Texas Hold’em Poker Probabilities & Odds

How often do you get aces, how often do you hit a set, how many different flops are there and how often do you flop a gutshot? Answers to these and similar questions about Texas Holdem poker probabilities and odds can be found here.

This collection of Texas Hold'em odds also contains the probabilities for several long-shot scenarios like set over set, flush over flush and other rather unlikely scenarios.

If you're missing a probability, just leave a comment below!

Preflop Odds

Probability of being dealt a certain starting hand

There are a total of exactly 1,326 different starting hand combinations in Texas Hold'em poker. However, many of them are practically identical, e.g. AK is exactly the same hand as AK before the flop. If you group these identical hands together, you get 169 different starting hand groups - 13 pairs, 78 suited combinations, and 78 off-suit combinations.

The following table shows the probabilities and odds of getting dealt specific hole cards:

Texas Hold'em Preflop: Odds of being dealt a certain starting hand

Being dealt ...ProbabilityFormula
Pairs
Aces (or another specific pair)0.452%
(1 : 220)
{6}\times{\binom{52}{2}}^{-1}
Aces or Kings0.905%
(1 : 110)
2\times{6}\times{\binom{52}{2}}^{-1}
A premium pair (QQ+)1.36%
(1 : 73)
3\times{6}\times{\binom{52}{2}}^{-1}
Any pair5.88%
(1 : 16)
13\times{6}\times{\binom{52}{2}}^{-1}
Suited cards
AK (or another specific suited hand)0.302%
(1 : 331)
{4}\times{\binom{52}{2}}^{-1}
Any suited connector (54s - JTs)2.11%
(1 : 46)
7\times4\times{\binom{52}{2}}^{-1}
Any suited hand23.5%
(1 : 3.3)
\binom{13}{2}\times{4}\times{\binom{52}{2}}^{-1}
Off-suit cards
AK (or another specific unpaired off-suit hand)0.905%
(1 : 110)
{12}\times{\binom{52}{2}}^{-1}
Any offsuit hand70.6%
(2.4 : 1)
\binom{13}{2}\times{12}\times{\binom{52}{2}}^{-1}
Other combinations
Ace-King (suited or off-suit) (or any specific other two unpaired cards)1.21%
(1 : 82)
{16}\times{\binom{52}{2}}^{-1}
Queens or better or ace-king2.56%
(1 : 38)
\left(3\times6+16\right)\times{\binom{52}{2}}^{-1}
2 cards ten or better ("broadway cards")14.3%
(1 : 6.0)
{\binom{5\times4}{2}}\times{\binom{52}{2}}^{-1}
2 cards nine or lower37.4%
(1 : 1.7)
{\binom{8\times4}{2}}\times{\binom{52}{2}}^{-1}

Download: Texas Hold'em Preflop Odds, ( PDF)

How do you calculate poker hand percentages?

To calculate preflop probabilities you just have to do some combinatorics. There are \frac{52\times51}{2} = \binom{52}{2}=1,326 ways to deal 2 hole cards. So that's the total number of possible preflop combinations. The symbol in the middle of the formula is the so called Binomial Coefficient. It calculates the number of ways of picking 2 cards from a deck of 52 cards if the order of the cards doesn't matter.

Now let's say you want to know the probability of being dealt aces preflop. We already know there are 1,326 different two-card-combinations. Exactly 6 of those are pocket aces, namely AA, AA, AA, AA, AA and AA. This means the probability of being dealt aces preflop is exactly \frac{6}{1,326}=0.452%.

For all other possible hands and ranges you can calculate the probability in the same way. Just count the number of combinations and divide by the number of total possible preflop combinations.

The formulas in the tables above and below show how each probability is calculated.

Odds of running into better hands

It is one of the biggest fears poker players have when holding queens or kings before the flop: another player wakes up with aces and takes down the pot.

If you are playing against a single opponent those events will occur very rarely. If you're holding kings for example, the probability of your opponent holding aces is less than 0.5 percent.

But the more players there are left to act behind you the more likely it is that one of them has your premium pair beaten. Another example: if you'er holding jacks under the gun at a full ring table, the chances of at least one opponent behind you holding queens or better are already more than 11 percent.

