Poker Probability

[Lorence.xD]

Poker by far is my favorite when it comes to gambling, it is a good mixture of skill with the ability to bluff other players to the point that you can win a zilch even though he got a better hand, and luck by the probability of getting the better hand than your opponents. In this post, I will focus on the probability of getting a higher hand. The table below only presents 5-card hand poker game. If you are curious as to what is the math behind poker, hopefully this post will satisfy you.

|

Hand

|

Distinc Hands

|

Frequency

|

Probability

|

Cumulative Probability

|

Odds Against

|

|

Royal Flush

|

1

|

4

|

0.000154%

|

0.000154%

|

649,738:1

|

|

Straight Flush

|

9

|

36

|

0.00139%

|

0.0015%

|

72,192 1/3:1

|

|

Four of a Kind

|

156

|

624

|

0.0240%

|

0.0256%

|

4,156:1

|

|

Full House

|

156

|

3,744

|

0.1441%

|

0.17%

|

693.17:1

|

|

Flush

|

1,277

|

5,108

|

0.1965%

|

0.367%

|

508.8:1

|

|

Straight

|

10

|

10,200

|

0.3925%

|

0.76%

|

253.8:1

|

|

Three of a Kind

|

858

|

54,912

|

2.1128%

|

2.87%

|

46.33:1

|

|

Two Pair

|

858

|

123,552

|

4.7539%

|

7.62%

|

20:1

|

|

One Pair

|

2,860

|

1,098,240

|

42.2569%

|

49.9%

|

1.366:1

|

|

No Pair/High Card

|

1,277

|

1,302,540

|

50.1177%

|

100%

|

0.995:1

|

Additional Notes
Cumulative probability refers to the probability of drawing a hand as good as or better than the specified one. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand at least as good as three of a kind is about 2.87%. The cumulative probability is determined by adding one hand's probability with the probabilities of all hands above it.
Odds are defined as the ratio of the number of ways not to draw the hand, to the number of ways to draw it. In statistics, this is called odds against. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. The formula for establishing the odds can also be stated as (1/p) - 1 : 1, where p is the aforementioned probability.
Straight Flush is different from Royal Flush which is comprised of 10, Jack, Queen, King and Ace while the former is comprised of the other card numbers.
Flush is different from Straight Flush and Royal Flush where Straight Flush and Royal Flush has a condition that there is a order of cards of the same suit whereas Flush just needs to have the same suit.
Straight is different from Straight Flush and Royal Flush where both has the condition of Same suit and order for both hands, Straight on the other hand just needs to be in order even if not the same suit.

Source:https://en.m.wikipedia.org/wiki/Poker_probability

[1714806203]

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[1714806203]

1714806203

[swogerino]

Poker by far is my favorite when it comes to gambling, it is a good mixture of skill with the ability to bluff other players to the point that you can win a zilch even though he got a better hand, and luck by the probability of getting the better hand than your opponents. In this post, I will focus on the probability of getting a higher hand. The table below only presents 5-card hand poker game. If you are curious as to what is the math behind poker, hopefully this post will satisfy you.

Hand	Distinct Hands	Frequency	Probability	Cumulative Probability	Odd Against
Royal Flush	1	4	0.000154%	0.000154%	649,739:1
Straight Flush	9	36	0.00139%	0.0015%	72,192 1/3: 1
Four of a Kind	156	624	0.0240%	0.0256%	4,156:1
Full House	156	3,744	0.1441%	0.17%	693.17:1
Flush	1,277	5,108	0.1965%	0.367%	508.8:1
Straight	10	10,200	0.3925%	0.76%	253.8:1
Three of a Kind	858	54,912	2.1128%	2.87%	46.33:1
Two Pair	858	123,552	4.7539%	7.62%	20:1
One Pair	2,860	1,098,240	42.2569%	49.9%	1.366:1
No Pair/High Card	1,277	1,302,540	50.1177%	100%	0.995:1

Additional Notes
Cumulative probability refers to the probability of drawing a hand as good as or better than the specified one. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand at least as good as three of a kind is about 2.87%. The cumulative probability is determined by adding one hand's probability with the probabilities of all hands above it.
Odds are defined as the ratio of the number of ways not to draw the hand, to the number of ways to draw it. In statistics, this is called odds against. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. The formula for establishing the odds can also be stated as (1/p) - 1 : 1, where p is the aforementioned probability.
Straight Flush is different from Royal Flush which is comprised of 10, Jack, Queen, King and Ace while the former is comprised of the other card numbers.
Flush is different from Straight Flush and Royal Flush where Straight Flush and Royal Flush has a condition that there is a order of cards of the same suit whereas Flush just needs to have the same suit.
Straight is different from Straight Flush and Royal Flush where both has the condition of Same suit and order for both hands, Straight on the other hand just needs to be in order even if not the same suit.

