probabilities we can ignore the distinction between the two opponents holding A♠J♥ and
♥
♣ and the opponents holding ♥
♣ and A♠J♥. The number of ways that hands can be distributed between n opponents is n! (pronounced n factorial). So the number of unique hand combinations H against two opponents is
H = (50/2)(48/2) -: 2! = 690,900
and against three opponents is
H = (50/2)(48/2)(46/2) -: 3! = 238,360,500
and against n opponents is
H = nk=1Π((50-2k)/2) -: kor alternately H = (50/2n) x (2n-1)!!
where (2n − 1)!! (!! is the double factorial operator) is the number of ways to distribute 2n cards between n hands of two cards each. The following table shows the number of hand combinations for up to nine opponents.
Opponents
Number of possible hand combinations
1
1,225
2
690,900
3
238,360,500
4
56,372,258,250
5
≈9.7073 × 1012 (more than 9.7 trillion)
6
≈1.2620 × 1015 (more than 1.2 quadrillion)
7
≈1.2674 × 1017 (more than 126 quadrillion)
8
≈9.9804 × 1018 (almost 10 quintillion)
9
≈6.2211 × 1020 (more than 622 quintillion)
An exhaustive analysis of all of the match ups in Texas Hold ‘em of a player against nine opponents requires evaluating each possible board for each distinct starting hand against each possible combination of hands held by nine opponents, which is
69 x (50/18) x 17!! x (32/5) ~ 2.117 x 1028 (more than 21 octillion.)
If you were able to evaluate one trillion (1012) combinations every second, it would take over 670 million years to evaluate all of the hand/board combinations. While it is possible to significantly reduce the total number of combinations by pruning combinations with identical properties, the total number of situations is still well beyond the number that can be evaluated by brute force. For this reason, most software programs compute probabilities and expected values for Hold ‘em poker hands against multiple opponents by simulating the play of thousands or even millions of hands to determine statistical probabilities.
Play begins with each player being dealt one card face down, followed by one card face up (beginning as usual with the player to the dealer’s left). If played with a bring-in, the player with the lowest-ranking upcard must pay the bring in, and betting proceeds after that. If two players have equally ranked low cards, suit rankings may be used to break the tie. If there is no bring-in, then the first betting round begins with the player showing the highest-ranking upcard, who may check. In this case, suit should not be used to break ties; if two players have the same high upcard, the one first in clockwise rotation from the dealer acts first.
After the first betting round is complete, another face-up card is dealt to each player (after a burn card, starting with the player to the dealer’s left, as will all subsequent rounds). Betting now begins with the player whose upcards make the best poker hand (since fewer than five cards are face up, this means no straights, flushes, or full houses). On this and subsequent betting rounds, the player to act first may check or bet up to the game’s limit. The second betting round is followed by a third upcard to each player and a third betting round, again starting with the player with the best poker hand showing (thus, the first player to act on each round may change). A fourth face-up card and fourth betting round is followed by a showdown, if necessary (it usually won’t be–most deals of five-card stud end early when a player bets and gets no calls).
Here’s a sample deal. Assume that a game is being played by four players: Alice, who is dealing, Bob, who is sitting to her left, Carol to his left, and David to Carol’s left. Alice deals one card face down to each player, followed by one card face up to each player, beginning with Bob and ending with herself. Bob is dealt the ♠, Carol the K♦, David the ♦, and Alice the ♣. Because they had earlier agreed to play with a $1 bring-in, David is required to start the betting with a $1 bring-in (his ♦ is lower than Bob’s ♠ by suit). He has the option to open the betting for more, but he chooses to bet only the required $1. The bring-in sets the current bet amount to $1, so Alice cannot check. She decides to call. Bob folds, indicating this by turning his upcard face down and discarding his cards. Carol raises to $3. David folds (forfeiting his bring-in), and Alice calls. Alice now deals a second face-up card to each remaining player: Carol is dealt the J♣, and Alice the K♥. Alice’s two face-up cards make a poker hand of no pair, K-9 high, and Carol has K-J high, so it is Carol’s turn to bet. She checks, as does Alice, ending the betting round. Another face up card is dealt: Carol gets the ♥, and Alice gets the K♣. Alice now has a pair of kings showing, and Carol still has no pair, so Alice bets first. She bets $5, and Carol folds. Alice wins the pot without a showdown.
