Texas Holdem Hand Probability

8 min readJan 30, 2021

Playing poker is about playing the odds. The following list gives the odds for outcomes in Texas Hold’em hands. When you realize how heavily the odds are stacked against you, you may want to rethink going all-in before the flop with two suited cards. Use the odds to your advantage: 1. The Texas Hold’em odds of how likely hands are to unfold after the flop will help guide almost every action you make on the flop Odds On the Flop in Texas Hold’em. The flop is the turning point of a Hold’em hand. This is where you’re going to make your biggest and most expensive decisions.

Poker hands odds & outs: a crash course-guide on poker odds, pot odds, probabilities & odds charts so you can win at Texas Hold’em at the tables or online. One of the most important things that a poker player should know is what their poker odds are in a given situation. There are no betting lines to choose from in poker like you get in the sportsbook or on a racing form, so.
And the odds of making a royal flush is 649,739 to 1. This is correct assuming that every game plays to the river. In poker terms, the river is the name for the fifth card dealt, face-up on the board. In total, there are 2,598,960 possible poker hands with 52 cards. The odds of getting four of a kind in Texas Hold ‘Em.

Ever wondered where some of those odds in the odds charts came from? In this article, I will teach you how to work out the probability of being dealt different types of preflop hands in Texas Holdem.

It’s all pretty simple and you don’t need to be a mathematician to work out the probabilities. I’ll keep the math part as straightforward as I can to help keep this an easy-going article for the both of us.

Probability calculations quick links.

Texas Holdem Probability Calculator

A few probability basics.

When working out hand probabilities, the main probabilities we will work with are the number of cards in the deck and the number of cards we want to be dealt. So for example, if we were going to deal out 1 card:

The probability of dealing a 7 would be 1/52 — There is one 7 in a deck of 52 cards.
The probability of dealing any Ace would be 4/52 — There four Aces in a deck of 52 cards.
The probability of dealing any would be 13/52 — There are 13 s in a deck of 52 cards.

In fact, the probability of being dealt any random card (not just the 7) would be 1/52. This also applies to the probability being dealt any random value of card like Kings, tens, fours, whatever (4/52) and the probability of being dealt any random suit (13/52).

Each card is just as likely to be dealt as any other — no special priorities in this game!

The numbers change for future cards.

A quick example… let’s say we want to work out the probability of being dealt a pair of sevens.

The probability of being dealt a 7 for the first card will be 4/52.
The probability of being dealt a 7 for the second card will be 3/51.

Notice how the probability changes for the second card? After we have been dealt the first card, there is now 1 less card in the deck making it 51 cards in total. Also, after already being dealt a 7, there are now only three 7s left in the deck.

Always try and take care with the numbers for future cards. The numbers will change slightly as you go along.

Working out probabilities.

Whenever the word ‘and’ is used, it will usually mean multiply.
Whenever the word ‘or’ is used, it will usually mean add.

This won’t make much sense for now, but it will make a lot of sense a little further on in the article. Trust me.

Probability of being dealt two exact cards.

Multiply the two probabilities together.

So, we want to find the probability of being dealt the A and K. (See the ‘and’ there?)

Probability of being dealt A — 1/52.
Probability of being dealt K — 1/51.

Now let’s just multiply these bad boys together.

P = (1/52) * (1/51)
P = 1/2652

So the probability of being dealt the A and then K is 1/2652. As you might be able to work out, this is the same probability for any two exact cards, as the likelihood of being dealt A K is the same as being dealt a hand like 7 3 in that order.

But wait, we do not care about the order of the cards we are dealt!

When we are dealt a hand in Texas Hold’em, we don’t care whether we get the A first or the K first (which is what we just worked out), just as long as we get them in our hand it’s all the same. There are two possible combinations of being dealt this hand (A K and K A), so we simply multiply the probability by 2 to get a more useful probability.

