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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.

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View Notes - Exam1Solution from EE 30363 at University of Notre Dame. A poker hand consists of ﬁve cards drawn at random from a deck of 52 cards (13 differ- ent denominations, each of 4.

What is Probability?

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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:

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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.

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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.

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Frank Op de WoerdMo Nuwwarah

'Ok, people, ante up!'

It's how most hands begin, as the dealers asks the players to put in their ante. Usually, this all goes well and doesn't take too much time. Sometimes, though, a player forgets to ante. Sometimes, a player makes change with other players to speed up the game, but the dealer misses what has happened which makes for a delay. Sometimes, there's an argument about who did ante, and who didn't, resulting in people wanting to rip other people's heads off, as seen in the video above.

In short, anteing slows down the game, while still being necessary to keep the game itself moving.

For years, anteing was seen as a necessary evil. It was just the way things were. That was until some bright minds started to think about it and came up with an alternative. In ARIA's big buy-in live events, the big blind pays the ante for everyone at the table. In partypoker live events, the button now pays the ante for all the players at the table. This way, just a single player has to be paying attention, instead of a full table worth of players.

@partypoker tournament at @Dusk_Till_Dawn_ starting with 'button ante' from this week. Button pays ante for everybdy each hand. Quicker. gto

— Patrick Leonard (@plenopads)

After the initial discussion following Leonard's above tweet back in June, PokerStars is now playing with the idea of introducing a similar rule for their future events. There aren't any plans to alter the structure sheets for the events in Prague, but changes for the PCA might just happen.

@RealKidPoker I like btn ante more, with the Bb ante change really a lot your utg/1 spot.

— Mustapha Kanit (@Mustacchione)

We asked around in the €50,000 Super High Roller and the €1,100 PokerStars National Championship to see what some of the players think about this issue.

Vincent van der Fluit: 'The reasons for a button ante are obvious, one person anteing just saves time. The only problem is what to do when there's a dead button. The downside to going with a big blind ante is the massive hit to one's stack when you're the big blind. My solution would be for the button to ante. In case of a dead button, everyone at the table just antes. That sounds to me like the best solution, but it doesn't matter too much really.'

Timothy Adams: 'I like the big blind ante because it just speeds up the game and it's more efficient. Everyone anteing just sucks. I've played the big blind ante format, and it's just way quicker. I would definitely want that moving forward in poker tournaments. It's just easier for everybody.'

Christopher Kruk: 'The big blind and button antes are clearly better I think. It speeds up the game, saves the dealer some time. Online it works really well to have everyone anteing because it's automatic. The efficiency [of one person anteing] is just unbelievably high. I think it's much better to go with the big blind ante. Saves you the times where you have a dead button but you can never have a dead big blind, so just making it out of the big blind makes a whole lot of sense. Just keeps the game flowing, there's no confusion.'

Jeff Sarwer: 'I actually haven't thought about it. The big blind ante option looks just so extreme. How much is that going to affect the big blind's defending range? It just changes the entire dynamic completely. Everything about poker strategy changes based on that, that's crazy. I don't know. My first instinct is that people already defend from the big blind very, very wide, so they're going to have to defend even wider. You're forced to be out of position with even more invested in the pot. I'm not sure on this one.'

Georgios Zisimopoulos: 'I would say regular antes first, button second and big blind third. It's more fair when everybody is putting in an ante, or the best position.'

Mike Wang: 'I guess [the button ante] does speed up play a lot. It's just easier on the dealers. Anything that helps out the dealers, I'm definitely for. I know they have a lot to work with already. I just worry that it impacts the strategy. I know it does deep in the tournament when there's less players at the table. Say, there's a final table of six or seven, there's less antes than there normally would be. It does have a slight effect. I've played in tournaments where they compensated for that by adjusting the amount of the button ante. So, if implemented correctly, it could be the same, strategy-wise. If that's the case, I don't see any real argument against it. That also gets into a lot of work. I'm down for either format. (On big blind ante) That's pretty harsh for the big blind. That seems really punishing for short stacks. Under the gun or under-the-gun-plus-one, you have to get it in lighter because that's gonna be a huge thing that hits you. I think the button ante seems to be the best compromise, personally.'

Ludovich Geilich: 'This is a good discussion. Personally, with me, I play old school. I don't mind putting the antes in. I don't think it slows down the action as much as people think. I'm also happy with the button ante as well. The problem with it is, there's a few flaws. When there's a dead button, whoever wins that pot got penalized. Obviously, it's totally random. It hardly ever happens. I don't mind either way, but if there's a button ante, the only way you could fix it is for everyone to post as normal [when there's a dead button].'

Yan Tsang: 'I think the button ante. The button has the best position, so he can ante. It makes more sense if one player will ante. If you'd let me choose, I'd still take a regular ante.'