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Poker Math & Statistics

The main underpinning of poker is math – it is essential. For every decision you make, while factors such as psychology have a part to play, math is the key element. Knowing the odds is what it’s all about in poker. It has also been said that in poker, there are good bets and bad bets. The game just determines who can tell the difference. That statement relates to the importance of knowing and understanding the math of the game.

In the next module (Key Concepts in Poker) we’ll talk about the importance of using pot odds, which helps you determine if there is enough in the pot to call a bet. But for this lesson we’ll be keeping it relatively simple, and focus on basic hold’em math, and other poker statistics which you should find useful. You don’t need to memorize the information given in this lesson, but re-reading it a few times will help commit it to memory, which you will definitely find worthwhile in the long run.

Basic Hold’em Math

In the previous lesson, “Starting Hands in Hold’em” we mentioned some of the odds against being dealt specific starting hands, such as 3-to-1 for suited cards, and 220-to-1 for any specific pair, such as Aces. Now let’s look at the chances of certain events occurring when playing certain starting hands. The following table lists some interesting and valuable hold’em math:

The Probability That… Percent % The Odds
Non-pairs will pair at least one card on the flop 32 3-1
Two suited cards will make a flush 6.5 15-1
Two suited cards you will flop a flush 0.85 118-1
Two suited cards flop a four flush 10.9 9-1
A pair will flop a set 12 8-1
A pair will flop four of a kind 0.25 400-1

The purpose of showing you these odds is to get you thinking even more about your starting hand selection. Many beginners to poker overvalue certain starting hands, such as suited cards. As you can see, suited cards don’t make flushes very often. Likewise, pairs only make a set on the flop 12% of the time, which is why small pairs are not always profitable.

The above statistics can be very useful for helping you to make good decisions pre-flop in hold’em. But remember that hold’em isn’t a game of just two cards, so it’s useful to know the likelihood of what might happen after the flop. The following table gives a breakdown of the odds and outs after the flop:

    Flop to the River Turn to the River
Outs Common Draws Percent Odds Percent Odds
20   67.5 0.48-1 43.5 1.30-1
19   65.0 0.54-1 41.3 1.42-1
18   62.4 0.60-1 39.1 1.56-1
17   59.8 0.67-1 37.0 1.71-1
16   57.0 0.75-1 34.8 1.88-1
15 Straight & Flush 54.1 0.85-1 32.6 2.07-1
14   51.2 0.95-1 30.4 2.29-1
13   48.1 1.08-1 28.3 2.54-1
12   45.0 1.22-1 26.1 2.83-1
11   41.7 1.40-1 23.9 3.18-1
10   38.4 1.60-1 21.7 3.60-1
9 Flush 35.0 1.86-1 19.6 4.11-1
8 Straight 31.5 2.17-1 17.4 4.75-1
7   27.8 2.60-1 15.2 5.57-1
6   24.1 3.15-1 13.0 6.67-1
5   20.3 3.93-1 10.9 8.20-1
4 Pair or Inside Straight Draw 16.5 5.06-1 8.7 10.50-1
3   12.5 7.00-1 6.5 14.33-1
2   8.4 10.90-1 4.3 22.00-1
1   4.3 22.26-1 2.2 45.00-1

An "out" is a card which will make your hand. If you are on a flush draw, with four hearts in your hand, then there will be nine hearts (outs) remaining in the deck to give you a flush. Remember there are 13 cards in a suit, so this is easily worked out by 13 – 4 = 9. As you can see in the above table, if you’re holding a flush draw after the flop you have a 35% chance of hitting with the turn and river. After the turn with only one card remaining you have almost 20% chance of hitting the flush. These odds and outs will become very important in the next module when we discuss the importance of pot odds, and whether it is right to call a bet.

Heads-up Match-ups in Hold’em

So far in this lesson we’ve talked about the odds of being dealt certain starting hands, and the odds against these cards improving on the flop, turn or river. This is all highly valuable, but whether you play cash games or tournaments, if you’re playing no-limit hold’em it’s essential that you know where you stand against a range of starting hands on which you put your opponent. Let’s start by looking at hand match-ups when holding a pair:

Pair vs. Pair

jcjd vs. 8s8h [82% vs. 18%]

The higher pair is an 82 percent or 4.5-to-1 favourite. We can get very technical and highlight the fact that if the underpair has clean suits and/or the maximum number of straight outs it helps.

Pair vs. Overcards

5h5d vs. acad [55% vs. 45%]

This is the classic coin flip hand that we see many times late in the televised tournaments with one player being all-in. The term coin flip indicates an even money situation which is really a 55 to 45 percent situation. The pair is the slight favourite.

Pair vs. Undercards

5h5h vs. 8s5h [80% vs. 20%]

In this situation the pair is normally about a 5-to-1 favourite and can vary depending on whether the two undercards are suited and/or connectors.

Pair vs. an Overcard and an Undercard

8h5h vs. 8h8s [69% vs. 31%]

The pair is about a 69 percent or not quite 2.5-to-1 favourite. An example of this holding would be JJ against A9. The underdog non paired hand has three outs while the favourite has redraws.

Pair vs. an Overcard and One of That Pair

kckd vs. ac8h [70% vs. 30%]

The classic example of this situation is the confrontation between Big Slick, AK and pocket cowboys, KK. Big Slick has three outs and it becomes a 70-30 percent situation or a 2.3-to-1 dog for the cowboys. This is a far cry from the next situation where even though one of the pair is matched the other card is lower.

Pair vs. an Undercard and One of That Pair

acad vs. ah5h [90% vs. 10%]

This is domination personified! The non pair has to hit its undercard twice or make a straight or flush to prevail. The pair is better than a 90 percent favourite or slightly better than 10-to-1 odds. I’ll take those odds anytime.

Pair vs. Lower Suited Connectors

adqh vs. 8s8s [77% vs. 23%]

You see this match-up late in tournaments when a player is getting desperate and pushes all-in with middle suited connectors. A hand such as QQ against 76 suited would be a prime example. The pair is about a 77 percent or a 3.3-to-1 odds favourite to win.

Pair vs. Higher Suited Connectors

8s8h vs. adjd [50% vs. 50%]

Here is the real coin flip situation. A pair of eights heads-up against a suited QJ is a fifty-fifty proposition. The higher suited cards would have an edge against a lower pair, such as 2’s or 3’s, since the board itself can sometimes destroy little pairs.

Common Pre-Flop Match-Ups (Non Pairs)

The following heads-up confrontations contain no pairs.

Two High Cards vs. Two Undercards

kcqh vs. 5h5d [65% vs. 35%]

The two higher cards are usually a 65% favourite to win, but it can vary depending on whether any of the cards are suited and/or connectors.

High Card, Low Card vs. Two Middle Cards

ac8s vs. ad5h [57% vs. 43%]

In this match-up the high card gives it the edge. But it’s only a marginal winner, approximately 57% to the hand containing the high card.

High Card, Middle Card vs. Second Highest, Low Card

kd8h vs. 5h8s [62% vs. 38%]

The edge is increased by around 5% when the low card becomes the third highest card, as shown in this example, which gives approx 62% to 38% for high card/middle card combination.

High Card, Same Card vs. Same Card, Low Card

ahjd vs. jc8s [74% vs. 26%]

In this example the AJ is in a very strong position. If we discount any flush or straight possibilities, it only leaves the player holding J8 with three outs (the three remaining 8’s).

Same High Card, High Kicker vs. Same Card, Low Kicker

ackd vs. ah5h [75% vs. 25%]

The high kicker gives this hand a fairly big edge. It’s very common for AK to run into AQ, AJ, and lower, and it’s why AK is such a powerful hand in NL hold’em, particularly at the business end of tournaments when people move all-in with any sort of Ace.

Statistical Variations

For any math maniacs reading this who do not find these odds precise enough, I acknowledge that the math is rounded and for the most part does not take into account the possibilities of ties and back door straights and flushes. What players need to be equipped with is the general statistical match-up – not the fact that in the example of a pair of eights against a suited QJ the percents are exactly 50.61 for the eights to 48.99 for the suited connectors with the balance going to potential ties. I call that a fifty-fifty proposition.

Of greater importance than quibbling over tenths of a percent is the fact that in most heads-up confrontations you can never be a prohibitive underdog. That is one reason the game is so challenging and fun. Of course, while true, I’m not attempting to embolden the reader to ignore the odds and become a maniac. Math, as stated in the introduction is the underpinning of the game and if you regularly get your money into the middle with the worst of it you will go broke.

Conclusion

Poker statistics can be a valuable tool as well as fun. One statistic that hasn’t been mentioned, and it’s one that I particularly like is this – the odds of both players being dealt Aces when playing heads up (one on one) is 270,724-to-1. It’s my favourite statistic because it provides me with almost total confidence when I’m playing heads up and receive pocket Aces that I’m the boss! That confident feeling lasts right up to the river when my Aces get cracked by some rotten piece of cheese which my opponent elected to play. As stated earlier in this lesson, rarely are you a prohibitive underdog. Remember that to keep those losing hands in perspective.

Depending on where you stand on the knowledge curve of this fascinating subject, pick out and study what will help you in this lesson. Come back and visit this lesson again when you need a re-cap. While it’s not essential that these statistics be committed to memory, it won’t hurt you if you do. Remember that the foundation upon which to build an imposing knowledge of hold’em starts and ends with the math. I’ll end this lesson by simply saying – "The Math is Essential".

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