The following table shows the probabilities of running into better hands when you're holding a premium hand and how often you can expect certain scenarios to happen in the long run (e.g. on a full ring table you can expect to be dealt kings and run into aces every 5,737 hands):

Probabilities of running into better hands preflop

ScenarioProbabilityFormula
Double aces
Being dealt aces preflop0.452%
(1 : 220)
{6}\times{\binom{52}{2}}^{-1}
If you have aces preflop your opponent has aces as well (heads-up)0.0816%
(1 : 1,224)
\binom{50}{2}}^{-1}
If you have aces preflop an opponent has aces as well (full-ring)0.651%
(1 : 153)
1-\left(1-\binom{50}{2}^{-1}\right)^8
Kings vs. aces
If you have kings preflop your opponent has aces (heads-up)0.490%
(1 : 203)
6\times\binom{50}{2}}^{-1}
If you have kings preflop an opponent has aces (full-ring)3.85%
(1 : 25)
1-\left(1-6\times\binom{50}{2}^{-1}\right)^8
You are dealt kings and your opponent has aces (heads-up)0.00222%
(1 : 45,120)
6\times\binom{52}{2}^{-1}\times6\times\binom{50}{2}^{-1}
You are dealt kings and someone has aces (full-ring)0.0174%
(1 : 5,737)
6\times\binom{52}{2}^{-1}\times\left(1-\left(1-6\times\binom{50}{2}^{-1}\right)^8\right)
Queens vs. aces or kings
If you have queens preflop your opponent has kings or aces (heads-up)0.980%
(1 : 101)
2\times6\times\binom{50}{2}}^{-1}
If you have queens preflop an opponent has kings or aces (full-ring)7.57%
(1 : 12)
1-\left(1-2\times6\times\binom{50}{2}^{-1}\right)^8
You are dealt queens and your opponent has aces or kings (heads-up)0.00443%
(1 : 22,559)
6\times\binom{52}{2}^{-1}\times2\times6\times\binom{50}{2}^{-1}
You are dealt queens and someone has aces or kings (full-ring)0.0343%
(1 : 2,917)
6\times\binom{52}{2}^{-1}\times\left(1-\left(1-2\times6\times\binom{50}{2}^{-1}\right)^8\right)
Jacks vs. better pairs
If you have jacks preflop your opponent has a better pair (heads-up)1.31%
(1 : 76)
3\times6\times\binom{50}{2}}^{-1}
If you have jacks preflop an opponent has a better pair (full-ring)11.2%
(1 : 8.0)
1-\left(1-2\times6\times\binom{50}{2}^{-1}\right)^8
You are dealt jacks and your opponent has a better pair (heads-up)0.00665%
(1 : 15,039)
6\times\binom{52}{2}^{-1}\times3\times6\times\binom{50}{2}^{-1}
You are dealt jacks and someone has a better pair (full-ring)0.0505%
(1 : 1,978)
6\times\binom{52}{2}^{-1}\times\left(1-\left(1-3\times6\times\binom{50}{2}^{-1}\right)^8\right)
Ace-king vs. aces or kings
If you have ace-king preflop your opponent has kings or aces (heads-up)0.490%
(1 : 203)
2\times3\times\binom{50}{2}}^{-1}
If you have ace-king preflop an opponent has kings or aces (full-ring)3.85%
(1 : 25)
1-\left(1-2\times3\times\binom{50}{2}^{-1}\right)^8
Ace-queen vs. queens+ or ace-king
If you have ace-queen preflop your opponent has queens+ or ace-king (heads-up)1.96%
(1 : 50)
\left(3+6+3+12\right)\times\binom{50}{2}}^{-1}
If you have ace-queen preflop an opponent has queens+ or ace-king (full-ring)14.6%
(1 : 5.8)
1-\left(1-\left(3+6+3+12\right)\times\binom{50}{2}^{-1}\right)^8
Ace-jack vs. jacks+ or ace-queen+
If you have ace-jack preflop your opponent has jacks+ or ace-queen (heads-up)3.43%
(1 : 28)
\left(3+2\times6+3+2\times12\right)\times\binom{50}{2}}^{-1}
If you have ace-jack preflop an opponent has jacks+ or ace-jack (full-ring)24.4%
(1 : 3.1)
1-\left(1-\left(3+2\times6+3+2\times12\right)\times\binom{50}{2}^{-1}\right)^8
heads-up: playing against one opponent; full-ring: playing at a table with 9 players

Download: Probabilities of running into better hands preflop, ( PDF)

Preflop match-up odds

If only two players are remaining in a Texas Hold'em Poker hand before the flop, the odds of one player winning can range from 5% up to 95%.

