Poker Starting Hand Probability Chart

카테고리 없음 2021. 8. 18. 01:09

Poker Hands Probability
Probabilities In Poker
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Poker Starting Hand Chart

Ever wondered where some of those odds in the odds charts came from? In this article, I will teach you how to work out the probability of being dealt different types of preflop hands in Texas Holdem.

The following is a passage from Wikipedia on starting hands probability. The 1,326 starting hands can be reduced for purposes of determining the probability of starting hands for Hold 'em—since suits have no relative value in poker, many of these hands are identical in value before the flop. Three-of-a-Kind is the next poker hand down on the poker rankings ladder. This hand often confuses some people. It can mean a “Set”, which is where you have a pair in your hand, like 7-7, matched with another 7 on the board. “Trips” is a pair on the board, like 5-5-4, with another 5 in your starting hand. A series of poker lessons providing general guidance and strategic advice on how to play certain starting hands in no-limit hold’em.

It's all pretty simple and you don't need to be a mathematician to work out the probabilities. I'll keep the math part as straightforward as I can to help keep this an easy-going article for the both of us.

Probability calculations quick links.

A few probability basics.

When working out hand probabilities, the main probabilities we will work with are the number of cards in the deck and the number of cards we want to be dealt. So for example, if we were going to deal out 1 card:

The probability of dealing a 7 would be 1/52 - There is one 7 in a deck of 52 cards.
The probability of dealing any Ace would be 4/52 - There four Aces in a deck of 52 cards.
The probability of dealing any would be 13/52 - There are 13 s in a deck of 52 cards.

In fact, the probability of being dealt any random card (not just the 7) would be 1/52. This also applies to the probability being dealt any random value of card like Kings, tens, fours, whatever (4/52) and the probability of being dealt any random suit (13/52).

Each card is just as likely to be dealt as any other - no special priorities in this game!

The numbers change for future cards.

A quick example... let's say we want to work out the probability of being dealt a pair of sevens.

The probability of being dealt a 7 for the first card will be 4/52.
The probability of being dealt a 7 for the second card will be 3/51.

Notice how the probability changes for the second card? After we have been dealt the first card, there is now 1 less card in the deck making it 51 cards in total. Also, after already being dealt a 7, there are now only three 7s left in the deck.

Always try and take care with the numbers for future cards. The numbers will change slightly as you go along.

Working out probabilities.

Whenever the word 'and' is used, it will usually mean multiply.
Whenever the word 'or' is used, it will usually mean add.

This won't make much sense for now, but it will make a lot of sense a little further on in the article. Trust me.

Probability of being dealt two exact cards.

Multiply the two probabilities together.

So, we want to find the probability of being dealt the A and K. (See the 'and' there?)

Probability of being dealt A - 1/52.
Probability of being dealt K - 1/51.

Now let's just multiply these bad boys together.

P = (1/52) * (1/51)
P = 1/2652

So the probability of being dealt the A and then K is 1/2652. As you might be able to work out, this is the same probability for any two exact cards, as the likelihood of being dealt A K is the same as being dealt a hand like 7 3 in that order.

But wait, we do not care about the order of the cards we are dealt!

When we are dealt a hand in Texas Hold'em, we don't care whether we get the A first or the K first (which is what we just worked out), just as long as we get them in our hand it's all the same. There are two possible combinations of being dealt this hand (A K and K A), so we simply multiply the probability by 2 to get a more useful probability.

P = 1/2652 * 2
P = 1/1326

You might notice that because of this, we have also worked out that there are 1,326 possible combinations of starting hands in Texas Holdem. Cool huh?

Probability of being dealt a certain hand.

Two exact cards is all well and good, but what if we want to work out the chances of being dealt AK, regardless of specific suits and whatnot? Well, we just do the same again...

Multiply the two probabilities together.

So, we want to find the probability of being dealt any Ace andany King.

Probability of being dealt any Ace - 4/52.
Probability of being dealt any King - 4/51 (after we've been dealt our Ace, there are now 51 cards left).