This is a great example that chances to have 4 of a kind is really low.However it shows that the more down we come like one pair or two pairs chances are generally pretty high.Beside these statements which I think everyone should copy to a notepad or word document to keep always present during poker gameplays let’s also not forget that also skill and patience also play a big role if you want to be a successful poker player.

[Mauser]

Poker by far is my favorite when it comes to gambling, it is a good mixture of skill with the ability to bluff other players to the point that you can win a zilch even though he got a better hand, and luck by the probability of getting the better hand than your opponents. In this post, I will focus on the probability of getting a higher hand. The table below only presents 5-card hand poker game. If you are curious as to what is the math behind poker, hopefully this post will satisfy you.

Hand	Distinct Hands	Frequency	Probability	Cumulative Probability	Odd Against
Royal Flush	1	4	0.000154%	0.000154%	649,739:1
Straight Flush	9	36	0.00139%	0.0015%	72,192 1/3: 1
Four of a Kind	156	624	0.0240%	0.0256%	4,156:1
Full House	156	3,744	0.1441%	0.17%	693.17:1
Flush	1,277	5,108	0.1965%	0.367%	508.8:1
Straight	10	10,200	0.3925%	0.76%	253.8:1
Three of a Kind	858	54,912	2.1128%	2.87%	46.33:1
Two Pair	858	123,552	4.7539%	7.62%	20:1
One Pair	2,860	1,098,240	42.2569%	49.9%	1.366:1
No Pair/High Card	1,277	1,302,540	50.1177%	100%	0.995:1

Additional Notes
Cumulative probability refers to the probability of drawing a hand as good as or better than the specified one. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand at least as good as three of a kind is about 2.87%. The cumulative probability is determined by adding one hand's probability with the probabilities of all hands above it.
Odds are defined as the ratio of the number of ways not to draw the hand, to the number of ways to draw it. In statistics, this is called odds against. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. The formula for establishing the odds can also be stated as (1/p) - 1 : 1, where p is the aforementioned probability.
Straight Flush is different from Royal Flush which is comprised of 10, Jack, Queen, King and Ace while the former is comprised of the other card numbers.
Flush is different from Straight Flush and Royal Flush where Straight Flush and Royal Flush has a condition that there is a order of cards of the same suit whereas Flush just needs to have the same suit.
Straight is different from Straight Flush and Royal Flush where both has the condition of Same suit and order for both hands, Straight on the other hand just needs to be in order even if not the same suit.

Great summary of the chances to good poker hands. I played poker for a long time and never got a royal flush, I have seen a few and managed to get a few straight flushes over the years myself. 4 of a kind is the best in my opinion. If there are 3 times the same card on the board no one really thinks of a 4 of kind. Most people would expect someone to have a full house. These are the setups where you can earn a fortune in cash games. To make money when playing poker you want your opponent to have a good hand so he calls you down and then of course you need to have the better hand.

[crwth]

The math is pretty straight forward for sure. What deceives us is how people manage to see only the highlights of poker games and see the "legendary hands" and think that it's always going to be possible, but obviously, the math doesn't lie. It's just by luck to really get that strong hand.

Do you think the professional poker players memorized this? Does this increase the chances of determining the correct hands of the other players?

[Lorence.xD]

Do you think the professional poker players memorized this? Does this increase the chances of determining the correct hands of the other players?

In my opinion, professional players focus more on the skill side of the game more than the maths. The reason for that is that they do not have a control of the card stack, unless there is something underhanded happening then things will be different. So my answer to your question is no, they do not memorize this table but they do know about it for sure, they just don't pay it heed and I do not think it raises the chance of determining the other players hand except Negreneau, that dude seems to have x-ray vision.