We have listed the most important preflop match-up probabilities below:

Texas Hold'em Preflop: Common hand match-up probabilities

MatchupOddsProbability
Pair against two higher cards ("coin flip")1.1 : 146% (44 vs. JT)
up to
57% (QQ vs. AK)
Pair against higher and lower card2.4 : 168% (66 vs. 75)
up to
73% (QQ vs. K2)
Pair against two lower cards4.9 : 177% (KK vs. 87)
up to
89% (KK vs. 77)
Pair against higher and equal rank1.9 : 160% (44 vs. 54)
up to
70% (88 vs. K8)
Pair against equal and lower rank7.3 : 181% (55 vs. 54)
up to
95% (KK vs. K2)
Two higher against two lower cards1.8 : 158% (K8 vs. 54)
up to
71% (JT vs. 72)
High and low card against two inbetween1.5 : 158% (K2 vs. 87)
up to
63% (A2 vs. 83)
Same high card, different kicker1.8 : 153% (A3 vs. A2)
up to
76% (KQ vs. K2)
Interlocked cards1.6 : 156% (K8 vs. 97)
up to
66% (A9 vs. T4)

Download: Texas Hold'em Preflop Match-Ups, ( PDF)

Odds and probabilities on the flop

Probabilities of flopping certain hands

The following table shows the probabilities and odds of hitting specific hands and draws on the flop:

Hitting the flop

Flopping things ...ProbabilityFormula
Flopping things with a pair
Flopping a set or better with a pair11.8%
(1 : 7.5)
1-\binom{48}{3}\times\binom{50}{3}^{-1}
Flopping quads with a pair0.245%
(1 : 407)
48\times\binom{50}{3}^{-1}
Flopping an overpair or better with KK77.4%
(3.4 : 1)
\binom{46}{3}\times\binom{50}{3}^{-1}
Flopping an overpair or better with QQ58.6%
(1.4 : 1)
\binom{42}{3}\times\binom{50}{3}^{-1}
Flopping an overpair or better with JJ43.0%
(1 : 1.3)
\binom{38}{3}\times\binom{50}{3}^{-1}
Flopping an overpair or better with TT30.5%
(1 : 2.3)
\binom{34}{3}\times\binom{50}{3}^{-1}
Flopping things with suited cards
Flopping a flush with suited cards0.842%
(1 : 118)
\binom{11}{3}\times\binom{50}{3}^{-1}
Flopping a flush draw with suited cards10.9%
(1 : 8.1)
39\times\binom{11}{2}\times\binom{50}{3}^{-1}
Flopping a backdoor flush draw with suited cards41.6%
(1 : 1.4)
\binom{39}{2}\times11\times\binom{50}{3}^{-1}
Flopping straights and straight-draws
Flopping a straight with a connector (54 - JT)1.31%
(1 : 76)
4\times4^3\times\binom{50}{3}^{-1}
Flopping a straight draw with a connector9.71%
(1 : 9.3)
\left(144+1632+128\right)\times\binom{50}{3}^{-1}
Flopping a straight with a one-gapper (53 - QT)0.980%
(1 : 101)
3\times4^3\times\binom{50}{3}^{-1}
Flopping a straight draw with a one-gapper7.67%
(1 : 12)
\left(96+1088+128+192\right)\times\binom{50}{3}^{-1}
Flopping things with unpaired cards
Flopping quads with two unpaired cards0.0102%
(1 : 9,799)
2\times\binom{50}{3}^{-1}
Flopping a full house with two unpaired cards0.0612%
(1 : 1,632)
2\times6\times\binom{50}{3}^{-1}
Flopping trips with two unpaired cards1.35%
(1 : 73)
2\times6\times44\times\binom{50}{3}^{-1}
Flopping two pair with two unpaired cards (no pair on the board)2.02%
(1 : 48)
3\times3\times44\times\binom{50}{3}^{-1}
Flopping at least one pair32.4%
(1 : 2.1)
1-\binom{44}{3}\times\binom{50}{3}^{-1}

Download: Probability of flopping a set, flush or straight, ( PDF)

Odds for setups on the flop

Sometimes two players flop very string hands. The most common example for this is certainly the set over set scenario. The following table shows the probabilities for several scenarios where two or more players hit very strong hands:

Probability of two or more players flopping strong hands

Flopping things ...ProbabilityFormula
Set over set
Flopping a set or better with a pair11.8%
(1 : 7.5)
1-\binom{48}{3}\times\binom{50}{3}^{-1}
Being dealt a pair and flopping a set0.691%
(1 : 144)
13\times6\times\binom{52}{2}\times\left(1-\binom{48}{3}\times\binom{50}{3}^{-1}\right)
If two players have a pair, both flop a set1.06%
(1 : 93)
2\times2\times46\times\binom{50}{3}^{-1}
Two players are dealt a pair and both flop a set (heads-up)0.003518%
(1 : 28,423)
4^3\times\binom{13}{3}\times\binom{52}{3}^{-1}\times9\times\binom{49}{2}^{-1}\times6\times\binom{47}{2}^{-1}
Two players are dealt a pair and both flop a set (full-ring)0.127%
(1 : 789)
4^3\times\binom{13}{3}\times\binom{52}{3}^{-1}\times\binom{9}{2}\times9\times\binom{49}{2}^{-1}\times6\times\binom{47}{2}^{-1}
Set over set over set
If three players have a pair, all flop a set0.0463%
(1 : 2,161)
2\times2\times2\times\binom{48}{3}^{-1}
Three players are dealt a pair and all flop a set (3 player table)0.00001066%
(1 : 9,379,926)
4^3\times\binom{13}{3}\times\binom{52}{3}^{-1}\times9\times\binom{49}{2}^{-1}\times6\times\binom{47}{2}^{-1}\times3\times\binom{45}{2}^{-1}
Two players are dealt a pair and both flop a set (full-ring)0.0008955%
(1 : 111,665)
4^3\times\binom{13}{3}\times\binom{52}{3}^{-1}\times\binom{9}{3}\times9\times\binom{49}{2}^{-1}\times6\times\binom{47}{2}^{-1}\times3\times\binom{45}{2}^{-1}
Quads over quads
Hitting quads with a pair until the river0.816%
(1 : 122)
\binom{48}{3}\times\binom{50}{5}^{-1}
If two players have a pair, both hit quads until the river0.002077%
(1 : 48,153)
44\times\binom{50}{5}^{-1}
Two players have a pocket pair and make quads (heads-up)0.000008884%
(1 : 11,255,912)
\combin{13}{2}\times\binom{4}{2}^2\times44\times\binom{52}{3}^{-1}\times2\times\binom{47}{2}^{-1}\times\binom{45}{2}^{-1}
Two players have a pocket pair and make quads (full-ring)0.0003198%
(1 : 312,663)
\combin{13}{2}\times\binom{4}{2}^2\times44\times\binom{52}{3}^{-1}\times\combin{9}{2}\times2\times\binom{47}{2}^{-1}\times\binom{45}{2}^{-1}
Flush over flush
Flopping a flush with two suited cards0.842%
(1 : 118)
\binom{11}{3}\times\binom{50}{3}^{-1}
Being dealt suited cards and flopping a flush0.198%
(1 : 504)
\binom{13}{2}\times{4}\times{\binom{52}{2}}^{-1}\times\binom{11}{3}\times\binom{50}{3}^{-1}
If two players have suited cards, both flop a flush0.486%
(1 : 205)
\binom{9}{3}\times\binom{48}{3}^{-1}
Two players are dealt suited cards and both flop a flush (heads-up)0.005131%
(1 : 19,490)
\binom{13}{3}\times4\times\binom{52}{3}^{-1}\times\binom{10}{2}\times\binom{49}{2}^{-1}\times\binom{8}{2}\times\binom{47}{2}^{-1}
Two players are dealt suited cards and both flop a flush (full-ring)0.185%
(1 : 540)
\binom{13}{3}\times4\times\binom{52}{3}^{-1}\times\binom{9}{2}\times\binom{10}{2}\times\binom{49}{2}^{-1}\times\binom{8}{2}\times\binom{47}{2}^{-1}
Flush over flush over flush
If three players have suited cards, all flop a flush0.231%
(1 : 433)
\binom{7}{3}\times\binom{46}{3}^{-1}
Three players are dealt suited cards and all flop a flush (3 player table)0.00007774%
(1 : 1,286,389)
\binom{13}{3}\times4\times\binom{52}{3}^{-1}\times\binom{10}{2}\times\binom{49}{2}^{-1}\times\binom{8}{2}\times\binom{47}{2}^{-1}\times\binom{6}{2}\times\binom{45}{2}^{-1}
Three players are dealt suited cards and all flop a flush (full-ring)0.006530%
(1 : 15,313)
\binom{13}{3}\times4\times\binom{52}{3}^{-1}\times\binom{9}{3}\times\binom{10}{2}\times\binom{49}{2}^{-1}\times\binom{8}{2}\times\binom{47}{2}^{-1}\times\binom{6}{2}\times\binom{45}{2}^{-1}
heads-up: playing against one opponent; full-ring: playing at a table with 9 players

Download: Probability of two or more players flopping strong hands, ( PDF)

Odds and probabilities after the flop

How often do you hit a straight, what are the odds of making a full house, and what's the probability of making a flush? The following table shows all common scenarios after the flop and the probabilities of improving your hand.