Five-card draw is often the first poker variant learned by most players, and is very common in home games although it is now rare in casino and tournament play. The lowball variations make more interesting games and are more commonly played in casinos. Two to eight players can play.
The descriptions below assume that you are familiar with the general game play of poker, and with hand values. They also make no assumptions about what betting structure is used. In casino play, it is common to use blinds; the first betting round thus begins with the player to the left of the big blind, and subsequent rounds begin with the player to the dealer’s left. In home games, it is typical to use an ante; the first betting round begins with the player to the dealer’s left, and the second round begins with the player who opened the first round.
Play begins with each player being dealt five cards, one at a time, all face down. The remaining deck stub is placed aside, often protected by placing a chip or other marker on it. Players pick up the cards and hold them in their hands, being careful to keep them concealed from the other players. The first “before the draw” betting round occurs at this point, starting with the player to the dealer’s left (or to the left of the big blind if blinds are used).
If more than one player remains after the first round, the “draw” phase begins. Each player specifies how many of his cards he wishes to replace, and discards that many from his hand. The deck stub is retrieved, and after a burn card is dealt, each player in turn beginning at the dealer’s left is dealt from the stub the same number of cards he discarded, so that each player again has five cards. It is important that each player discards the cards he wishes to replace before he takes any replacements, and that he take the same number of replacements as he discarded.
A second “after the draw” betting round occurs after the draw phase, beginning with the player to the dealers left or else beginning with the player who opened the first round (the latter is common when antes are used instead of blinds). This is followed by a showdown if more than one player remains, in which the player with the best hand wins the pot.
A common “house rule” in some places is that a player may not replace more than three cards, unless he draws four cards while keeping an ace (or wild card). This rule is only needed for low-stakes social games where many players will stay for the draw, and will help avoid depletion of the deck stub. In more serious games such as those played in casinos it is unnecessary and generally not used. A rule that is used by many casinos is that a player is not allowed to draw five consecutive cards from the deck stub. In this case, if a player wishes to replace all five of his cards, he is given four of them in turn, the other players are given their draws, and then the dealer returns to that player to give him his fifth replacement (if no later player drew, it is necessary to deal a burn card first).
Another common house rule is that the bottom card of the deck is never given as a replacement, to avoid the possibility of someone who might have seen it during the deal using that information. If the deck stub is depleted during the draw before all players have received their replacements, the last players can receive cards chosen randomly from among those discarded by previous players. For example, if the last player to draw wants three replacements but there are only two cards remaining in the deck stub, the dealer gives the player the one top card he can give, then shuffles together the bottom card of the deck, the burn card, and the earlier players’ discards (but not the player’s own discards!), and finally deals two more replacements to the last player.
Sample deal
The sample deal below assumes that a game is being played by four players: Alice, who is dealing in the examples; Bob, who is sitting to her left; Carol to his left; and David to Carol’s left.
All four players ante $.25. Alice deals five cards to each player and places the deck stub aside. Bob opens the betting round by betting $1. Carol folds, David calls, and Alice calls, closing the betting round. Bob now declares that he wishes to replace three of his cards, so he removes those three cards from his hand and discards them. Alice retrieves the deck stub, deals a burn card, then deals three cards directly to Bob, who puts them in his hand. David discards one card, and Alice deals one card to him from the deck stub. Alice now discards three of her own cards, and replaces them with three from the top of the deck stub (Note: in a player-dealt casino game there is often a rule that the dealer must discard before picking up the deck stub, but this is a home game so we won’t worry about such details). Now a second betting round begins. Bob checks, David bets $3, Alice calls, and Bob folds, ending the second betting round. David shows a flush, and Alice shows two pair, so David takes the pot.