P = 1/2652 * 2
P = 1/1326

You might notice that because of this, we have also worked out that there are 1,326 possible combinations of starting hands in Texas Holdem. Cool huh?

Probability of being dealt a certain hand.

Two exact cards is all well and good, but what if we want to work out the chances of being dealt AK, regardless of specific suits and whatnot? Well, we just do the same again…

Multiply the two probabilities together.

So, we want to find the probability of being dealt any Ace andany King.

Texas Holdem Probability Chart

Probability of being dealt any Ace — 4/52.
Probability of being dealt any King — 4/51 (after we’ve been dealt our Ace, there are now 51 cards left).

P = (4/52) * (4/51)
P = 16/2652 = 1/166

However, again with the 2652 number we are working out the probability of being deal an Ace and then a King. If we want the probability of being dealt either in any order, there are two possible ways to make this AK combination so we multiply the probability by 2.

P = 16/2652 * 2
P = 32/2652
P = 1/83

The probability of being dealt any AK as opposed to an AK with exact suits is more probable as we would expect. A lot more probable in fact. Also, as you might guess, this probability of 1/83 will be the same for any two value of cards like; AQ, JT, 34, J2 and so on regardless of whether they are suited or not.

Probability of being dealt a range of hands.

Work out each individual hand probability and add them together.

What’s the probability of being dealt AA or KK? (Spot the ‘or’ there? — Time to add.)

Probability of being dealt AA — 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt KK — 1/221 (4/52 * 3/51 = 1/221).

P = (1/221) + (1/221)
P = 2/221 = 1/110

Easy enough. If you want to add more possible hands in to the range, just work out their individual probability and add them in. So if we wanted to work out the odds of being dealt AA, KK or 7 3…

Probability of being dealt AA — 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt KK — 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt 7 3–1/1326 ([1/52 * 1/51] * 2 = 1/1326).

P = (1/221) + (1/221) + (1/1326)
P = 359/36465 = 1/102

This one definitely takes more skill with adding fractions because of the different denominators, but you get the idea. I’m just teaching hand probabilities here, so I’m not going to go in to adding fractions in this article for now! This fractions calculator is really handy for adding those trickier probabilities quickly though.

Overview of working out hand probabilities.

Hopefully that’s enough information and examples to allow you to go off and work out the probabilities of being dealt various hands and ranges of hands before the flop in Texas Holdem. The best way to learn how to work out probabilities is to actually try and work it out for yourself, otherwise the maths part will just go in one ear and out the other.

I guess this article isn’t really going to do much for improving your game, but it’s still pretty interesting to know the odds of being dealt different types of hands.

I’m sure that some of you reading this article were not aware that the probability of being dealt AA were exactly the same as the probability of being dealt 22! Well, now you know — it’s 1/221.

Introduction

In Texas Hold ’Em a hand is said to be dominated if another player has a similar, and better, hand. To be more specific, a dominated hand is said to rely on three or fewer outs (cards) to beat the hand dominating it, not counting difficult multiple-card draws. There are four types of domination, as follows.

Texas Holdem Hand Statistics

A pair is dominated by a higher pair. For example J-J is dominated by Q-Q. Only two cards help the J-J, the other two jacks.
A non-pair is dominated by a pair of either card. For example, Q-5 is dominated by Q-Q or 5–5. In the case of 5–5, three cards only will help the Q-5, the other three queens.
A non-pair is dominated by a pair greater than the lower card. For example, Q-5 is dominated by 8–8. Only three cards will help the Q-5, the other three queens.
A non-pair is dominated by another non-pair if there if there is a shared card, and the rank of the opponent’s non-shared card is greater the dominated non-shared card. For example Q-5 is dominated by K-5 or Q-7. In the former case (K-5 over Q-5) only three cards can help Q-5, the other three queens.

That said, the following tables present the probability of every two-card hand being dominated, according to the total number of players.

Probability of Domination — PairsExpand

Cards2 Players3 Players4 Players5 Players6 Players7 Players8 Players9 Players10 Players2,20.05880.11420.16590.21500.26090.30440.34490.38350.41953,30.05400.10490.15320.19830.24190.28260.32120.35760.39224,40.04890.09560.14000.18200.22200.26020.29660.33130.36405,50.04410.08620.12650.16530.20210.23760.27100.30310.33456,60.03920.07670.11330.14810.18160.21360.24480.27450.30367,70.03440.06750.09960.13060.16050.18950.21770.24470.27098,80.02950.05810.08580.11290.13910.16480.18940.21380.23699,90.02460.04850.07200.09470.11730.13910.16040.18130.2017T,T0.01960.03890.05780.07650.09470.11260.13000.14780.1649J,J0.01470.02930.04350.05770.07190.08560.09920.11320.1262Q,Q0.00980.01950.02920.03890.04830.05790.06740.07660.0861K,K0.00490.00980.01470.01960.02450.02940.03410.03910.0439A,A0.00000.00000.00000.00000.00000.00000.00000.00000.0000

Probability of Domination — Non-PairsExpand

Cards2 Players3 Players4 Players5 Players6 Players7 Players8 Players9 Players10 Players3,20.27420.47850.62890.73890.81870.87530.91560.94380.96294,20.26450.46340.61240.72270.80360.86260.90490.93500.95624,30.24960.44170.58770.69860.78150.84330.88880.92200.94595,20.25460.44870.59560.70600.78810.84890.89340.92550.94865,30.23990.42630.57010.68050.76450.82790.87540.91080.93675,40.22530.40360.54390.65390.73930.80500.85560.89370.92276,20.24500.43380.57860.68850.77180.83440.88090.91520.94036,30.23020.41100.55250.66200.74700.81180.86140.89860.92666,40.21540.38810.52540.63440.71990.78690.83940.87960.91056,50.20080.36470.49750.60470.69110.75990.81460.85810.89197,20.23500.41860.56110.67090.75500.81910.86760.90420.93117,30.22040.39550.53400.64300.72850.79480.84610.88540.91557,40.20570.37240.50650.61380.70000.76810.82200.86420.89717,50.19100.34840.47760.58330.66930.73880.79510.84020.87617,60.17630.32440.44780.55100.63650.70710.76510.81280.85148,20.22550.40340.54340.65260.73750.80320.85360.89230.92138,30.21050.38000.51570.62370.70950.77710.83000.87140.90348,40.19590.35630.48700.59320.67910.74810.80370.84780.88288,50.18120.33230.45740.56140.64670.71680.77430.82080.85868,60.16660.30780.42720.52770.61220.68290.74160.79040.83118,70.15180.28290.39520.49220.57500.64530.70560.75630.79929,20.21560.38780.52500.63380.71940.78620.83880.87930.91049,30.20100.36430.49680.60390.68950.75830.81300.85640.89049,40.18620.34020.46740.57200.65770.72740.78430.83000.86689,50.17140.31570.43710.53880.62340.69370.75230.80030.83989,60.15690.29110.40610.50360.58680.65730.71670.76670.80889,70.14190.26580.37340.46690.54760.61740.67760.72890.77309,80.12740.24030.34000.42820.50610.57420.63420.68670.7320T,20.20570.37220.50660.61430.70050.76880.82290.86540.8987T,30.19100.34850.47720.58310.66910.73870.79500.84020.8762T,40.17640.32400.44740.55010.63520.70550.76380.81110.8499T,50.16170.29950.41630.51530.59910.66960.72860.77840.8196T,60.14700.27420.38430.47900.56060.63050.69040.74130.7847T,70.13230.24870.35120.44110.51960.58810.64780.69960.7448T,80.11760.22270.31690.40080.47540.54180.60090.65320.6993T,90.10300.19650.28170.35860.42860.49230.54920.60100.6473J,20.19