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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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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.

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?

Ridiculous Poker Hands

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

There are many spots these days in six-max no-limit hold'em where online poker players are turning hands far too strong into bluffs.

Most often, players should have played such hands more slowly at an earlier juncture, then from there they get themselves into a 'win the pot at all costs mentality' after having forsaken pot control — or impulse control (it is sometimes hard to distinguish).

Just as a general reminder, you should be making the biggest bluffs with the stone bottom of your range. Therefore, often, when it comes to absolute hand strength, when those bluffs get looked up, they will look potentially ridiculous.

Wow... what a preamble, you say, waiting on the edge of your seat for the example hand that is about to exemplify the concepts discussed above. Wait no more.

Three-Betting Preflop, Then Continuing Small

It started out as just another ordinary hand of 100NL 6-max. ($0.50/$1). They all glide by so casually, each indistinguishable from the next. How then, do players suddenly let loose and lose their mind and with that their money? Who is to say? But this hand began simply enough with the middle position player raising to $3.50 after under the gun folded.

It folded to a loose-aggressive recreational player in the small blind who was on a card rush — the kind of player you think you want to play against until they are putting you all in — who put in a three-bet to $9. The action returned to the initial raiser, playing $125, who just called. Rule out aces and kings.

The flop came . The small blind continuation bet $9 — an interesting size, since one would think most of the small blind's ace-king combinations would give up here, save perhaps three of the 16 that have backdoor flush draws.

In other words, small blind should not be betting that often on this texture, since the player who called the three-bet does not often fold for one bet. When someone has a lower betting frequency, you'll often see a bigger size. But instead he chose $9.

Things Get Interesting

There are some derisive terms sometimes used to describe recreational players going to war. Those terms will not be repeated here.

Suffice to say the turn was the and the small blind decided to bet huge — the exact pot, which was $35 and change after rake.

Ridiculous Poker Hands Clip Art

This bet seems really strangely advised, since the in-position player is more likely to have hands like -suited and -suited as well as blockers to ten-nine like pocket tens and nines. Indeed, one would imagine from a theory perspective, this bet size is a mistake.

If the small blind has pocket aces, he is about to hate life if any more money goes into the pot. Meanwhile if he has ace-king he is probably not getting quite enough folds as he would like. It's a fine line, and it is not that often that just jamming on the 'bet pot' button threads it.

The three-bet caller, the defender, the player on the back of his heels, called this pot-sized wager.

Cue Fireworks

With roughly $105 in the middle the river brought the , making the final board with a turned flush draw that didn't come in. Time for a small digression.

There is an argument to be made in favour of over representing ace-king when bluffing, in so many spots, because ace-king is the hand that all players put their opponents on most often, far more often than they should, when facing aggression.

Whether that argument applies here in this spot is hard to say. In small stakes games, we should focus on bluffing weakness, not strength. It is indisputable that the player here calling the preflop three-bet, the small flop bet, and the big turn bet has a strong range. Calling is stronger than betting, and becomes even stronger the bigger the pot gets. And by the time we reach the river in this exact hand, this is a very big pot.

In terms of bluffing candidates, our small blind kamikaze wants to have a hand that blocks , and therefore ace-king in general, but isn't so strong to have a pair that can win at showdown. is a value hand with a straight, and should check the flop to call more often than bomb away. Meanwhile, it is hard to think of a hand that could even justify three-betting preflop and then would not have a piece of this board with all its Broadway cards.

Cards Up

Have you put together where all this is going? On the river the small blind let loose with for $70, putting his opponent to the test and clearly shouting he was bluffing on the turn for pot with ace-king. I think that the size the small blind bet on the turn is a mistake, because after all, what hand is going to fold for $35 that won't fold for $25? And — to look at the other side of the coin — what hand will call $25 but fold for pot? Recall how so many one-pair hands like -suited, -suited, and picked up straight draws.

With that being said, seems an excellent candidate with which the small blind can empty the clip here. It checks three boxes. First, the river is scary, presenting a clear threat to his opponent's range. Second, it blocks his opponent's easiest calling hands. Third, it has absolutely, positively, no hope of winning when the action goes check-check on the river.

The recreational player in position had a tough spot to sort out with a hand as absolutely strong as . He of course called in the end. Even so, in this little corner of the internet, the small blind's heroism got recorded by history.

Ridiculous Poker Hands Games

^{This strategy article by Gareth Chantler for PokerNews is sponsored by partypoker.}

Ridiculous Poker Hands Game

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