In no-limit or pot limit games the ratio of stack to the blinds and/or antes. For example 8-way in a no limit game hold’em game with blinds of $50/$100 and an ante of $10 a stack of $23,000 has an M-ratio of 100. To further clarify, M = $23,000 / [$100 + $50 + ($10 * 8)] = 100.
maniac
A loose and aggressive player. A player who bets constantly and plays many inferior hands.
match the pot
To put in an amount equal to all the chips in the pot.
misdeal
A deal which is ruined for some reason and must be redealt.
move in
In a no-limit game, to “move in” or to “go all in” means to bet one’s entire stake on the hand in play.
muck
To fold.
To discard one’s hand without revealing the cards. Often done after winning without a showdown or at a showdown when a better hand has already been revealed.
For any given starting hand, there are 50 × 49 ÷ 2 = 1,225 hands that an opponent can have before the flop. (After the flop, the number of possible hands an opponent can have is reduced by the three community cards revealed on the flop to 47 × 46 ÷ 2 = 1,081 hands.) Therefore, there are
(52/2)(50/2) -: 2 = 812.175
possible head-to-head match ups in Hold ‘em. (The number of total number of match ups is divided by the two ways that two hands can be distributed between two players to give the number of unique match ups.) However, since there are only 169 distinct starting hands, there are 169 × 1,225 = 207,025 distinct head-to-head match ups.[2]
It is useful and interesting to know how two starting hands compete against each other heads up before the flop. In other words, we assume that neither hand will fold, and we will see a showdown. This situation occurs quite often in no limit and tournament play. Also, studying these odds helps to demonstrate the concept of hand domination, which is important in all community card games.
This problem is considerably more complicated than determining the frequency of dealt hands. To see why, note that given both hands, there are 48 remaining unseen cards. Out of these 48 cards, we can choose any 5 to make a board. Thus, there are
(48/5) = 1.712.304
possible boards that may fall. In addition to determining the precise number of boards that give a win to each player, we also must take into account boards which split the pot, and split the number of these boards between the players.
The problem is trivial for computers to solve by brute force search; there are many software programs available that will compute the odds in seconds. A somewhat less trivial exercise is an exhaustive analysis of all of the head-to-head match ups in Texas Hold ‘em, which requires evaluating each possible board for each distinct head-to-head match up, or 1,712,304 × 207,025 = 354,489,735,600 (≈354 billion) results.[2]
Head-to-head starting hand matchups
When comparing two starting hands, the head-to-head probability describes the likelihood of one hand beating the other after all of the cards have come out. Head-to-head probabilities vary slightly for each particular distinct starting hand matchup, but the approximate average probabilities, as given by Dan Harrington in Harrington on Hold’em [p.125], are summarized in the following table.
Favorite-to-underdog matchup
Probability
Odds for
Pair vs. 2 undercards
0.83
4.9 : 1
Pair vs. lower pair
0.82
4.5 : 1
Pair vs. 1 overcard, 1 undercard
0.71
2.5 : 1
2 overcards vs. 2 undercards
0.63
1.7 : 1
Pair vs. 2 overcards
0.55
1.2 : 1
These odds are general approximations only derived from averaging all of the hand matchups in each category. The actual head-to-head probabilities for any two starting hands vary depending on a number of factors, including:
Suited or unsuited starting hands;
Shared suits between starting hands;
Connectedness of non-pair starting hands;
Proximity of card ranks between the starting hands (lowering straight potential);
Proximity of card ranks toward A or 2 (lowering straight potential);
Possibility of split pot.
For example, A♠A♣ vs. K♠Q♣ is 87.65% to win (0.49% to split), but A♠A♣ vs. ♦ ♦ is 76.81% to win (0.32% to split).
The mathematics for computing all of the possible matchups is quite complex. However, a computer program can perform a brute force evaluation of the 1,712,304 possible boards for any given pair of starting hands in seconds.