[Wexnident]

Well, the math is pretty straightforward tbh. It's also rather easy to understand since the truth that numbers don't lie is oddly the truth in most luck-based games. Just that said numbers can go on and on until it meets the said odds, so it isn't really a helpful notion to rely on at times. Heck, it shouldn't even be used as a basis whenever playing since knowing doesn't really help in bringing those cards into your hand or into the table. Most players would rather learn how to bluff or learn how to decide when to fold or whatnot. Social Engineering/Human Engineering would be the term I'd use if I was asked what it is.

[Astvile]

This is why I never win playing with poker, bad odds, bad at bluffing ain't got no future for this game lol.

Do you think the professional poker players memorized this? Does this increase the chances of determining the correct hands of the other players?

Surely they don't memorize this in order to increase your chance of winning. I've watched some poker matches where the poorest hands are able to get the other player with better cards to fold with their bluffing skills. Even Kevin Hart did it succesfuly. The odds will be the same no matter how you memorize this table, it's a mix game of skill and luck as OP said, skills to bluff and luck for cards, that's what makes you win poker base on what I learn from watching videos and reading OPs post.

[kryptqnick]

Thanks for sharing this. Remembering the odds is very important to assess how good your hand is and how likely someone else is to have a better one. I found this data and while the probabilities are strikingly close, there's a tiny difference. Could you or someone else explain why?
I wondered some time ago whether the following improvement to poker websites could be implemented: each player sees the probability of her/his hand and the probability of someone else having a better one. Can this be properly counted, taking into account not the cards which are open on the table and those the player currently holds? Would it be useful?

[Cnut237]

Well, the math is pretty straightforward tbh.

I'd disagree with this. Whilst we can calculate the static odds of a certain hand, it is extremely difficult to calculate on-the-fly, in the middle of a game, what your chances are of getting a certain hand - and of other players getting certain hands. The people-reading is certainly a key part of being a successful player, but the ability to even approximate the odds in the middle of a game is a vital vital skill. I think I'm an okay poker player, not bad but not great, and I do try to estimate very rough odds whilst games are in progress - it is an extremely difficult challenge.

I wondered some time ago whether the following improvement to poker websites could be implemented: each player sees the probability of her/his hand and the probability of someone else having a better one. Can this be properly counted, taking into account not the cards which are open on the table and those the player currently holds? Would it be useful?

I think this is a cheat, really, I'd not be in favour as it takes away some of the player's expertise. It's like if there was also an on-screen counter showing running percentages of how often each player has bluffed.

[Reid]

This is why I never win playing with poker, bad odds, bad at bluffing ain't got no future for this game lol.

I'm also like that when I started.
Once you play more experience will teach you.
You can also copy how other players bluff. Somehow, it works too. It's a battle of non-blinking eyes with smiles of evil intent. 

OP, great post. Thanks for the numbers. I don't really rely much on this but it may help once in a while.
Will be keeping this in my notes. 

[Cnut237]

~

Not sure if people will find it useful, but I've expanded that table to include how likely each hand is compared to the immediately better hand (in the row above) and to all  better hands (all rows above).
We can see for example that the biggest jump in likelihood is Four of a Kind, which is x17 more likely than a Straight Flush, and x15.6 more likely than any better hand (Straight Flush or Royal Flush).
It's interesting to note that High Card is roughly as likely as all better hands... so everything being equal, if you have High Card and you're playing against a single opponent, it's roughly 50-50 whether he is also relying on High Card.

Hand    |	Distinct Hands |	Frequency |	Times more Likely than Above Hand |	Cum Freq |	Times more Likely than ALL Above Hands
Royal Flush	1	4		4
Straight Flush	9	36	9.0	40	9
Four of a Kind	156	624	17.3	664	15.6
Full House	156	3744	6.0	4408	5.6
Flush	1277	5108	1.4	9516	1.16
Straight	10	10200	2.0	19716	1.07
Three of a Kind	858	54912	5.4	74628	2.8
Two Pair	858	123552	2.3	198180	1.7
One Pair	2860	1098240	8.9	1296420	5.5
No Pair/High Card	1277	1302540	1.2	2598960	1.0

[ashmodeus]

to be honest, i've seen that table calculation since 2016 , and for sure, its not helping much. since its back to your mentality as well, low pair card got raise by 1/2 total chips, i just folded, eventhough i know chance for get 3 of kind or two pair is close already.