Draws and outs on the flop and turn

ImprovementOutsFlop → TurnTurn → RiverFlop → River
Improving set to quads
e.g. KK on KQJ
12.1%
(1 : 46)
2.2%
(1 : 45)
4.3%
(1 : 22)
Improving pair to trips
e.g. AK on AT9
24.3%
(1 : 22)
4.3%
(1 : 22)
8.4%
(1 : 11)
Hitting a gutshot
e.g. 98 on T62
48.5%
(1 : 11)
8.7%
(1 : 10)
16.5%
(1 : 5.1)
Improving one pair to two pair or trips
e.g. QT on T76
510.6%
(1 : 8.4)
10.9%
(1 : 8.2)
20.4%
(1 : 3.9)
Making a pair with an unpaired hand
e.g. AJ on 973
612.8%
(1 : 6.8)
13.0%
(1 : 6.7)
24.1%
(1 : 3.1)
Improving set to full house or quads
e.g. 66 on T86
7 / 10 *14.9%
(1 : 5.7)
21.7%
(1 : 3.6)
33.4%
(1 : 2)
Hitting an open-ended straight draw
e.g. 87 on 962
817.0%
(1 : 4.9)
17.4%
(1 : 4.8)
31.5%
(1 : 2.2)
Hitting a flush
e.g. A2 on J84
919.1%
(1 : 4.2)
19.6%
(1 : 4.1)
35.0%
(1 : 1.9)
Hitting a gutshot or improving to a pair
e.g. AK on QT7
1021.3%
(1 : 3.7)
21.7%
(1 : 3.6)
38.4%
(1 : 1.6)
Hitting a gutshot or a flush
e.g. Q9 on T86
1225.5%
(1 : 2.9)
26.1%
(1 : 2.8)
45.0%
(1 : 1.2)
Hitting an open-ended straight draw or improving to a pair
e.g. KQ on JT7
1429.8%
(1 : 2.4)
30.4%
(1 : 2.3)
51.2%
(1 : 1)
Hitting an open-ended straight draw or a flush
e.g. 87 on T94
1531.9%
(1 : 2.1)
32.6%
(1 : 2.1)
54.1%
(1.2 : 1)
Hitting a flush or improving to a pair
e.g. AK on 976
1531.9%
(1 : 2.1)
32.6%
(1 : 2.1)
54.1%
(1.2 : 1)
Hitting a gutshot or a flush or improving to a pair
e.g. JT on 974
1838.3%
(1 : 1.6)
39.1%
(1 : 1.6)
62.4%
(1.7 : 1)
Hitting an open-ended straight draw or a flush or improving to a pair
e.g. KQ on JT5
2144.7%
(1 : 1.2)
45.7%
(1 : 1.2)
69.9%
(2.3 : 1)
*7 Outs for a set to improve on the flop, 10 outs on the turn.

Download: Draws and outs on the flop and turn, ( PDF)

Odds for specific board textures

How often does the flop show a pair, how often is the flop single suited and what are the odds of the board not allowing a flush draw on the turn? The following table shows the odds and probabilities for many common (and some uncommon) Texas Hold'em board textures:

Probabilities for specific board textures

Board textureProbabilityFormula
Flop
The flop contains a pair17.2%
(1 : 4.8)
1-\binom{13}{3}\times4^3\times\binom{52}{3}^{-1}
The flop contains trips0.235%
(1 : 424)
13\times4\times\binom{52}{3}^{-1}
The flop is single-suited5.18%
(1 : 18)
\binom{13}{3}\times4\times\binom{52}{3}^{-1}
The flop contains two different suits55.1%
(1.2 : 1)
1-\left(4\times13^3+\binom{13}{3}\times4\right)\times\binom{52}{3}^{-1}
The flop contains three different suits (rainbow flop)39.8%
(1 : 1.5)
4\times13^3\times\binom{52}{3}^{-1}
The flop is single coloured (all black or all red)23.5%
(1 : 3.3)
2\times\binom{26}{3}\times\binom{52}{3}^{-1}
The flop contains at least one ace (or any other specific rank)21.7%
(1 : 3.6)
1-\binom{48}{3}\times\binom{52}{3}^{-1}
The flop contains at least one ace or king (or any two other specific ranks)40.1%
(1 : 1.5)
1-\binom{44}{3}\times\binom{52}{3}^{-1}
The flop contains the A (or any other specific card)5.77%
(1 : 16)
1-\binom{51}{3}\times\binom{52}{3}^{-1}
Flop and Turn
The board contains a pair32.4%
(1 : 2.1)
1-\binom{13}{4}\times4^4\times\binom{52}{4}^{-1}
The board contains trips0.922%
(1 : 107)
13\times4\times48\times\binom{52}{4}^{-1}
The board contains quads0.004802%
(1 : 20,824)
13\times\binom{52}{4}^{-1}
The board is single-suited1.06%
(1 : 94)
\binom{13}{4}\times4\times\binom{52}{4}^{-1}
The board contains three cards of the same suit16.5%
(1 : 5.1)
4\times\binom{13}{3}\times39\times\binom{52}{4}^{-1}
The board contains two cards of the same suit71.9%
(2.6 : 1)
-
The board contains four different suits (rainbow board)10.5%
(1 : 8.5)
13^4\times\binom{52}{4}^{-1}
The board is single coloured (all black or all red)11.0%
(1 : 8.1)
2\times\binom{26}{4}\times\binom{52}{4}^{-1}
Full board (flop, turn and river)
The board contains a pair49.3%
(1 : 1.0)
1-\binom{13}{5}\times4^5\times\binom{52}{5}^{-1}
The board is single-suited0.198%
(1 : 504)
\binom{13}{5}\times4\times\binom{52}{5}^{-1}
The board is single coloured (all black or all red)5.06%
(1 : 19)
2\times\binom{26}{5}\times\binom{52}{5}^{-1}
All probabilities in this table are assuming you don't know anything about the 52 cards (e.g. have not seen your hole cards).

Download: Probabilities for specific board textures, ( PDF)

Poker Odds and Probabilities FAQ

Preflop: Starting Hands

How many starting hands are there in Texas Holdem?

1,326

There are 1,326 distinct starting hands in Texas Hold'em Poker. They can be grouped into 13 pairs, 78 off-suit hands and 78 suited hands.

There are \frac{52\times51}{2} = \binom{52}{2}=1,326 ways to deal 2 hole cards from a deck of 52 cards.

There are 6 different ways to form a specific pair (e.g. AA, AA, AA, AA, AA, AA). For a specific suited hand there are 4 possible combinations and for a specific off-suit hand there are 12 possible combinations.

What are the odds of being dealt aces in Texas Hold'em Poker?

0.45249% or 1 : 220

There are 6 ways to deal pocket aces preflop and the probability is 0.452%. The odds for that are 220 : 1. The probabilities are the same for each specific pair.

What is the probability of getting a pocket pair?

5.8824% or 1 : 16

The probability of being dealt a pair in Texas Hold'em is 5.88%, or odds of 1 : 16. There are 13 pairs in Hold'em (22 - AA) and for each there are 6 ways to be dealt.

There are 6 different ways to form a specific pair and there are 13 different pairs. Meaning there are 6\times13=78 unique hole card combinations that are a pair. The total number of starting hand combinations is 1,326. Thus the probability of being dealt a pair is \frac{78}{1,326}=5.88%.

How many combinations does Ace King have?

16

There are 16 ways to deal ace-king in poker. 4 different aces can each be matched with 4 different kings. The are four combinations of ace-king-suited and 12 combinations of ace-king offsuit.

Preflop: Matchups

What are the odds of pocket aces vs pocket kings?

81.946% or 4.5 : 1

The odds of pocket Aces winning against pocket Kings are 4.5 : 1. The aces win roughly 82% of the time. If you are holding Kings on a full ring table (9 players), the odds of one of your opponents holding Aces are 1:25 (or roughly 4%).

What is a coin flip in Poker?

A situation where where a player with two high cards (e.g. Ace-Queen) is all-in preflop against another player with a lower pair (e.g. Jacks) is called a coin flip. Each player has roughly a 50% chance of winning the hand.

In most cases is the pair the slight favourite to win the showdown. The most extreme case is QQ against AK where the queens are a 57 : 43 favourite.

There is no perfect coinflip in Texas Hold'em before the flop, one hand is always slightly favoured. The match-up that is closest to a perfect coinflip is AT vs. 33. This is a 49.99% : 50.01% match-up.

What is the worst preflop matchup in Texas Hold'em?

5.0763% or 1 : 19

A pair versus a an equal an lower card is the most uneven matchup in Texas Hold'em. In the extreme case kings vs king-deuce the king-deuce only has 5% equity.

What are the odds of running into aces with pocket kings?

3.8518% or 1 : 25

The odds of one opponent holding aces (AA) when you're holding pocket kings (KK) preflop, depend on the number of opponents. Against one opponent it's 0.49% (1:203), against 8 opponents it's 3.85% (1:25).

These are the probabilities of running into aces with kings preflop depending on the number of players at the table:

# of opponentsProbabilityOdds
10.48980%1 : 203
20.97719%1 : 101
31.4622%1 : 67
41.9448%1 : 50
52.4251%1 : 40
62.9030%1 : 33
73.3786%1 : 29
83.8518%1 : 25

What are the odds of an opponent having a better pair when you have a pair?

The odds of having an opponent having a better pair than you before the flop in Texas Hold'em depend on your pair and the number of opponents you face. The probabilities range from 0.49% (you have kings against one opponent) to 42% (deuces against 9 opponents).

