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.

Thus, the total number of four-of-a-kinds is: Full house — The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and consists of three of the four suits. The pair can be any one of the remaining twelve ranks, and consists of two of the four suits. 11 rows  Four of a kind, also known as quads, is a hand that contains four cards of one rank and one. You might also get a four of a kind in one of the other 8 ranks besides aces and the four you discarded. So the total ways to get a four of a kind on the deal is 44+8=52. The total number of combinations on the deal is combin(47,4)=178365. So the probability of a four of a kind is 52/178365 = 1 in 3430.

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

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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This 7+ hour course gives you applicable rules for continuation betting, barreling, raising, and easy ratios so that you ALWAYS have the right number of bluffing combos. Take the guesswork out of your strategy, and begin playing like the top-1%.

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.

The odds of flopping Four of a Kind or better with a pocket pair is 0.24% or 1 in 416

Definition of Four of a Kind (also known as Quads) –

We hold four cards of equal rank.

Example – AdAhAsAcKh

Four of a Kind Aces is the strongest Four of a Kind hand in poker and is also referred to as “Quad Aces”.

Odds of Making a Four of a Kind (Quads) on the Flop

Making Four of a Kind on the flop is an extremely rare occurrence.

We’ll focus solely on the odds of making Four of a Kind or better.

Probability Of Getting Four Of A Kind In Poker

Odds of flopping Four of a Kind or better with any starting hand = 0.03%

Odds of flopping Four of a Kind or better with a pocket pair = 0.24%

Odds of flopping Four of a Kind or better with AKo = 0.01%

To put this into context, assuming any starting hand, we’ll flop Quads or better roughly once every 3,333 flops.

Assuming we play roughly 25% of our starting hands, we can expect to flop Quads or better approximately once every 13,332 hands.

Odds of Making Four of a Kind on the Later Streets

The most common draw to make Four of a Kind is where we already hold Three of a Kind on the current street. Since there are only 4 cards of each rank in the deck, it means that we only have one out to make Quads on each street.

Odds of hitting Quads on the turn = 1/47 = 0.0213 or roughly 2.1%

Odds of hitting Quads on the river = 1/46 = 0.0217 or roughly 2.2%

To calculate the odds of hitting Quads on either the turn or river, we can use a simple trick.

We’ll calculate the probability of not hitting and then subtract from 100.

Odds of not hitting Quads on the turn = 46/47

Odds of not hitting Quads on the river = 45/46

Odds of not hitting Quads on either the turn or the river = 46/47 * 45/46 = 0.9574 or roughly 95.7%

Hence, the chance of hitting Quads by the river after flopping trips is (100 – 95.7) = roughly 4.3%

So even after making Three of a Kind, we are statistically unlikely to make Four of a Kind by the river.

Poker Four Of A Kind Probability Worksheet

Implied Odds Analysis of Four of a Kind

Four of a Kind always carries excellent implied odds provided we use at least one of our hole cards.

Poker Four Of A Kind Probability Calculator

If the four cards of equal rank are on the board, then every player at the table has Quads, so the implied odds are much less relevant. Any player with an Ace in the hole will win, once there are four cards of the same rank on the board.

Unfortunately, it’s not that likely our opponent will invest a large amount of his stack with the King kicker, so we can’t expect excellent implied odds.

Using one of our hole-cards to make Quads (i.e. three cards of equal rank on the board) carries excellent implied odds. If our opponent has a big pair in the hole, he’ll often assume he has the nuts with his full house and never fold.

He might be aware of the fact that we could hold Quads, but it will rarely be correct for him to make the laydown unless we are playing with very deep effective stacks.

Using two of our hole-cards to make Quads (i.e. two cards of identical rank on the board) is also an extremely valuable configuration in terms of implied odds since our Quads will be relatively disguised and we can frequently “cooler” any full houses our opponent may have.

It does, however, vary by board texture. One issue is that there will be two cards of identical rank on the board, and our opponent may give us credit for having trips even if he never suspects that we might have Quads.

For example, we are holding 55 on 5-5-4-2, and our opponent makes some laydowns, concerned we might hold a 5x. We also know that our opponent can never have a 5x himself due to card removal effects.

Holding 55 on a board texture like 5-5-6-6 is extremely valuable, however, since our opponent will never fold the 6x overfull.

Poker Four Of A Kind Probability Worksheets

In other words, the exact implied odds of Quads made with a pocket pair depends on the board structure.

Basic Strategy Advice

Quads are basically the nuts. Never fold Quads when 100bb deep (provided we use at least one of our hole cards in formulating the four of a kind).

If the four cards of equal rank are on the board itself, then we should proceed cautiously unless we specifically have the Ace. In such circumstances, we should usually look to continue betting for value even though it’s unlikely we’ll get paid off by worse.

Since this is an uncommon scenario, newer players may make call downs after incorrectly determining the strength of their hand. (For example, they might not realise that their QQ has been counterfeited on the KKKKx board and that any Ace will now win the pot.)