Probability Of Poker Hands Explained

24-05-2021

Probability Of Poker Hands Explained

Poker Hand Probabilities Explained
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Probability Of Poker Hands Explained
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Probabilities Of Poker Hands
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Before you can begin to calculate your poker odds you need to know your “outs”. An out is a card which will make your hand. For example, if you are on a flush draw with four hearts in your hand, then there will be nine hearts (outs) remaining in the deck to give you a flush. Learning odds will expand your poker IQ in a way that makes learning advanced strategies and theory much easier. But there’s a problem. Up until now, poker odds was only taught by a handful of pros and books, and most of the time it’s been explained in a way that’s too complex to understand. Dec 29, 2008 But, if you expect your opponent to call a bet or raise on the river if you make your hand, your implied odds are 6-1 or 7-1. A Simple Rule of Thumb for Hold'em and Omaha Every out gives you an approximate 4% chance of hitting on the turn and river combined.

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If you’ve ever watched live poker you may have wondered how TV programmes and live streams know the exact odds of every player at the table winning? Simply put, it’s by calculating outs and generating odds on the basis of them.

If you’re an amateur player or want to improve your game it’s imperative that you learn how to count your outs and calculate your odds of a winning hand.

Contents

What is a poker out?

A poker out is quite simply any unseen card that will improve a player’s hand and increase their chance of winning that round. If you want to progress beyond being a complete beginner you will have to know how many outs you have at the start of every hand.

Example

A hand with 4 clubs has 9 potential outs to make a flush. There are 13 clubs in a deck and 4 have been seen.

To learn outs you need to know every poker hand (pictured below) and how they rank against each other. Once you have committed these to memory you need to practice reading the table, working out if there are possible flushes or pairs.

As with all things, the more time you spend counting, the better you will be, and the simpler it will become.

How to count your outs

Calculating poker outs is basic maths, the more outs available to you the better the chances you have of winning.

Counting your odds is all about assessing which cards are left in the deck and what hands you can make with the remaining cards.

Example

If you have 2 diamonds and you get 2 on the flop, there are still 9 diamonds left in the deck. This means you have a total of 9 outs.

What are poker odds?

Poker is essentially a game of odds: every single hand you play, every card you receive, etc. has a big effect on your odds and the probability of winning that particular hand.

The odds are a player’s likelihood of winning a round with the cards in their possession.

How to calculate your odds

Poker Hand Probabilities Explained

Once you have mastered counting your outs then it is time to calculate what your chances are of the next card out of the deck being one you need. When it comes to odds there’s a simple formula you can use to get a decent estimation of what your chances are to improve your hand.

You multiply your outs by 3 and add 8 to the result. (Out * 3) + 8 =% chance of winning the hand (or at least getting the hand you wanted).

Example: if your current hand contains a pair of 3s and you want to make three of a kind then you know there are two 3s left in the deck, so you have two outs. You have two cards in your hand are 4 are visible on the table, from the turn and the flop. This means there are 46 cards left unseen.

Apply the formula: your chances of getting your 3 of a kind are: (2 x 3) + 8 = 14%.

While it may feel wrong, it’s important to ignore the possible cards your opponents are holding, as out and betting calculations are only judged by the cards you hold and what is exposed on the table.

What constitutes good odds

Learning what your odds are is just the first step, it’s equally important to know whether to call your bet, fold, or go all in.

Spotting good odds is simple, and the better the odds, the higher your chance of having a winning hand.

A good rule of thumb is:

Under 25% is risky and folding is your best bet.
Between 25% and 40% means it’s best to be cautious with your betting.
Over 40% give an excellent chance of winning, and going all in is a serious possibility.

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In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

Frequency of 5-card poker hands

The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)

The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields as above.

Hand	Frequency	Approx. Probability	Approx. Cumulative	Approx. Odds
Royal flush	4	0.000154%	0.000154%	649,739 : 1
Straight flush (excluding royal flush)	36	0.00139%	0.00154%	72,192.33 : 1
Four of a kind	624	0.0240%	0.0256%	4,164 : 1
Full house	3,744	0.144%	0.170%	693.2 : 1
Flush (excluding royal flush and straight flush)	5,108	0.197%	0.367%	507.8 : 1
Straight (excluding royal flush and straight flush)	10,200	0.392%	0.76%	253.8 : 1
Three of a kind	54,912	2.11%	2.87%	46.3 : 1
Two pair	123,552	4.75%	7.62%	20.03 : 1
One pair	1,098,240	42.3%	49.9%	1.36 : 1
No pair / High card	1,302,540	50.1%	100%	.995 : 1
Total	2,598,960	100%	100%	1 : 1

The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.

When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.

Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.

The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.

Derivation of frequencies of 5-card poker hands

of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

Straight flush — Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is:
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
  or simply . Note: this means that the total number of non-Royal straight flushes is 36.

Four of a kind — Any one of the thirteen ranks can form the four of a kind by selecting all four of the suits in that rank. The final card can have any one of the twelve remaining ranks, and any suit. 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. Thus, the total number of full houses is:

Flush — The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:

Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-10. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:

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Three of a kind — Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The remaining two cards can have any two of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of three-of-a-kinds is:

Two pair — The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:

Pair — The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is: