The game of poker has exploded in popularity recently. It’s hard to flip through the TV channels and not find a tournament being televised. One thing that you will notice if you watch these shows is that frequently the same names keep popping up again and again at the final table. Many of these players are referred to as “pros”(short for professional poker players). Yet how is it possible that many of these people consistently win time after time? The answer is that they understand gambling theory and the mathematics of poker. In the following paper I will attempt to get at the mathematical ideas that govern the results of this game.
Mathematical expectation,commonly known as EV (for expected value),is a very important idea in poker. Winning players understand it and losing players don’t. In the Theory of Poker, by David Sklansky, he states that“mathematical expectation is the amount a bet will average winning or losing.” It is the amount on average you expect to win or lose if the hand were played out infinitely many times. A play can be considered a good play if on average you are expect to win any amount over $0 over the long term. Say for example we decide to have a coin flipping a contest and I decide that every time it’s heads I’ll pay you $2 and every time it’s tails you’ll pay me $1. How do you figure out your EV? If the coin is fair and we flipped it 1000 times we would expect it to land heads 50% and tails 50%. So 500 times I would pay you two dollars and 500 times you would pay me $1. Your expectation would look like this.
(500*$2 – 500*$1)/(1000) = $.50 per one flip of the coin
For you this game is +EV. After 1000 flips you should expect to be ahead by $500. Your actual result might differ from this. Some reasons for this could be
- The coin is not fair- This could very well be the case. If someone offered to play this game with you they probably are stupid or are using a biased coin.
- Your sample size is too small. While you expect to win 500 times your actual result could be 475 heads and 525 tails.If the coin is fair this will eventually even out but you need to do more trials until your results approach statistical significance. It is vitally important differentiate between what you expect to get and what you actually get. I will discuss more about this concept in the results based thinking section of this paper.
Calculating the odds is another important skill that winning gamblers posses. Let’s look at the game of Texas Hold’em and one of the many complex situations that come up. Thisis a very math intensive section and some reader’s without statistics knowledge may wish to skip to the resulting table.Say for example that you are playing heads up against a lone opponent.
For example you have JT and your opponent holds QQ
The flop has been dealt 982. What are your chances of winning the hand?
Note that this is not as easy a situation as it first appears. There are two cards left to come.You will win with a 7 or Q which is six outs but there are also many other possibilities. Say the turn is a J. You can also win by catching another Jack. Getting the exact odds is not as easy as it appears. One way to figure out your odds is to use a computer program to run simulations. An excellent program is contained at I will use the program throughout this paper to estimate odds.Another way is to calculate it exactly using combinatorial mathematics.Professional player Barry Greenstein devised an excellent system for breaking down the odds which I will use here. This system is contained in more detail at
1. Count outs: Count the direct outs you have against your opponent
2. Calculate safe combinations: Calculate the number of ways two cards can come up that don’t give you the winning hand
3. Get base number: Subtract the number of safe combinations from the number of ways any two cards can come up.
4. Add back door draws: Add in the number of two-card outs.
5. Subtract redraws: Figure out the number of ways you will draw out on the turn but your opponent will hit an out on the river to beat your hand
6. Calculate odds: Divide the number of ways your opponent wins by the number of ways you win.
So for our example of JT vs. QQ Flop 982
Step 1: 6 outs (2 queens and 4 sevens)
Step 2: ((45 – 6) choose 2) = (39 choose 2) = (39 x 38)/2 = 741
Step 3: 990 – 741 = 249
Step 4: Backdoor flush: Eight different hearts on turn, seven different hearts on the river; (8 choose 2) = (8 x 7)/2 = 28.
Trip Jacks: 3 Jacks on turn, 2 Jacks on river, (3 choose 2) = (3 x 2)/2 = 3 Trip Tens: 3 Tens on turn, 2 Tens on river, (3 choose 2) = (3 x 2)/2 = 3
After step 4 our total is Current total is 249 + 28 + 3 +3 = 283
Step 5: Queens full: two different Q’s on the turn, 9 different pairing cards on the river; 2 x 9 = 18. Current total is 283 – 18 = 265.
FourQueens: 1 combination. Final total is 265 – 1 = 264.
Therefore, 264 is the number of wins for the JT
Step 6: 264 wins for theJT, so there are 990 – 264 = 726 wins forQQ.
726/264 = 2.75, so the odds 2.75 to 1 which is around 26.6%.
The following table gives odds for situations you're likely to encounter in Texas Hold'em. Two to come and one to come refers to the number of cards left to dealt.
