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?

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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:

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.

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.

Poker bluffing is a very important aspect of the game of poker.

In basic terms bluffing in poker is when a player fakes a move and also fakes his face against his desired or real motives, luring his opponents to act in a desired way.

To be successful in poker bluffing, the bluffer must know the thoughts of the other players so as to create a game plan for winning.

Life is not always a matter of holding good cards, but sometimes, playing a poor hand well.

The understanding and awareness of certain factors are very important in poker bluffing.

Some of which are the types of poker players at the table, the number of the opponents on the table, the ability of the bluffer to read the plays of his opponents, the cards on the board post-flop, the image already created by the bluffer for himself at the table, the size of the pot already on the table and his position in the hand.

Before I begin with my list I would like to share you my favorite bluffing video featuring Chris Moneymaker in the 2003 WSOP for some inspiration. He really shows us how to bluff in poker.

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Top 10 Poker Bluffing Tips

So there are many tips and how to guide on how to master the art of poker bluffing. Here is my very own top 10 of different ways to bluff in poker. I hope you enjoy it as much as I did writing it.

Winning with poker bluffing requires you to not play every hand, and you should always reserve your bluffing for the moments of better chances of winning.

Most people can’t skip a round even when they are given a lousy starting hand because they get bored when they fold.

They cannot stand to wait for the next hand and play anyway, leading to loss of money to them.

However, it will eventually be discovered by people if you never put any money on the table without having four of a kind first, leading to smaller pots on your winning hands because your opponents will definitely fold.

You should always bluff a maximum of 2 players at a time if you want to be successful at poker bluffing.

With many active players at the poker table, you are not likely to scare more than 2 away.

Going against this rule causes one of the unmoved players to call your bluff if he has a really good hand.

In most cases, pots with more than 2 players are not profitable for bluffing because the more the opponents you are betting into, the less willing a player would be to call.

You should not bluff when playing against bad or newbie players, as these people frequently call bets regardless of the strength of their hand.

They do this because they still don’t know what the game is all about, and are thus happy to call down bets with mediocre hands.

Inexperienced or bad players do not have the ability to fold, while a good player first tries to understand the strength represented by you.

This rule should be followed even if you feel you have a stronger hand.

Your bluffing should always be based on the pot odds, which means you should always display the attitude of a person with a winning hand each time you bluff.

In most cases, a player anticipates folding of a minimum of one hand when he bluffs.

But, you will be exposed if your bluff fails because there is no fold.

A player is not likely to fold if there is more money in the pot because he has more to gain by playing while your bluff will fold more hands with a small pot.

Therefore, a bluff should not be wasted on a large pot.

Bluffing is a good strategy which can be done when you actually hold aces full of kings.

It could even be done if you had low bets and a good hand, as that is a tactic which engages your opponents, forcing them to add to the pot for you to end up with a bigger pot when the hope of winning is lost.

You should also pay attention to the opponents who have folded already, because a player who is out of a hand tends to register more emotion on his face.

If a person folds an eight, it is easier for you to know before two eights are on the board.

In addition, if you notice that a player is upset, there are more chances of you being able to bluff the remaining opponents in the hand and take the pot.

Though bluffing is a common poker strategy, a long time is needed to really be good at it.

You should know when to bluff, making it a strategy that should be launched by experienced players.

Most inexperienced players think that to really become a winning poker player, they need to bluff.

This is not true, because it is possible to be a winning poker player without even bluffing once.

Thus, your bluffs should be limited as much as possible.

They are to be used only in situations where you are sure your opponent will be forced to fold.

You should never be compelled to bluff because you have not done that for a long time.

If all signs indicate that your opponent has a strong hand, you should not bluff.

If you cannot just sit without doing anything, using a semi-bluff is a good idea. This is the type of bluff backed up by a decent poker hand.

The semi-bluff works if you are both having the same hand, and though it is an aggressive move, it is not as risky as a true bluff.

It is a large bet that is being made by you to avoid a showdown, but you would not be helpless even if the showdown were to still occur.

If your betting shows a consistent trend, you should bluff.

For instance, if you had a pre-flop raised in a game holding 6-6, and the flop produced A Q 5 after getting a caller, it would be an indication of the best time to bluff because of the consistent trend.

At the same time, this should be done carefully and not too frequently during the course of the entire game, or you would be caught by the other players that you were just bluffing.

The secret of an effective bluff here is not making the others to be aware that you are following a specific trend or that you are bluffing or not during the game.

If you completely missed the flop after pre-flopping with a good hand, it could be a good idea to repeat that bet.

In some cases, this is a good time to bluff because a better hand could still be made with subsequent cards.

Now, many of the other players are aware that your bet pre-flopped the last time, and don’t know that you missed it.

If you represent your bet, pressure will be put on the other players because they will have the suspicion that you now have something better than you have actually.

Bluffing also works when there is a pair on the board.

The reason is that the chances of a person hitting a hand are reduced.

When you have 2 cards, and they are the same cards on the board, it means they are less of a flush possibility (not the same suit), less of a straight possibility (not ordered), and that out there, there are only 2 other cards which could match them to make a set.

It could be that those 2 cards had not yet come, or were either buried (especially in a low pair, would people often fold lower cards pre-flop).

In this situation, it is like having 2 cards coming on the flop, and statistically, this means only a few people would have a drawing hand or have hit their hand.

Therefore, you have a good opportunity for bluffing.

final thoughts

Poker bluffing is an art because the player needs to consider a lot of factors.

At the same time, the paradox of bluffing is that if nobody has ever caught you bluffing, you are either not bluffing enough or you are one of the best bluffers.

You would be bluffing too frequently if you were caught almost every time you bluff.

Late positioning and sensing weaknesses should form part of your bluffing strategies and though they do not usually win a pot, they narrow it down to a few people.

Good Poker Statistics To Know People

Now you should go and practice your newly learned poker bluffing skills by playing some real poker.

Good Poker Statistics To Know Life