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.
In this lesson we’re going to give an overview of probability and how it relates to poker. This will include the probability of being dealt certain hands and how often they’re likely to win. We’ll also cover how to calculating your odds and outs, in addition to introducing you to the concept of pot odds. And finally we’ll take a look at how an understanding of the math will help you to remain emotional stable at the poker table and why you should focus on decisions, not results.
What is Probability?
Probability is the branch of mathematics that deals with the likelihood that one outcome or another will occur. For instance, a coin flip has two possible outcomes: heads or tails. The probability that a flipped coin will land heads is 50% (one outcome out of the two); the same goes for tails.
Probability and Cards
Poker Win Rate Statistics
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When dealing with a deck of cards the number of possible outcomes is clearly much greater than the coin example. Each poker deck has fifty-two cards, each designated by one of four suits (clubs, diamonds, hearts and spades) and one of thirteen ranks (the numbers two through ten, Jack, Queen, King, and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 (7.7%), while the odds of getting any spade as your first card are 1 in 4 (25%).
Unlike coins, cards are said to have “memory”: every card dealt changes the makeup of the deck. For example, if you receive an Ace as your first card, only three other Aces are left among the remaining fifty-one cards. Therefore, the odds of receiving another Ace are 3 in 51 (5.9%), much less than the odds were before you received the first Ace.
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Pre-flop Probabilities: Pocket Pairs
In order to find the odds of getting dealt a pair of Aces, we multiply the probabilities of receiving each card:
(4/52) x (3/51) = (12/2652) = (1/221) ≈ 0.45%.
To put this in perspective, if you’re playing poker at your local casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an average of once every 7.5 hours.
The odds of receiving any of the thirteen possible pocket pairs (twos up to Aces) is:
(13/221) = (1/17) ≈ 5.9%.
In contrast, you can expect to receive any pocket pair once every 35 minutes on average.
Pre-Flop Probabilities: Hand vs. Hand
Players don’t play poker in a vacuum; each player’s hand must measure up against his opponent’s, especially if a player goes all-in before the flop.
Here are some sample probabilities for most pre-flop situations:
Post-Flop Probabilities: Improving Your Hand
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:
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.
PDF Chart
We have created a poker math and probability PDF chart (link opens in a new window) which lists a variety of probabilities and odds for many of the common events in Texas hold ‘em. This chart includes the two tables above in addition to various starting hand probabilities and common pre-flop match-ups. You’ll need to have Adobe Acrobat installed to be able to view the chart, but this is freely installed on most computers by default. We recommend you print the chart and use it as a source of reference.
Odds and Outs
If you do see a flop, you will also need to know what the odds are of either you or your opponent improving a hand. In poker terminology, an “out” is any card that will improve a player’s hand after the flop.
One common occurrence is when a player holds two suited cards and two cards of the same suit appear on the flop. The player has four cards to a flush and needs one of the remaining nine cards of that suit to complete the hand. In the case of a “four-flush”, the player has nine “outs” to make his flush.
A useful shortcut to calculating the odds of completing a hand from a number of outs is the “rule of four and two”. The player counts the number of cards that will improve his hand, and then multiplies that number by four to calculate his probability of catching that card on either the turn or the river. If the player misses his draw on the turn, he multiplies his outs by two to find his probability of filling his hand on the river.
In the example of the four-flush, the player’s probability of filling the flush is approximately 36% after the flop (9 outs x 4) and 18% after the turn (9 outs x 2).
Pot Odds
Another important concept in calculating odds and probabilities is pot odds. Pot odds are the proportion of the next bet in relation to the size of the pot.
For instance, if the pot is $90 and the player must call a $10 bet to continue playing the hand, he is getting 9 to 1 (90 to 10) pot odds. If he calls, the new pot is now $100 and his $10 call makes up 10% of the new pot.
Experienced players compare the pot odds to the odds of improving their hand. If the pot odds are higher than the odds of improving the hand, the expert player will call the bet; if not, the player will fold. This calculation ties into the concept of expected value, which we will explore in a later lesson.
Bad Beats
A “bad beat” happens when a player completes a hand that started out with a very low probability of success. Experts in probability understand the idea that, just because an event is highly unlikely, the low likelihood does not make it completely impossible.
A measure of a player’s experience and maturity is how he handles bad beats. In fact, many experienced poker players subscribe to the idea that bad beats are the reason that many inferior players stay in the game. Bad poker players often mistake their good fortune for skill and continue to make the same mistakes, which the more capable players use against them.