Notes
^ ab By removing reflection and applying aggressive search tree pruning, it is possible to reduce the number of unique head-to-head hand combinations from 207,205 to less than 50,000. Reflection eliminates redundant calculations by observing that given hands h1 and h2, if w1 is the probability of h1 beating h2 in a showdown and s is the probability of h1 splitting the pot with h2, then the probability w2 of h2 beating h1 is w2 = 1 − (s + w1), thus eliminating the need to evaluate h2 against h1. Pruning is possible, for example, by observing that Q♥ J♥ has the same chance of winning against both 8♦ 7♣ and 8♦ 7♠ (but not the same probability as against 8♥ 7♣ because sharing the heart affects the flush possibilities for each hand.)
Five-card stud is probably the earliest form of the card game, stud poker, originating during the American Civil War, but is less commonly played today than seven-card stud and other games. It is still a popular game in a few locations such as South Africa (where it is played with a stripped deck). In Finland a specific version of five-card stud called Sökö (Canadian stud or Scandinavian stud) is still quite popular. The word sökö is also used for checking in Finland (“I check” = “minä sökötän”).Unlike seven-card stud, five-card stud plays very well at no limit and pot limit, though fixed limit and spread limit games are still more common (with higher limits in the later betting rounds). It is typical to use a small ante and a bring-in.
High-low and other variants
The game can be played with low hand values, in which case the best low hand showing starts each betting round instead of the best high hand showing. Also, the highest-ranking card must pay the bring-in if it is played with a bring-in. If played high-low split, the highest showing hand always acts first.
The fifth and final card is dealt face down in some games. Otherwise play is identical (the player who acted first on round three will therefore act first again on round four since no one’s exposed hand has changed). This game is described as “one down, three up, one down” or simply “1-3-1″, while traditional five-card stud is called “one down, four up”.
Here is a sample deal involving our four players. The players’ individual hands will not be revealed until the showdown, to give a better sense of what happens during play:
Compulsory bets: Alice is the dealer. Bob, to Alice’s left, posts a small blind of $1, and Carol posts a big blind of $2.
First betting round: Alice deals four cards face down to each player, beginning with Bob and ending with herself. Ted must act first because he is the first player after the big blind. He cannot check, since the $2 big blind plays as a bet, so he folds. Alice calls the $2. Bob adds an additional $1 to his $1 small blind to call the $2 total. Carol’s blind is “live”, so she has the option to raise here, but she checks instead, ending the first betting round. The pot now contains $6, $2 from each of three players.
First draw: Each player may now opt to draw up to four cards in an attempt to improve their hands. Bob, who is to the dealers immediate left, is given the first chance to draw. Bob discards two cards and receives two replacement cards from the top of the deck. Bob’s discarded cards are not added to the deck, but removed from play. Carol now chooses to also draw two. Finally, Alice chooses to draw one.
Second betting round: Since there are no forced bets in later betting rounds, Bob is now first to act. He chooses to check, remaining in the hand without betting. Carol bets, adding $2 to the pot. Alice and Bob both call, each adding $2 to the pot. The pot now contains $12.
Second draw: Bob draws one. Carol opts not to draw any cards, keeping the four she has (known as standing pat). Alice draws one.
Third betting round: Bob checks again and Carol bets $4. Alice, this round, raises making the total bet $8. Bob folds and Carol calls the additional $4. The pot now contains $20.
Third draw: Since Bob has folded Carol is now first to act, she opts to draw one. Alice stands pat (does not draw).
Last betting round: Carol checks and Alice bets $4. Carol calls.
Showdown: Alice shows 2♠4♣6♦9♥ for a nine-high badugi (or four card hand). Carol has 3♠5♦7♣8♥, an eight-high badugi. Carol wins the $28 pot.
A tough choice to fold a good hand in anticipation of superior opposition.
limit
The minimum or maximum amount of a bet.
limp, limp in
To enter a pot by simply calling instead of raising.
live
.Still raisable. A live bet is one which a player can raise even if they’ve already bet and everyone else has made a call, typically because the player posted a blind or straddle.
. A game full of unskilled or bad players.
lock up
To “lock up” a seat in a cash game means to place a poker chip, player’s card, or other personal effect on the table in front of the seat, to signify that the seat is occupied even though the player may not be present.
loose cannon
A player who is not afraid to put money in the pot; one who is “gambling” a lot and liable to lose all his money at any given time.
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The blinds for this example hand
