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• Texas Holdem Statistics
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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

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

Texas Holdem Statistics

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.

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.

Good to Know Before MIT 15.S50 Begins

Poker / Texas Hold’em Rules

Obviously, you should know the ranking of the poker hands, and how cards are dealt in texas hold’em. Know the terms straight flush, four-of-a-kind (or quads), full house (or boat), flush, straight, three-of-a-kind, two-pair, pair, high card; know preflop, postflop, flop, turn, river.

Betting Rules

• You should know how betting in poker works. Understand the terms bet, raise, call, check, fold, and all-in. Check-raising means checking and then later raising in the same betting round.
• Know what a blind is. There will be a small blind and a big blind each hand.
• Know that preflop, the big blind is last to act. On each postflop betting round, the small blind is first to act, and the dealer is last to act. Know when a betting round ends (eg. if all players check, then the betting round ends).
• Other good terms to know for positions are cutoff (right of the dealer), hijack (right of the cutoff), under-the-gun (the person left of the big blind; first to act preflop).

Mathematical Concepts

In class we will run through an example to illustrate the terminology you should know.

• Suppose the pot has $500 in it, and your opponent bets another $250.
• You may call his $250, in which case cards are flipped over:
  • If your cards beat his, you win the whole pot of $500 + $250 (that he just put in) + $250 (that you just put in) = $1000. You profited $750 from this gamble.
  • If his cards beat yours, you get $0 back. You lost $250 from this gamble.
• Or, you can fold, resulting in a payoff of $0.
• In this example, we say that you are getting 3-to-1 odds to call. When you win, you profit 3 times what you risk losing.
• Suppose the probability that your cards beat his is 10%. Then your expectation for calling is 0.1(+750)+0.9(-250)=-150. By calling, you expect to lose $150 in the long run. Your expectation for folding is always $0. Therefore, you should fold, since your expectation for calling is negative.
• Suppose the probability that your cards beat his is 50%. Then your expectation for calling is 0.5(+750)+0.5(-250)=250. By calling, you expect to earn $250 in the long run. Therefore, you should call, since your expectation for calling is positive. If you were to play this game a large number of times, your average payoff per game would be $250, with 100% certainty. This is called the Law of Large Numbers.
• We say that a gamble has high variance if it takes a large number of trials to converge to the expected average payoff per game. A gamble has low variance if it converges quickly. In general, high variance means high risk, and high risk usually means higher reward, ie. higher expectation.

Book Recommendations

Miller, Ed, David Sklansky and Mason Malmuth. Small Stakes Holdem: Winning Big With Expert Play. Two Plus Two, 2004. ISBN: 978-1880685327. Only for limit hold’em, but still one of the classic books in poker and written by mathematicians.

Slightly outdated, but very good:

• Harrington, Dan. Harrington on Hold 'em Expert Strategy for No Limit Tournaments, Vol. 1: Strategic Play. Two Plus Two, 2004. ISBN: 978-1880685334.
• Harrington, Dan. Harrington on Hold 'em Expert Strategy for No Limit Tournaments, Vol. 2: Endgame. Two Plus Two, 2005. ISBN: 978-1880685358.

Rodman, Blair, Lee Nelson, Steven Heston, and Phil Hllmuth, Jr. Kill Phil: The Fast Track to Success in No-Limit Hold 'em Poker Tournaments. Huntington Press, 2009. ISBN: 978-1935396314.

Nelson, Lee, Tyson Steib, Steven Heston, Joe Hachem, and Bertrand Grospellier. Kill Everyone: Advanced Strategies for No-Limit Hold 'em Poker Tournaments and Sit-n-Go's. Huntington Press, 2009. ISBN: 978-1935396307.

More entertaining than educational:
Hansen, Gus. Every Hand Revealed. Kensington Publishing Corp., 2008. ISBN: 978-0818407277. [Preview with Google Books]

Not that practical, but theoretically very interesting:
Chen, Bill and Jerrod Ankenman. The Mathematics of Poker. Conjelco, 2006. ISBN: 978-1886070257.

Nazarewicz, Pawel. Building a Bankroll. Pawel Nazarewicz, 2012. ISBN: 978-0615589886. Mostly for full ring cash games.

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Texas Holdem Odds And Statistics

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