Texas Hold’em Outs (Figure 1)
Outs / Example Holding / Drawing To / 2 to come / 1 to come21 / .43:1 / 1.2:1
20 / .48:1 / 1.3:1
15 / open straight flush draw / straight, flush, straight flush / .85:1 / 2.07:1
14 / .96:1 / 2.28:1
13 / 1.08:1 / 2.54:1
12 / gutshot straight flush draw / straight, flush, straight flush / 1.22:1 / 2.83:1
11 / 1.40:1 / 3.18:1
10 / 1.61:1 / 3.60:1
9 / four flush / flush / 1.86:1 / 4.11:1
8 / open straight draw / straight / 2.18:1 / 4.75:1
7 / 2.59:1 / 5.57:1
6 / 3.14:1 / 6.67:1
5 / 3.91:1 / 8.20:1
4 / gutshot straight / straight / 5.07:1 / 10.50:1
3 / 7.01:1 / 14.33:1
2 / pocket pair / 3 of a kind / 10.88:1 / 22.0:1
1 / 3 of a kind / 4 of a kind / 22.5:1 / 45:1
You’ll notice that our JT vs. QQ was a 1 to 2.75 underdog. This falls in between the numbers for six and seven outs. So we could estimate that we had approximately 6.5 outs.
Any odds ratio can also be expressed as a percentage. For example a hand that is a 2:1 underdog to win the pot has a 1/(1+2) = .3333 or 33% chance of improving to the winning hand. You should also notice that as the number of outs increase so does the chance of improving. This chart is vital to understanding the mathematics behind whether it is correct to call, raise, or fold to any given bet in a poker game.
Let’s put the ideas of calculating odds and expectation together and show how they apply to poker. The game we’ll discuss is $2/$4 Limit Hold’em. If you are unfamiliar with the rules refer to the procedures section at the back of the text. I was last to act before the blinds (referred to as on the button) and four people had called in front of me and I looked down at JT. I made a marginal play and raised to $4. Both of the blinds folded and all four of the other players called. There was now $23 in the pot. The flop came Q 5 2. The first player checked and the second player bet $2 and the next two players called. There was now $29 in the pot and it cost me $2 to call giving me 14.5 to 1 pot odds. With my four flush I had 9 outs, assuming the other player had a pair of queens, I only needed 4.11 to 1(see odds table) to make my call be +EV. The turn card was the A. The first player checked as did the others. I decided to bet $4 to represent the Ace hoping the player with the possible pair of queens would fold. He disappointing called. The player after him folded and now the player on my right check-raised to $8. There was now $51 in the pot and it cost me $4 to call. The pot was offering me 13.75 to 1 and I had picked up additional outs. I had a gutshot straight and flush draw. All of which would beat my opponents probable pair of Aces if I hit on the river. I now only need 2.83 to 1 to call so again I called. The river was the gorgeous K giving me an unbeatable straight. My opponent bet $4 and I raised to $8. He reraised to $12 and I reraised to $16 which capped the betting and he called. He flipped up 4 3 giving him a straight. His best hand was A-2-3-4-5 for a low straight. Mine was A-K-Q-J-T giving me the highest possible straight and the $83 in the pot.
Now that we know my opponents hand on the turn we can figure out my mathematical expectation of calling. We will assume for the sake of this exercise that I will win no additional bets after I call on the turn. There was $51 in the pot and it was $4 to call. I will win with the 9 clubs and 3 Kings giving me 12 outs. Say we play the same situation 1000 times. Approximately 270 times(.27*1000) I will end up with best hand and win $55 and 730 times I will end up with the worst hand and fold and will lose my $4. My expectation looks like this:
($55*(270) – $4*(730))/1000= +$11.93 per time
Even if my opponent were to flip his cards up and show me he had the straight it would still be correct for me to call as my expectation would be positive. Folding would wrong as I expect to win $11.93 every time I call. As you can see I also made $16 extra after I made my hand. I hope this example helped in understanding the idea of mathematical expectation.
The next concept I want to discuss is by far the most important. The idea is what Mason Malmuth, author of Gambling Theory and Other Topics, refers to as the “statistical concept of self-weighting versus non-self weighting”. The ability to practice non-self weighting is the reason why games such as poker and blackjack are beatable and games like craps and roulette are not.
Winning players practice a non-self weighting strategy.Let’s usean extreme example of flipping a coin. Let’s say as before I’m paying you $2 for every head and you’re paying me $1 for every tails. We’re also going to add another unusual rule. On every 1001st flip of the coin I have to pay you $1,000,000 for every head and you pay me $4,000,000 if it’s tails. Quite a jump in stakes but it shows a good point.
Your expectation for the first 1000 flips is
(500*$2 – 500*($1))= $500
Your expectation for the 1001st flip is
((1/2)*($1,000,000)-(1/2)*($4,000,000))= -$1,500,000.
If you were asked what your results would be in playing this game they would depend almost entirely on the 1001st flip of the coin. Even though I was taking –EV bets for 1000 flips I was more than able to make up for it on the single flip of the coin when my expectation was enough to make up for the amount I had lost on the previous 1000 flips.
