1. Probability Of Pocket Pair
2. Probability Of Flopping A Set With Pocket Pair
3. Probability Of Being Dealt Pocket Pair

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Pocket Pairs and Over Cards Chart. Pot Odds and Outs Chart. Common Flop Odds. Expected Value Hand Charts – 2 Players. Expected Value Hand Charts – 6 Players. Expected Value Hand Charts – 9 Players. Expected Value Hand Charts – Statistics. For our complete list of poker charts, visit our Poker Odds Chart page. For Art the probability of a pocket pair is 3/51 = 1/17. Given that Art has a pocket pair, the conditional probability that Bob has the same pocket pair (ignoring suits, of course) is (2/50). (1/49). The conditional probability that Bob has a particular different pocket pair is (4/50). (3/49). The answer to my question is therefore. According to Wikipedia, the probability of being dealt any pocket pair as the starting hand is 6% (78/1326=0.0588), but yet the probability of facing another player holding a larger pair when holding a pocket pair yourself in pre-flop is significantly high especially when you're playing with 6 or more players. This is strange to me given the likelihood of getting a pocket pair for anyone is only 6%. It is notable that the probability of a no-pair hand is less than the probability of a one-pair or two-pair hand. The Ace-high straight flush or royal flush is slightly more frequent (4324) than the lower straight flushes (4140 each) because the remaining two cards can have any value; a King-high straight flush, for example, cannot have the Ace.

As a poker beginner, you might be already envisioning the money made when you look down at your cards and see pocket aces. While it is true that AA will win more often than not, you still need to understand you will not win every time. So how often can you expect to win with aces?

Pocket aces win 85% of the time against one opponent holding the random hand in Texas Holdem. Although the percentage varies greatly depending on the opponent’s hand and the number of opponents. 85% is the odds to win when AA goes all-in preflop. Odds change on later streets.

It might feel unfair when we loose holding the best hand but keep in mind if we are not 100% to win, we will lose sometimes.

Chances of winning with pocket aces preflop

Every time you get dealt a hand in Texas Holdem your odds of getting pocket aces are 1/221, which is roughly 0.9%.

First, let’s check how often do pocket aces win preflop. For better representation, I will take a sample of 100 examples. So next 100 times you receive an AA, 15 times you will lose and win 85 times.

We should not forget about variance. If we experience negative variance (if we are unlucky), then we may lose 30 times and win 70 times only. On the other hand, if we get a positive side of variance (if we are lucky), we could win 95 out of 100 times!

Pocket aces odds for winning when all-in preflop change depending on which hand we are up against. See how well does AA against specific hands in terms of poker percentages (chance to win). For new players to poker, let me elaborate on the table below.

• 72o means any combination of 7 and 2, o after means the hand is off-suit (meaning that suits on 7 and 2 don’t match)
• 22+ means any pocket pair of 2 and higher (which is all pocket pairs from 22 to and including AA)
• T+ means any card T and higher (T, J, Q, K, A)
• JTs stands for JT suites, which means that suits of J and T match
• broadway hands are the hands that include any combination of hands where the lowest card is T or higher (TQ, TJ, AA, KK, TT, AT)

The best hand against aces preflop is 76s or 87s. They both win 22.5% of the time, so still a big underdog, but it is the best you can hope for.

How odds change against more players

The odds of winning reduce if more players are involved in the hand. Against two opponents holding random hands, AA wins 73.5% of the time. Against three, the odds of winning are now merely 64%.

The tool I am using to show you the % is called Equilab. And you can download it for free from Pokerstrategy’s website. Click on Poker Tools in the menu, the software and scroll down to the end where it says Free training software. Make sure you download the one for Hold’em (it’s written only Equilab).

Chances of winning with AA on later streets

In poker, there is more than just preflop. We know four streets. Those are preflop, flop, turn, and river. It’s easy to calculate your odds to win preflop. When it comes to postflop, things start to get more complex. There are many possible flops. In fact 22100 possible flops. Now some of those could be grouped, but that would still leave us with 1755 different scenarios of flops. On turn and river, that number only increases.

So I can’t show you the pocket aces win percentage against those. But I encourage you to download yourself Equilab I linked to above and play around with different flops, turns, and rivers. As a general rule of thumb, more players, the more cautious you need to be. It will be more likely at least one player has something good.

Chances of winning pre-flop with other hands

Pocket aces are the strongest hand in poker. But we shouldn’t ignore others. The second strongest hand in Hold’em are pocket kings followed by pocket queens.

Chances to win with pocket kings

From the table, we can notice that equity (% to win) with KK against random hand preflop is a bit different for some cards and pretty much the same for others. The biggest difference comes from broadway hands and 22+. The difference is because 22+ and broadways also include AA and that crushes us (AA is 82% to win against KK).

KK has 78.5% chance to win preflop against a random hand. So if our opponent holds a random hand they will still win roughly 1 out of five times. 21.5% of the time to be exact.

Important note. In scenarios above for AA and KK, where I compared them to random holding, I assumed the opponent has a random hand he is willing to go all-in with preflop. In real poker games, opponents will not be going all-in with random hands.

Lets now check the case for QQ

Same as for KK our equity with QQ now falls lower, when we are up against broadway hands and pocket pairs.

Lastly let’s check how we do against those hands with AK as many players like to overplay AK.

Many beginner players are shocked when they see how poorly AK does. That’swhere seasoned poker players will make a lot of money from beginners. Any experienced player knows newbie often overplays hands that look nice but don’t do that good against stronger holdings. With QQ our equity was a lot higher for all of the scenarios. Even against the worst combination of 72o, AK is supposed to win only 2 out of 3 times. This means you will lose 32 times and win 62 times. And that is if you run just average. And this if there would be no variance involved.

I will go deeper into the strategy in another article as it is a massive topic on its own. For now, know when the opponent goes all-in preflop, he will have a decent hand unless they are aggressive players willing to bluff their stack. While some players play tighter (they don’t play many hands) than others a good rule of thumb is pocket kings and aces are always strong enough to get all the money in the middle preflop. With QQ we should be good against the majority, but the tightest people. When it comes to AK, it depends on the opponent we are up against. Often it comes down to personal playing style. I prefer to play my AK a bit more passively, and I will not be blindly throwing my stack in the middle to gamble for what is around 50% to win against what opponent wants to go with all-in.

Conclusion

While pocket aces are a definite favorite against any other hand preflop, it doesn’t mean it’s guaranteed to win. Odds change significantly if more players are involved, but at the end of the day, if we get all the money in preflop, we should be happy. In the worst-case scenario, we will have 77.5% to win. When we have either KK or QQ we should still do good. But with AK we need to start being cautiouspreflop.

Related Links

• Ace King vs Pocket Jacks: Winrate and Examples
  
• Odds of Winning With Pocket Kings: Including the Winrate
  

Brian Alspach

Poker Digest Vol. 5, No. 1, December28-January 10, 2002

Every now and then in hold'em, a table is treated to two players who'vebeen dealt the same pocket pair. Determining the probability of thishappening is an easy application of inclusion-exclusion. Let's gothrough the steps outlining the method but excluding the details.

Since we are interested only in the outcome of two players being dealtthe same pocket pair and not which players are dealt the pairs, we arediscussing player ``semideals'. The number of player semideals in a10-handed game is easy to determine. There are ways ofchoosing the 20 cards for the players to receive. Once the 20 cardsare chosen, there are 19!! ways of breaking the 20 cards into 10 handsof two cards each. Recall that .

The product of and 19!! is then the number of player semideals.This product is 82,492,346,176,096,206,475,125 and is the last huge numberI'll throw at you in this article.

In order to get the exact probability, and the reason we useinclusion-exclusion, we have to take into account the fact there couldbe more than two people having the same pocket pair as someone else. Infact, it is possible (though highly unlikely) that five sets of identicalpairs have been dealt. Let's count how many ways this could happen.

There are choices for the ranks. Each set of four cardsmay be split intotwo hands in three ways. Thus, we multiply by and obtain312,741 player-semideals with five sets of identical pairs. Use to denote 312,741. The probability of such a player-semideal has 17 zerosfollowing the decimal point. I said it was highly unlikely.

We do one more example to illustrate the procedure.

Let's determine howmany player semideals have exactly four sets of identical pairs. Thereare choices for the ranks of the pairs; there are three ways tosplit the four cards of each rank into two hands; there are choices for the remaining four cards; and there are three choices forsplitting the four remaining cards into two hands. Multiplying all thesenumbers gives us the number of player semideals with at least four setsof identical pairs.

However, any player semideal counted in iscounted five times in the preceding product. Thus, we subtract and obtain player semideals with precisely foursets of identical pairs.

We determine and in the same way. We then take thesum to obtain the number ofplayer semideals with at least two people being dealt pairs of the samerank. We divide by the total number of player semideals given above andobtain a probability of .00215899 that, upon randomly dealing two cards toeach of ten players, there is at least one set of pairs of the same rankdealt to two players. This is about once in every 463 deals.

The preceding numbers have no restriction on the rank of the pairs. Whatcauses the most drama at the table is when two players both are dealtpocket aces or pocker kings. So let's work out the probability that twoplayers are dealt pocket aces.

Here we can simply use proportionality arguments. The hardest work isobtaining the numbers , but once they have been obtained we can usethem as follows. Now is the number of player semideals with precisely one rank for which two players are dealt pairs of that rank.Since each rank occurs equally often, we divide by 13 to obtainthe number of player semideals with two players being dealt pocket acesand no other rank with two players being dealt pairs of rank .

Continuing, is the number of player semideals with exactly tworanks for which four players have pairs of these two ranks. There are78 combinations of two ranks chosen from 13, and there are 12 combinationsof ranks including ace. That is, 2/13 of the combinations involve ace.So we multiply by 2/13 to get those which involve two players withpocket aces.

Similarly, we multiply by 3/13, by 4/13, and by 5/13.Doing this, then summing, and then dividing by the total number ofplayer semideals gives us a probability of .000166 that two players havebeen dealt pocket aces (the same is true for any fixed rank). This meansabout once every 6,025 deals two players should be dealt pocket aces.

Let's take another viewpoint now and ask the following: Suppose you areplaying a 10-handed hold'em game and upon looking at your two hole cards,you find A-A. What is the probability another player also has beendealt pocket aces?

This is now a question involving conditionalprobability and you must be careful. Even though the probability, as wejust saw, for two people to be dealt pocket aces is small, the result maybe quite different once the condition that someone has aces is given.

Probability Of Pocket Pair

The total number of completions to player semideals is given by theproduct of and 17!! because we are choosing 18 cards from 50to be dealt to the other nine players, and there are 17!! ways topartition the 18 cards into nine hands of two cards each. Then set asidethe remaining two aces to go into another hand. For the other eight hands,we are choosing 16 cards from 48 and partitioning the 16 cards in 15!!ways to form completions.

Probability Of Flopping A Set With Pocket Pair

This gives us completions toplayer semideals with another pair of pocket aces. Dividing the secondnumber by the first gives us a probability of 9/1,225 that someone elsealso has been dealt pocket aces. Expressed as a decimal, the probabilityis about .00735, and expressed as an Egyptian fraction, the probabilityis about 1/136.

Therefore, if you are dealt pocket aces, the odds against another playeralso having pocket aces is about 135-to-1.

Probability Of Being Dealt Pocket Pair