P = (4/52) * (4/51)
P = 16/2652 = 1/166

However, again with the 2652 number we are working out the probability of being deal an Ace and then a King. If we want the probability of being dealt either in any order, there are two possible ways to make this AK combination so we multiply the probability by 2.

P = 16/2652 * 2
P = 32/2652
P = 1/83

The probability of being dealt any AK as opposed to an AK with exact suits is more probable as we would expect. A lot more probable in fact. Also, as you might guess, this probability of 1/83 will be the same for any two value of cards like; AQ, JT, 34, J2 and so on regardless of whether they are suited or not.

Probability of being dealt a range of hands.

Work out each individual hand probability and add them together.

What's the probability of being dealt AA or KK? (Spot the 'or' there? - Time to add.)

Probability of being dealt AA - 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt KK - 1/221 (4/52 * 3/51 = 1/221).

P = (1/221) + (1/221)
P = 2/221 = 1/110

Easy enough. If you want to add more possible hands in to the range, just work out their individual probability and add them in. So if we wanted to work out the odds of being dealt AA, KK or 7 3...

Probability of being dealt AA - 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt KK - 1/221 (4/52 * 3/51 = 1/221).
Probability of being dealt 7 3 - 1/1326 ([1/52 * 1/51] * 2 = 1/1326).

Poker Hands Probability

P = (1/221) + (1/221) + (1/1326)
P = 359/36465 = 1/102

This one definitely takes more skill with adding fractions because of the different denominators, but you get the idea. I'm just teaching hand probabilities here, so I'm not going to go in to adding fractions in this article for now! This fractions calculator is really handy for adding those trickier probabilities quickly though.

Overview of working out hand probabilities.

Hopefully that's enough information and examples to allow you to go off and work out the probabilities of being dealt various hands and ranges of hands before the flop in Texas Holdem. The best way to learn how to work out probabilities is to actually try and work it out for yourself, otherwise the maths part will just go in one ear and out the other.

I guess this article isn't really going to do much for improving your game, but it's still pretty interesting to know the odds of being dealt different types of hands.

I'm sure that some of you reading this article were not aware that the probability of being dealt AA were exactly the same as the probability of being dealt 22! Well, now you know - it's 1/221.

Essentials[edit]

There are 1326 distinct possible combinations of two hole cards from a standard 52-card deck in hold 'em, but since suits have no relative value in this poker variant, many of these hands are identical in value before the flop. For example, A♥J♥ and A♠J♠ are identical in value, because each is a hand consisting of an ace and a jack of the same suit.

Therefore, there are 169 non-equivalent starting hands in hold 'em, which is the sum total of : 13 pocket pairs, 13 × 12 / 2 = 78 suited hands and 78 unsuited hands (13 + 78 + 78 = 169).

These 169 hands are not equally likely. Hold 'em hands are sometimes classified as having one of three 'shapes':

Pairs, (or 'pocket pairs'), which consist of two cards of the same rank (e.g. 9♠9♣). One hand in 17 will be a pair, each occurring with individual probability 1/221 (P(pair) = 3/51 = 1/17).

An alternative means of making this calculation

First Step As confirmed above.

There are 2652 possible combination of opening hand.

Second Step

There are 6 different combos of each pair. 9h9c, 9h9s, 9h9d, 9c9s, 9c9d, 9d9s

To calculate the odds of being dealt a pair

2652 (possible opening hands) divided by 12 (the number of any particular pair being dealt. As above)

2652/12 = 221

Suited hands, which contain two cards of the same suit (e.g. A♣6♣). Four hands out of 17 will be suited, and each suited configuration occurs with probability 2/663 (P(suited) = 12/51 = 4/17).

Offsuit hands, which contain two cards of a different suit and rank (e.g. K♠J♥). Twelve out of 17 hands will be nonpair, offsuit hands, each of which occurs with probability 2/221 (P(offsuit non-pair) = 3*(13-1)/51 = 12/17).