Decisions, Not Results
One of the most important reasons that novice players should understand how probability functions at the poker table is so that they can make the best decisions during a hand. While fluctuations in probability (luck) will happen from hand to hand, the best poker players understand that skill, discipline and patience are the keys to success at the tables.
A big part of strong decision making is understanding how often you should be betting, raising, and applying pressure.
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Conclusion
A strong knowledge of poker math and probabilities will help you adjust your strategies and tactics during the game, as well as giving you reasonable expectations of potential outcomes and the emotional stability to keep playing intelligent, aggressive poker.
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.
Related Lessons
By Gerald Hanks
Gerald Hanks is from Houston Texas, and has been playing poker since 2002. He has played cash games and no-limit hold’em tournaments at live venues all over the United States.
Related Lessons
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What Is A Poker Run
Many players have strong opinions about the impact rake has on our no-limit hold'em earn rate. Since actual data is the best way to answer such a question, I gathered hand histories from Las Vegas $1/$2 and $2/$5 no-limit hold'em cash games. Overall, I analyzed nearly a thousand hands taken from four Las Vegas cardrooms at various times of the day and on various days of the week. There were differences among the cardrooms, but I will only summarize the averages here.
$1/$2 No-Limit Hold'em Rake
Stats for Las Vegas $1/$2 NLH games are summarized the first column below. This is based on a 10 percent rake, up to a maximum of $4, as long as there was a flop. A $1 jackpot drop was also taken whenever a rake was taken. A total of 71 percent of these hands were raked, so there was also a jackpot drop 71 percent of the time.
| $1/$2 NLH: $4 Rake | $1/$2 NLH: $5 Rake | $1/$2 NLH: $6 Rake | $2/$5 NLH: $4 Rake | |
|---|---|---|---|---|
| Rake: $ / Hand | 2.16 | 2.46 | 2.72 | 2.92 |
| Jackpot-1: $ / Hand | 0.71 | 0.71 | 0.71 | 0.85 |
| Jackpot-2: $ / Hand | 1.33 | 1.33 | 1.33 | 1.46 |
| Avg. Pot: $ / Hand | 50.39 | 50.09 | 49.83 | 153.10 |
| Est. Avg. Profit: $ / Hand | 33.26 | 33.06 | 32.89 | 101.05 |
| Equity Rake: $ / Hour | 9.86 | 11.22 | 12.40 | 13.99 |
The average rake was $2.164 per hand, the average number of players was 8.78 per hand, and the average dealing rate was 40 hands per hour with an auto-shuffler. Therefore, the average player won 4.56 hands per hour. Accordingly, the average amount of rake paid per player was $9.86 per hour. I call this the 'equity rake' for the game.
Equity rake is the average rake the players paid per hour. But individual players vary from this depending on their playing styles. For example, a tight player will play fewer hands and probably win fewer pots per hour, so his equity rake is probably smaller. Nevertheless, we can learn a lot about the impact rake has on our win rate by studying Mr. Average.
Let's consider Tom Terrific, an extremely good player earning $20 per hour. He would then make about $30 per hour if he could play without a rake. So rake takes an incredible 33 percent from Tom's pre-rake profit, a big factor in his earn rate.
Now let's consider Solid Sam, a decent player making $5 per hour. His pre-rake profit would be $15 per hour and his rake tax would be about 67 percent, double Tom's tax rate. Talk about a regressive tax!
Or we can consider Mediocre Max, a break-even player. His pre-rake profit would be $10 per hour and the rake tax takes all of it.
So the impact rake has on our profit depends on how profitable we are.
Suppose the bell curve distribution of win rates for all NLH cash game players is symmetrical. In that case, the average Vegas $1/$2 player loses nearly $10 per hour, or 5 big blinds per hour, entirely due to the rake. So Max must be much better than mediocre in just to break even. This clearly illustrates that we must be significantly better than average in order to overcome the rake.
$2/$5 No-Limit Hold'em Rake
One possible solution to our rake problem is to move up in stakes. The right-most column in the table shows the stats for Vegas $2/$5 NLH. We can see that in those games the average rake is about one-third higher in terms of dollars. But the average rake is much smaller in BBs or as a percentage of the average winning pot size.
