Thursday, May 17, 2007

Probability of Poker Hands - Essential Poker Hands Probabilities for Texas Holdem Part #1

Probability of Poker Hands is a complex area. Understanding of this can give you a strategic edge over your opponents. This article will look at some of the key poker probablility statistics in texas holdem, both before the flop and after. This is part #1 which will focus on pre-flop poker hand probabilities... part #2 will cover the flop and part #3 will bring the whole thing together by looking at your chances of being outdrawn vs multiple opponents.

In texas holdem there are 1,326 different combinations of cards in any starting hand. There is a 1 in 52 chance for the first card multiplied by a 1 in 51 for the second. The question is how do we approach the subject of poker hand probabilies with such a large number of combinations?

Below is a chart for the common pre-flop poker hand probabilities, first we will look at how these are calculated using a pair of aces as an example.

There are 4 aces in our 52 card deck, this means that the probability of your first card being an ace is 4/52 or 1/13. For the second card there are 3 remaining aces in a deck of 51 unknown cards - your probability is now 3/51 or 1/17. Combining these two probabilities we get to (1/13)*(1/17) = 1/221, in percentage terms this is equal to 0.0045% of the time.

Since there are 13 different pairs we can use this information to calculate the chances of being dealt any pair pre flop. 0.0045% * 13 = approximately 0.6% this means you will be dealt a pair pre-flop in texas holdem around 1 in 17 times.

Here are some common pre-flop poker hand probabilities for Texas Holdem Poker.

- AA (or any specific pair) 0.045% or 1/221
- AK (suited or unsuited) 0.0121% or 1/82
- AA, KK or QQ 0.0136% or 1/73
- AA, KK, QQ, JJ or 10 10, 0.0226% or 1/43
- 2 Suited Cards higher than 10, 0.03% or 1/32
- Any Suited Connector, 0.04% or 1/24
- Any 2 cards Jack and Above, 0.09% or 1/10

Of course the real edge in understanding these poker hands probability rankings comes from knowing what the chances are of your opponents having specific hands against you. This, in turn is affected by the cards that you are dealt (we are looking pre-flop only today so community cards be factored in later). The significant number here is the number of opponents that you face... here are the probabilities of an opponent holding an overpair pre-flop:

You Hold 1 Opponent 3 Opponents 5 Opponents 9 Opponents
KK 0.0049 0.0147 0.0244 0.0439
QQ 0.0098 0.0292 0.0579 0.0859
TT 0.0196 0.0578 0.0946 0.1637
66 0.0392 0.1130 0.1799 0.2890
22 0.0588 0.1654 0.2546 0.3633

Ok, interesting numbers, but it is not always easy to remember a list of statistics when at the table. In Plan3t Gong style we will break it down into some easy-to remember poker maths bite-sized pieces starting from the bottom up.

- If you have 22 UTG on a full table there is a 36% (1 in 3) chance that someone will have a pair higher than you.

- QQ will meet with a higher pair on a full table approximately 1 in 12 times. However if 4 opponents have already folded then you will have the best hand 94% of the time (ok so this does not take into account the 'bunching' concept - but that is another post for another day!).

- If you hold KK in the small blind the chances of meeting AA are so tiny (the 1/221 already outlined) that you can rightfully consider yourself hard done by here!

Not quite finished yet... another common scenario is to run a pair into an AK pre flop. While any pair except 22 is a small favourite (KK and AA a big one) this is a such a common situation that it is worth looking at the probability angle separately here... so you have QQ, the probability of meeting AK for various numbers of opponents are:

One Opponent - 0.0121 (about 1 in 82)
Two Opponents - 0.0242 (now 1 in 41)
Three - 0.0363 (thats 1 in 27ish)
Four - 0.0484 (around 1/20 now)
Five - 0.0605 (1/16.4)
Six - 0.0726 (1/13.6)
Seven - 0.0874 (Nearly 1/12)
Eight - 0.0968 (1/11ish)
Nine - 0.1098 (just under 1/10)

Concluding now as my calculator finger is getting tired (!) on a full table with QQ you will face AK 10% of the time and KK or AA around 5% of the time... in other words you have a very strong (statistically speaking) starting hand.

Of course pre-flop is not everything, probabilities change dramatically once the flop is dealt. This is really where things get interesting - it is no longer the raw probabilities involved but affected by opponents bets.

In the next article in this series I will look at the probabilities of flopping various hands, will then tie it all together with opponents probabilitiy of outflopping you!

In the meantime there are plently of excellent poker strategy articles at sit and go planet, these cover Texas Holdem, Tournament Poker, SNGs and Stallite Qualifier Tournaments: http://www.sitandgoplanet.com

GL at the tables,
Mark

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