It is typical to abbreviate suited hands in hold 'em by affixing an 's' to the hand, as well as to abbreviate non-suited hands with an 'o' (for offsuit). That is,

QQ represents any pair of queens,

KQ represents any king and queen,

AKo represents any ace and king of different suits, and

JTs represents any jack and ten of the same suit.

There are 25 starting hands with a probability of winning at a 10-handed table of greater than 1/7.^[1]

Limit hand rankings[edit]

Some notable theorists and players have created systems to rank the value of starting hands in limit Texas hold'em. These rankings do not apply to no limit play.

Sklansky hand groups[edit]

David Sklansky and Mason Malmuth^[2] assigned in 1999 each hand to a group, and proposed all hands in the group could normally be played similarly. Stronger starting hands are identified by a lower number. Hands without a number are the weakest starting hands. As a general rule, books on Texas hold'em present hand strengths starting with the assumption of a nine or ten person table. The table below illustrates the concept:

Chen formula[edit]

The 'Chen Formula' is a way to compute the 'power ratings' of starting hands that was originally developed by Bill Chen.^[3]

Highest Card: Based on the highest card, assign points as follows:; Ace = 10 points, K = 8 points, Q = 7 points, J = 6 points.; 10 through 2, half of face value (10 = 5 points, 9 = 4.5 points, etc.)

Pairs: For pairs, multiply the points by 2 (AA=20, KK=16, etc.), with a minimum of 5 points for any pair. 55 is given an extra point (i.e., 6).

Suited: Add 2 points for suited cards.

Closeness: Subtract 1 point for 1 gappers (AQ, J9); 2 points for 2 gappers (J8, AJ).; 4 points for 3 gappers (J7, 73).; 5 points for larger gappers, including A2 A3 A4

Add an extra point if connected or 1-gap and your highest card is lower than Q (since you then can make all higher straights)

Phil Hellmuth's: 'Play Poker Like the Pros'[edit]

Probabilities In Poker

Phil Hellmuth's 'Play Poker Like the Pros' book published in 2003.

Tier	Hands	Category
1	AA, KK, AKs, QQ, AK	Top 12 Hands
2	JJ, TT, 99
3	88, 77, AQs, AQ
4	66, 55, 44, 33, 22, AJs, ATs, A9s, A8s	Majority Play Hands
5	A7s, A6s, A5s, A4s, A3s, A2s, KQs, KQ	Majority Play Hands
6	QJs, JTs, T9s, 98s, 87s, 76s, 65s	Suited Connectors

Statistics based on real online play[edit]

Statistics based on real play with their associated actual value in real bets.^[4]

Tier	Hands	Expected Value
1	AA, KK, QQ, JJ, AKs	2.32 - 0.78
2	AQs, TT, AK, AJs, KQs, 99	0.59 - 0.38
3	ATs, AQ, KJs, 88, KTs, QJs	0.32 - 0.20
4	A9s, AJ, QTs, KQ, 77, JTs	0.19 - 0.15
5	A8s, K9s, AT, A5s, A7s	0.10 - 0.08
6	KJ, 66, T9s, A4s, Q9s	0.08 - 0.05
7	J9s, QJ, A6s, 55, A3s, K8s, KT	0.04 - 0.01
8	98s, T8s, K7s, A2s	0.00
9	87s, QT, Q8s, 44, A9, J8s, 76s, JT	(-) 0.02 - 0.03

Nicknames for starting hands[edit]

In poker communities, it is common for hole cards to be given nicknames. While most combinations have a nickname, stronger handed nicknames are generally more recognized, the most notable probably being the 'Big Slick' - Ace and King of the same suit, although an Ace-King of any suit combination is less occasionally referred to as an Anna Kournikova, derived from the initials AK and because it 'looks really good but rarely wins.'^[5]^[6] Hands can be named according to their shapes (e.g., paired aces look like 'rockets', paired jacks look like 'fish hooks'); a historic event (e.g., A's and 8's - dead man's hand, representing the hand held by Wild Bill Hickok when he was fatally shot in the back by Jack McCall in 1876); many other reasons like animal names, alliteration and rhyming are also used in nicknames.

Poker Hands Probability Wiki

Notes[edit]