Basic Poker Math & Statistics


Continued from: Putting a Player on a Hand (Part II)

I feel that one of the most neglected aspects of poker is mathematics. I think a lot of people are intimidated by math and therefore they just chose to ignore that part of poker and are determined to "play by feel". When a person is playing "by feel" they are really attempting to use their subconscious to guide them through difficult decisions that are actually dictated by statistics and probability.

Poker Math

I want to start this section, which will address math and statistics, by reassuring you that poker math does not have to be complicated or boring. I would also like to stress the importance of having at least a basic understanding of poker statistics. There are a few numbers and mathematical concepts that are absolutely crucial to long-term success in any form of poker. Unfortunately for those right-brained players, No Limit Texas Hold'em is no different.

In NLHE there will be many times that you figure your hand is losing but it has a reasonable possibility of improving to the best hand if you hit one of your "outs" (an out is a card left in the deck that would give you the best hand). When someone makes a bet when you are, say, drawing to a flush, it is important that you take the correct steps in deciding whether or not to call. I'm going to show you precisely how to do that.

Normally, all one needs to do in order to determine the validity of a call is compare the odds of your making the winning hand to the pot odds that you are being offered. For example, if you have a 20% chance of improving to the best hand and someone bets $10 into a $70 pot, the pot is offering you 7 to 1 pot odds and you are only a 5 to 1 underdog to make your hand. This would normally be a trivially easy call. In the beginning of this paper I began to lay a foundation of simple pot odds and probabilities that I need to expand upon in order to instill an understanding of poker mathematics. It is my intention to give enough broad principles and ideas that you will be able to apply them in more situations than I could possibly list in examples. I will also give reasonable approximations of pot odds and probabilities for those situations that are most common in the game of NLHE.

We're going to be talking about some percentages and some ratios. I'd like you to be able to convert percentages to ratios and vice versa. First lets look at some ratios. Let's suppose you are offered a proposition bet. The bet goes something like this; a friend tells you that he will offer you a chance to risk one dollar to potentially gain two dollars. He wagers two dollars to your one dollar that you cannot roll a 6 on a standard six-sided die. If you roll the six he pays you two dollars; if you roll any other number you pay him $1. In this example you are being offered 2 to 1 on your bet. Now we need to know how to figure out if we should take the bet. We do that by figuring out how much of a favorite or dog you are to roll the 6. There are 6 numbers on the die and only one six. That means that there are 5 numbers you will lose with and only one that gives you a win. So the odds against you rolling a "6" are 5 to 1. Because the odds against you rolling the 6 are 5 to 1 and you are only getting 2 to 1 on your money, you should not take the bet.

However, if he agreed to give you six dollars each time you win and you paid him one dollar each time you lost, he would be offering you 6 to 1 odds and you are only 5 to 1 to lose. In that case you should definitely take the bet. On average you will lose one dollar 5 times and win 6 dollars once for every six rolls of the die for a net profit of one dollar. If you divide the one dollar into the six rolls it would take you on average to win it, you will find that you are winning a bit more than 16 cents on average each time you roll the die. Each time you roll the die you have a positive estimated value (EV) of about .167. That's a winning proposition. In poker, especially NLHE, you will be able to find many situations where you will have a much greater EV, but you will sometimes need to understand some basic poker math in order to assess the situation properly and decide whether your play does in fact have a positive EV. I'll go into much greater detail on that later. Let's look at some numbers.

I'm going to start with a simple list of common drawing situations and the odds that go along with them. In the following I will list the drawing situations on the right hand side followed by the approximate chance of improving to that specific hand with one card to come. It is important to note that, even if you have two cards left to come, you must treat each card as a specific situation when comparing your pot odds to your chances of improving to the best hand. If you have a about a 19% chance of improving to a flush on fourth street and another 19% on the river, you should not count those chances together to come up with a 38% chance of making your hand and call a bet as on fourth street as though you actually had a 38% chance of winning. Remember, if you miss your flush on the turn (fourth street) you could still be facing another bet before the next card comes off. The only time you can use the chances of improving to the best hand with both cards coming and compare them to the pot odds you get on your call is if all the money is going in after the flop. That would mean that there is no more betting that can occur and you can simply compare the pot odds you're getting to your chances of making your hand.

The list will include the draw you're on, the number of outs you have, then the percent chance of making the hand on the next card, and finally, your odds against making your hand. Roughly speaking, you can simply double the percentage to find your chances of hitting your hand if you have two cards to come. It's not exact because of other variables, but it should be sufficient for all practical situations. The chart will also allow for fluke occurrences like accidentally catching two pair while on a flush draw. Each situation is very specific, but these approximations are sufficient in practice. You may notice that one way to find your percent chance of improving is to multiply your outs by two and add one. That's not always exactly the case but it's definitely close enough to use as a very handy rule of thumb. And remember, these numbers are per card. Sometimes the stats are different after the flop and after the turn; in these cases the numbers are averaged out in a way that you can use them on either street.

  • 1) Flush draw. 9 outs. 19%. About 4 to 1 against.
  • 2) Open-ended straight draw. 8 outs. 17%. A bit worse than 4 to 1 against.
  • 3) Inside straight draw. 4 outs. 9%. Just about 9 to 1 against.
  • 4) Straight flush draw. 15 outs. 27%. A little better than 3 to 1 against.
  • 5) Pocket pair that need to make a set. 2 outs. 5%. 19 to 1 (not good).
  • 6) Pair that needs to hit its kicker. 3 outs. 7%. About 13 to 1 against.
  • 7) Two live over cards. 6 outs. 12%. Worse than 7 to 1 against.

♣ Continued at: Hold'em Practice Hands & Examples

♣ Back to the index of Dead Money's guide to hold'em strategy.