600.35660.48770.59440.68080.75050.80630.85080.8862J,30.18130.33240.45780.56170.64760.71800.77570.82270.8610J,40.16650.30780.42710.52750.61200.68280.74190.79110.8317J,50.15190.28270.39540.49160.57410.64410.70420.75490.7976J,60.13710.25730.36210.45370.53360.60260.66250.71430.7590J,70.12230.23140.32840.41420.49010.55720.61640.66880.7145J,80.10770.20500.29310.37250.44420.50830.56580.61740.6638J,90.09310.17850.25710.32890.39480.45530.51000.56010.6061J,T0.07830.15150.21990.28370.34270.39790.44930.49670.5409Q,20.18620.34060.46850.57390.66040.73120.78860.83520.8727Q,30.17130.31610.43790.54020.62550.69680.75570.80440.8445Q,40.15680.29100.40620.50450.58800.65900.71890.76960.8119Q,50.14220.26580.37360.46710.54820.61800.67830.72990.7744Q,60.12730.24000.34000.42800.50550.57340.63330.68570.7312Q,70.11260.21390.30480.38680.46000.52540.58350.63570.6818Q,80.09790.18750.26910.34350.41130.47300.52890.58000.6257Q,90.08330.16060.23210.29830.36000.41660.46890.51730.5619Q,T0.06870.13320.19400.25160.30520.35570.40320.44800.4894Q,J0.05400.10550.15470.20200.24740.29020.33130.37070.4082K,20.17630.32460.44910.55320.63950.71110.77020.81850.8579K,30.16160.29980.41780.51780.60270.67400.73430.78480.8269K,40.14690.27450.38510.48080.56330.63430.69480.74660.7908K,50.13220.24910.35170.44220.52110.59040.65090.70370.7494K,60.11750.22300.31710.40130.47630.54310.60250.65500.7016K,70.10290.19640.28140.35860.42850.49180.54900.60070.6473K,80.08810.16970.24470.31390.37770.43670.49050.53970.5853K,90.07340.14230.20690.26750.32380.37650.42590.47200.5148K,T0.05880.11460.16780.21830.26650.31200.35550.39610.4350K,J0.04410.08660.12770.16710.20580.24260.27800.31250.3452K,Q0.02940.05820.08650.11410.14140.16790.19400.21950.2444A,20.16650.30860.42940.53160.61770.69010.75050.80090.8425A,30.15170.28350.39700.49490.57910.65090.71200.76410.8080A,40.13720.25780.36360.45650.53760.60820.66950.72270.7684A,50.12240.23180.32940.41640.49340.56180.62230.67540.7225A,60.10770.20540.29400.37410.44620.51150.57020.62280.6701A,70.09310.17870.25750.33000.39630.45720.51290.56380.6101A,80.07830.15160.22000.28370.34280.39830.44980.49760.5418A,90.06370.12410.18100.23520.28660.33470.38040.42370.4647A,T0.04900.09590.14110.18470.22640.26640.30490.34170.3770A,J0.03430.06770.10030.13200.16290.19310.22230.25070.2784A,Q0.01950.03890.05820.07690.09560.11400.13200.15000.1676A,K0.00490.00980.01470.01950.02430.02920.03400.03880.0436

Texas Holdem Odds And Probabilities

Methodology: These tables were created by a random simulation. Each cell in the table above for pairs was based on 7.8 million hands, and 21.7 million for the non-pairs.

2-Player Formula

The probability of domination in a two player game is easy to calculate. For pairs it is 6×(number of higher ranks)/1225. For example, the probability a pair of eights is dominated is 6×6/1225 = 0.0294, because there are six ranks higher than 8 (9,T,J,Q,K,A).

Texas Hold’em Starting Hand Probabilities

For non-pairs the formula is (6+18×(L-1)+12×H)/1225, where

L=Number of ranks higher than lower card
H=Number of ranks higher than higher card