Let's consider Tom Terrific again, Tom makes $40 per hour in $2/$5, slightly worse in BBs per hour than he earned playing $1/$2. He would then make about $54 per hour if he could play without a rake. So rake takes a somewhat lower 26 percent of his pre-rake profit.
Perhaps Solid Sam can make $10 per hour playing $2/$5. His pre-rake profit would be $24 per hour so his tax rate would be 58 percent, also slightly better than his $1/$2 tax rate.
Another way to look at this is that the average $1/$2 player loses about 5 BB per hour, but the average $2/$5 player loses about 2.8 BB per hour. In other words, the rake has a smaller impact on his $2/$5 win rate.
But let's not lose sight of the most important parameter here: our hourly win rate. A player who can eke out a small $1/$2 profit may well be a loser in the tougher $2/$5 game.
Increasing the Rake
A rake increase at our local cardroom would be depressing to contemplate. Some Vegas cardrooms already take a $5 rake, and it can be even larger in other parts of the world. When we see the impact this has on our bottom line, we may wish to reconsider which cardroom we frequent.
The middle columns in the table estimate the cost of a 10 percent rake, up to a maximum of $5 or $6. At first glance, these higher rakes don't seem to be too terrible, increasing the rake per hand only modestly. However, when we consider the equity rake per hour, the average player loses and additional $1.36 per hour for just a $1 maximum rake increase. A $2 rake increase leads to an additional $2.54 per hour average loss in win rate.
We might also expect that a winning player will win a dollar or two less per hour with these higher rakes. But it could be even worse if the higher cost induces the worst players to leave the game, enriching the quality of the remaining player pool.
Jackpots
Our situation seems even bleaker when we consider the jackpot drop. The Jackpot-1 stat refers to taking a one dollar jackpot drop whenever there is a rake drop. We can see that this averages to about $0.71 per hand, which would be an 'equity jackpot' cost of about $3.24 per hour. This is even worse when a second jackpot dollar drops when the pot reaches $30, the Jackpot-2 stat.
Altogether, these 'taxes' cost the average $1/$2 player about $13 per hour with a $1 jackpot drop, and it costs him about $16 per hour with a $2 jackpot drop.
However, nearly all of the jackpot money is returned to the players by way of jackpot payouts. So our equity jackpot losses should not be part of our thinking as long as we participate in all the jackpot opportunities the cardroom provides.
Suppose the cardroom has a $300,000 'freeroll' tournament requiring 120 hours of play to qualify. This tourney is funded by the jackpot drop from as many as 430,000 dealt hands, so it represents a big chunk of the total jackpot money collected.
If we play insufficient hours to qualify or don't bother to play the freeroll, our share of the prize pool can never come our way. For 1,000 entrants, that would be $300 per player for perhaps 100 hours of qualifying play. Clearly, many players have subsidized the tourney players. So when a cardroom offers such a freeroll, we should make sure we qualify and play, otherwise a big portion of our jackpot 'contribution' can never come back to us.
Another common example is 'Quad-flopper Tuesdays,' where a cardroom pays a $500 jackpot for flopping quads on a Tuesday. If we never play on Tuesday, we are subsidizing those players who do. By piling up our playing hours on days with special jackpot promotions, we are the players being subsidized by players who don't.
Finally, we sometimes see a progressive bad beat jackpot that can become a huge windfall for some lucky player. I don't favor the progressive jackpots because they can take a huge chunk out of the total jackpot pool, and that jackpot money usually leaves the poker community. Furthermore, a large jackpot will automatically trigger a federal income tax liability. Although the live-changing nature of a huge jackpot will induce many players to chase it, I would prefer to see smaller payouts. The $500 quad-flopper jackpot won by a poor player will likely reach our own stack eventually.
Conclusion
Rake and jackpot drops have a major impact on our low-stakes NLH profitability. They are the driving force dictating that few players are winners. But since we don't generally have the option to play rake-free, all we can do is to play where the rake is smallest. And we can minimize our jackpot tax by exploiting every opportunity to win a portion of the jackpot prize pool.
Steve Selbrede has been playing poker for 20 years and writing about it since 2012. He is the author of five books, The Statistics of Poker, Beat the Donks, Donkey Poker Volume 1: Preflop, Donkey Poker Volume 2: Postflop, and Donkey Poker Volume 3: Hand Reading.
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cash game strategyno-limit hold’emrakewin ratebankroll managementjackpotslive poker