This table shows the probabilities of at least one opponent having a better pair before the flop depending on your pair and the number of opponents:

Number of opponents
Pair123456789
KK0.49%
(1 : 203)
0.98%
(1 : 101)
1.46%
(1 : 67)
1.94%
(1 : 50)
2.43%
(1 : 40)
2.90%
(1 : 33)
3.38%
(1 : 29)
3.85%
(1 : 25)
4.32%
(1 : 22)
QQ0.98%
(1 : 101)
1.95%
(1 : 50)
2.91%
(1 : 33)
3.86%
(1 : 25)
4.80%
(1 : 20)
5.74%
(1 : 16)
6.66%
(1 : 14)
7.57%
(1 : 12)
8.48%
(1 : 11)
JJ1.47%
(1 : 67)
2.92%
(1 : 33)
4.34%
(1 : 22)
5.75%
(1 : 16)
7.13%
(1 : 13)
8.50%
(1 : 11)
9.84%
(1 : 9.2)
11.17%
(1 : 8.0)
12.47%
(1 : 7.0)
TT1.96%
(1 : 50)
3.88%
(1 : 25)
5.76%
(1 : 16)
7.61%
(1 : 12)
9.42%
(1 : 9.6)
11.19%
(1 : 7.9)
12.93%
(1 : 6.7)
14.64%
(1 : 5.8)
16.31%
(1 : 5.1)
992.45%
(1 : 40)
4.84%
(1 : 20)
7.17%
(1 : 13)
9.44%
(1 : 9.6)
11.66%
(1 : 7.6)
13.82%
(1 : 6.2)
15.93%
(1 : 5.3)
17.99%
(1 : 4.6)
20.00%
(1 : 4.0)
882.94%
(1 : 33)
5.79%
(1 : 16)
8.56%
(1 : 11)
11.25%
(1 : 7.9)
13.86%
(1 : 6.2)
16.39%
(1 : 5.1)
18.84%
(1 : 4.3)
21.23%
(1 : 3.7)
23.54%
(1 : 3.2)
773.43%
(1 : 28)
6.74%
(1 : 14)
9.94%
(1 : 9.1)
13.02%
(1 : 6.7)
16.01%
(1 : 5.2)
18.89%
(1 : 4.3)
21.67%
(1 : 3.6)
24.35%
(1 : 3.1)
26.95%
(1 : 2.7)
663.92%
(1 : 25)
7.68%
(1 : 12)
11.30%
(1 : 7.8)
14.78%
(1 : 5.8)
18.12%
(1 : 4.5)
21.32%
(1 : 3.7)
24.41%
(1 : 3.1)
27.37%
(1 : 2.7)
30.21%
(1 : 2.3)
554.41%
(1 : 22)
8.62%
(1 : 11)
12.65%
(1 : 6.9)
16.50%
(1 : 5.1)
20.18%
(1 : 4.0)
23.70%
(1 : 3.2)
27.06%
(1 : 2.7)
30.28%
(1 : 2.3)
33.35%
(1 : 2.0)
444.90%
(1 : 19)
9.56%
(1 : 9.5)
13.99%
(1 : 6.2)
18.20%
(1 : 4.5)
22.21%
(1 : 3.5)
26.02%
(1 : 2.8)
29.64%
(1 : 2.4)
33.09%
(1 : 2.0)
36.36%
(1 : 1.8)
335.39%
(1 : 18)
10.49%
(1 : 8.5)
15.31%
(1 : 5.5)
19.87%
(1 : 4.0)
24.19%
(1 : 3.1)
28.27%
(1 : 2.5)
32.14%
(1 : 2.1)
35.79%
(1 : 1.8)
39.25%
(1 : 1.5)
225.88%
(1 : 16)
11.41%
(1 : 7.8)
16.62%
(1 : 5.0)
21.52%
(1 : 3.6)
26.13%
(1 : 2.8)
30.47%
(1 : 2.3)
34.56%
(1 : 1.9)
38.40%
(1 : 1.6)
42.03%
(1 : 1.4)

Preflop: Longshot Odds

What are the odds of getting pocket aces twice in a row?

0.0020475% or 1 : 48,840

The odds of being dealt aces twice in a row are 1 : 48,840 or 0.002%. The probability of being dealt aces in one specific hand is 0.45% and the probability of this happening twice in a row is this number squared.

The exact formula for the probability of being dealt aces twice in a row is \left(\frac{6}{1,326}\right)^2.

The odds of being dealt aces three times in a row are - of course - even smaller, namely 1 : 10,793,860.

How often do you see aces versus kings preflop in Texas Hold'em?

0.15957% or 1 : 626

At a full ring table (9 players) you will see the scenario AA vs. KK between any two players roughly every 600 hands. The odds are 1:626 and probability is 0.16%.

If you're playing poker long enough you will somewhat regularly encounter the aces vs. kings scenario at a table. A formula to estimate the probability for this to happen at a 9 player table is \binom{9}{2}\times2\times6\times\binom{52}{2}{-1}\times6\times\binom{50}{2}^{-1}. This formula slightly underestimates the actual probability which is a little bit higher.

