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Know the probability of the best poker hands

• Reference
• Probability Of Poker Hands
• Rules Of Card Games: Poker Hand Probabilities
• Cached

Then, the probability of each poker hand classification is simply its frequency divided by 2,598,960. The probabilities calculated below are based on drawing 5 cards from a shuffled poker deck. The likelihood of each type of hand determines its value. The less likely the hand, the more it is worth.

There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. Three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise. In Texas Hold'em, poker odds are THE probability tool you need as a poker player. In fact, you should always be thinking about poker odds - yours and your opponents' - when making decisions. In short, poker odds is the probability of you winning that hand, or the price it offers (pot odds). The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; =, ). For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is 4 / 2,598,960, or one in 649,740. Probabilities in Two-Player Texas Hold 'Em This page examines the probabilities of each final hand of an arbitrary player, referred to as player two, given the poker value of the hand of the other player, referred to as player one. Combinations shown are out of a possible combin (52,5)×combin (47,2)×combin (45,2) = 2,781,381,002,400.

Poker may take some number crunching, nerves of steel and a blank face but when it all comes down to it, poker is still a game of chance. That means that savvy players can figure out the probability of losing a hand, depending on what they’re holding. In a game like 5-card poker, in addition to poker tools these are the probabilities of you landing a winning hand:

Pair
You aren’t storming the gates with two of a kind, but it’s better than nothing. And having two Jacks, Queens, Kings or Aces will pack more than a little punch even though the probability of pulling a pair is 0.422569 – regardless of card value.

Two Pair
Everybody loves twins. Well maybe not poker players. Two pairs are pretty good, but not great and can sometimes lure you into a big pot, only to have the rug pulled out from underneath you. Probability of two pair, a sharp drop from a single pair to 0.047539.

Triple
Also known as “Three of a Kind”, this hand can get the job done if you’re playing at a smaller table. There are six hands that are going to beat the triple, so do a head check and bet accordingly. The probability of pulling out three of a kind drops significantly from two pair, to 0.021128.

Straight
High or low, the straight is a solid poker hand. But, much like the triple, you’ve got to size up your field and know the chances of an opponent pulling a better hand. The consecutive five-card sequence holds a probability of 0.00392465.

Reference

Flush
Getting five cards suited – but not in sequential order – holds a probability of 0.0019654, since there are just 13 cards of the same suit in a single 52-card deck. It’s a slightly behind the full house in terms of probability and rank, but stands well ahead of its cousin the straight.

Full House
Not just a great 80’s TV sitcom but also a tremendous poker hand. You’ve got three of a kind and managed to pull two pair along with it. Holding a probability of 0.001441, your opponents will need to pull out a miracle to beat the full house most hands.

Four of a kind
The mighty quintuplets are a Top-5 poker hand and if a miracle is needed to beat a full house, than you might need the Pope in your corner to top four of a kind, which holds a slim probability of 0.000240. If you’re not going all in with four on the floor, you’ve wasted a gift from the poker gods.

Straight Flush
You could play poker for years and years and never see this hand come up. It’s the unicorn of 5-card poker, with not only five cards suited but also those cards run in sequential order. The probability of a straight flush – excluding the Royal Flush – is 0.0000138517.

Royal Flush
This is the great white whale of poker hands. The odds of a common golfer scoring a hole-in-one are 12,500 to 1. The odds of pulling a Royal Flush are 649,739 to 1 or a probability of 0.00000153908. Luckily, for you the odds of getting hitting by lightning are only slightly longer at 960,000 to 1.

This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities

Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.

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Preliminary Calculation

Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.

These are the same hand. Order is not important.

The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.

The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.

Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is

This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.

The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.

If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.

Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.

Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:

One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.

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The Poker Hands

Here’s a ranking chart of the Poker hands.

The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.

Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.

The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.

Definitions of Poker Hands

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Probability Of Poker Hands

Counting Poker Hands

Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.

Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.

Rules Of Card Games: Poker Hand Probabilities

Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is

Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?

Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.

Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.

Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.

Two Pair and One Pair
These two are left as exercises.

High Card
The count is the complement that makes up 2,598,960.

The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.

Probabilities of Poker Hands

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2017 – Dan Ma