So how does this apply to poker? As we talked about earlier every play that you make has a mathematical expectation. The key with poker is that you can control the amount you bet or wager on anygiven hand. In games like poker, blackjack, sports betting you can weight your bets so that you bet heavily when your EV is positive and bet minimally when it is negative. If you understand the odds and game theory (which is a complex topic in itself) you can successfully practice a non-self weighting strategy. In games like roulette and crapsyou are forced use a self weighting strategy. The odds never change and you never have a positive expectation bet. You can vary your bet amount but all that does is vary the amount of your –EV bet.
The final topic that I would like to discuss is the mindset of results based thinking. This subject is important because it deals with idea of what you expect as a result versus what you actually end up getting. Many people will look at their gambling decisions like this; If a bet or move wins it was the right play and if it loses it was the right play. But this is completely the wrong way to view it. Let’s use a coin flipping contest as an example.Imagine this: At dawn tomorrow, all 6 billion people on Earth go out and on guess the flip of a coin. If they guess right they win $1 from those who guessed wrong. Next dawn, everyone who won does it again and wagers their $2, and so on, day after day always winning twice as much as the round before and always having half as many people. After 10 days about 6 million people are still left and they’ve each won $1024 dollars. After 25 days there would only 179 people left who all have won $33,554,432. At this point they’d probably all write a book about how they struck it rich flipping a coin and explain their methods for picking heads or tails. Yet they are no better than anyone at doing this activity because everyone had the same chance of winning because they were engaging a self-weighting game. They just happened to be the ones who actually did win. They assume because they’ve won they’ve been making positive expectation bets. The true expectation of every flip was break even. On average they should expect to be neither up nor down after any amount of flips.
So how does this idea apply to poker? Often you will make bets in poker with a positive expecation and you will still lose. This is simply due to the luck factor of the game. Just because you lose does not mean that any bets you made were wrong and just because you win it doesn’t mean that bet you make is right. I would like to use an example of a hand in which a poker acquaitance was bragging about a “good fold” that he made. The game was $1/$2 No-Limit Hold’em with a $100 buy-in. The player in question was on the button and before the action could even get to him five players had pushed their $100 into into the pot (a very unusual situation). The player looked down and saw two aces(the best starting hand). The player then folded those aces assuming that all of the other aces in the deck were gone and that there was no way he could win. The actual hands of the five other players were AK, AK, JJ, KK, and 22. The flop came with a 2 giving the player with 22 three twos which in the end was the winning hand. The player who folded aces was complementing himself on folding because had he called he would have lost. Yet his fold was wrong and I can prove it using the concept of mathematical expectation.
When the action get to our hero there is $500 in the pot and he has $100 so it costs him $100 to call. He is receiving $500 to $100 pot odds which means that his hand only needs to win 16.66% in order for his call to have a positive expectation. I’m going to use to simulate the actual chance of each hand winning the pot. The results were as follows:
cards win %win lose %lose tie %tie EV
As Ah 356638 54.20 294540 44.76 6830 1.04 0.544
Ks Ac 5646 0.86 633716 96.31 18646 2.83 0.020
Kc Ad 5682 0.86 633680 96.30 18646 2.83 0.020
Kd Kh 36 0.01 654114 99.41 3858 0.59 0.001
Js Jh 149486 22.72 504664 76.70 3858 0.59 0.228
2c 2d 121874 18.52 532276 80.89 3858 0.59 0.186
As you can see the EV of the AA is well above the 16.6% needed to make his expectation positive. The expectation on his call had he made it would have been:
(($600)*(.544)-($100)*(.456)) = +$280.8
Folding the aces was the near equivalent of setting fire to $280. Not something I like to get in the habit of doing. Folding in this spot was clearly wrong and the subsequent bragging was the result of mathematically unsound results based thinking. Yet these are the type of plays losing players typically make. They confuse a play than won with the winning play. One is mathematically correct and the other just happened to win.
The tournament players you see on television are distributed in the same way as the winner of a given poker hand. Some of them are consistently making the mathematically correct plays that are the underlying foundation to long term success. Others are like the coin flippers who simply happen to be the ones that won. Some of these winners write books, very bad books, that espouse incorrect concepts and strategies for the game of poker. The majority of them are from the former group and you will continue to see their faces year after year. The latter group you should expect to see very little of in the years to come.
Glossary of poker terms (Figure 2)
Texas Hold’em is a game that contains a lot of jargon. To someone new it can almost sound as if the players are speaking a different language. These definitions should help to clear up some of the confusion.
Action: The betting in a particular hand or game. A game with a lot of action is a game with a lot of betting. The player who starts the action is the player who makes the first bet.
All-In: Having all one’s money in the pot
Ante: A bet required from players before the start of a hand
Belly buster: A draw to an inside straight. Also called a gutshot
Bet: to put money in the pot before anyone else on any given round.
Blank: A card that is not of any value to a player’s hand