What are the odds of aces vs kings vs queens preflop in Texas Hold'em?

0.0059415% or 1 : 16,830

In Texas Hold'em a hand where aces, kings and queens pair up preflop is very rare. At a 9 player table this scenario unfolds roughly every 17,000 hands. The odds are 1:16,830 and the probability is 0.006%.

Aces vs. kings vs. queens does happen every now and then, for example during this hand at the Bike. A formula to estimate the probability for this happen at a 9 player table is \binom{9}{3}\times3\times6\times\binom{52}{2}^{-1}\times2\times6\times\binom{50}{2}{-1}\times6\times\binom{48}{2}^{-1}. This formula slightly underestimates the actual probability which is a little bit higher.

Flop: General

How many different flops are there in Texas Hold'em Poker?

22,100

There are 22,100 different flops in Texas Hold'em. For each combination of hole cards you are holding there are 19,600 different flops.

There are 52 cards in a Texas Hold'em deck and a flop consists of 3 cards. There are \frac{52\times51\times50}{3\times2\times1}=\binom{52}{3}=22,100 different ways to deal 3 cards and that's the total number of possible flops in Texas Hold'em.

But when you're playing a game, you already hold two of the 52 cards and only 50 cards remain to chose from for the flop. The total number of possible flops given that you are holding 2 cards is only 19,600 \left(=\binom{50}{3}\right).

How often do you flop a pair in Texas Hold'em Poker?

32.429% or 1 : 2.1

With two unpaired, unconnected cards the odds of flopping at least a pair are 1:2.1 or 32%. Roughly speaking: you will flop a pair or better once every third flop.

If you have two hole cards there are 50 cards left in deck. 6 of those will give you a pair, 44 wont. There are \binom{44}{3}=13,244 flops which will not pair any of your hole cards. There's a total of \binom{50}{3}=19,600 possible flops for your hole cards. The probability of you not hitting at least a pair is \frac{13,244}{19,600}=67.6\% and thus the probability of you hitting at least one pair is 1-\frac{13,244}{19,600}=32.4\%.

What is the probability of flopping a set?

11.755% or 1 : 7.5

If you're holding a pocket pair the probability of flopping a set (three of a kind) is 11.8%. The odds are 1 : 7.5.

If you have a pocket pair there are 50 cards left in deck. Exactly 2 of those will give you a set, 48 wont. There are \binom{48}{3}=17,296 flops which will not give you a set. There's a total of \binom{50}{3}=19,600 possible flops for your hole cards. The probability of you not hitting a set or better is \frac{17,296}{19,600}=88.2\% and thus the probability of you hitting a set or better is 1-\frac{17,296}{19,600}=11.8\%.

What are the odds of hitting a flush with 2 suited cards?

0.84184% or 1 : 118

If you're holding a suited cards the odds of flopping a flush are 1:118. That's a probability of 0.84% - rather unlikely.

If you have two suited cards there are 50 cards left in deck. 11 of those remaining cards are of your suit. There are \binom{11}{3}=165 flops which will give you a flush. There's a total of \binom{50}{3}=19,600 possible flops for your hole cards. The probability of you flopping a flush is \frac{165}{19,600}=0.84\%.

What are the odds of flopping a flush draw?

10.944% or 1 : 8.1

If you're holding a suited cards the odds of flopping a flush draw are 1:8.1. That's a probability of 10.9%.

What are the odds of flopping a backdoor flush draw?

41.587% or 1 : 1.4

With two suited cards the flop will contain one card of your suit and give you a backdoor flush draw 41.6% of the time.

What are the odds of getting 4 of a kind?

0.24490% or 1 : 407

If you're holding a pocket pair the probability of flopping quads (four of a kind) is 0.24%. The odds are 1 : 407 - very unlikely.

If you have a pocket pair there are 50 cards left in deck. The flop needs to contain the two other cards matching the rank of your pair and one of 48 other random cards. Meaning, there are 48 different flops which will give you quads. There's a total of \binom{50}{3}=19,600 possible flops for your hole cards. The probability of you hitting quads is \frac{48}{19,600}=0.24\%.

What are the chances of flopping a straight flush?

0.020408% or 1 : 4,899

The odds of flopping a straight flush with a suited connector are 1 : 4,899 or 0.02%.

If you're holding a suited connector like JT there are exactly 4 flops which will give you a straight flush: AKQ (for a royal flush), KQ9, Q98, and 987. There are 19,600 possible flops in total. Thus the probability of you flopping a straight flush is \frac{4}{19,600}=0.020\%.

If you're holding a suited one-gapper (e.g. 97) the chances to down to 0.015%, with a two-gapper (e.g. KT) to 0.010% and with a three-gapper (e.g. T6) to 0.005%.

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