[acroman08]

Poker by far is my favorite when it comes to gambling, it is a good mixture of skill with the ability to bluff other players to the point that you can win a zilch even though he got a better hand, and luck by the probability of getting the better hand than your opponents.
-snip

This is why I feel poker is one the best card game out there. different people have different gameplay for what card they would get every round which what makes it exciting.

@OP I hope you don't mind or find it negative but I fixed the table to make it cleaner than the current one on the original post. you can just copy the table and post it on the OP. great Idea btw. showing the probability of getting a higher hand.

|

Hand

|

Distinc Hands

|

Frequency

|

Probability

|

Cumulative Probability

|

Odds Against

|

|

Royal Flush

|

1

|

4

|

0.000154%

|

0.000154%

|

649,738:1

|

|

Straight Flush

|

9

|

36

|

0.00139%

|

0.0015%

|

72,192 1/3:1

|

|

Four of a Kind

|

156

|

624

|

0.0240%

|

0.0256%

|

4,156:1

|

|

Full House

|

156

|

3,744

|

0.1441%

|

0.17%

|

693.17:1

|

|

Flush

|

1,277

|

5,108

|

0.1965%

|

0.367%

|

508.8:1

|

|

Straight

|

10

|

10,200

|

0.3925%

|

0.76%

|

253.8:1

|

|

Three of a Kind

|

858

|

54,912

|

2.1128%

|

2.87%

|

46.33:1

|

|

Two Pair

|

858

|

123,552

|

4.7539%

|

7.62%

|

20:1

|

|

One Pair

|

2,860

|

1,098,240

|

42.2569%

|

49.9%

|

1.366:1

|

|

No Pair/High Card

|

1,277

|

1,302,540

|

50.1177%

|

100%

|

0.995:1

|

[Lorence.xD]

~snip

Appreciate the help, wanted to that in the first place but I am new to doing tables in my posts. I edited it out.

This is why I never win playing with poker, bad odds, bad at bluffing ain't got no future for this game lol.

I'm also like that when I started.
Once you play more experience will teach you.
You can also copy how other players bluff. Somehow, it works too. It's a battle of non-blinking eyes with smiles of evil intent. 

OP, great post. Thanks for the numbers. I don't really rely much on this but it may help once in a while.
Will be keeping this in my notes. 

That is what I love about poker, besides the math behind the game. The battle of attrition when it comes to emotion is what makes this game exciting, it also helps you sharpen your people reading skill. You don't have to rely on this, my goal for this post is to show the probability behind my/our beloved game, even professional don't pay no heed so you can sure that you are not doing something wrong.

[chaser15]

Thanks for the share OP.

I learned poker and successfully improved my strategy as I progress but I'm not aware of that.

But yes, just to show the probability of numbers about this game.

Do you think the professional poker players memorized this? Does this increase the chances of determining the correct hands of the other players?

Even they somehow know the probability to win, the key to win is proper execution. That's why even with good cards, it's not an assurance of an easy win.

[Fundamentals Of]

Do you think the professional poker players memorized this? Does this increase the chances of determining the correct hands of the other players?

Even they somehow know the probability to win, the key to win is proper execution. That's why even with good cards, it's not an assurance of an easy win.

That's correct. Professional poker players do not just automatically fold if they have very poor hands. As always mentioned, poker games are not just about luck. If you are given 3 of diamonds and 9 of hearts, it doesn't mean you no longer have to play the game. But they must also have rough ideas as to their opponent cards.

[sunsilk]

I play poker but not competitively and I'm not a professional. But to see this stats about the cumulative probability, it makes sense as to why I always get the high card. From my experience, a kicker can change the result and can beat a confident player.

to be honest, i've seen that table calculation since 2016 , and for sure, its not helping much. since its back to your mentality as well, low pair card got raise by 1/2 total chips, i just folded, eventhough i know chance for get 3 of kind or two pair is close already.

It's one of the most exciting card to have especially if only two guys are remaining are on that phase. You can win with the weakest card that you have by simply bluffing, that's all. I mostly bluff if my cards are weak, it's fun yet there are also good players that you can't scare.

[crwth]

So my answer to your question is no, they do not memorize this table but they do know about it for sure, they just don't pay it heed and I do not think it raises the chance of determining the other players hand except Negreneau, that dude seems to have x-ray vision.

I saw his masterclass in an ad and it seems pretty convincing to learn how he does stuff. Knowing that he has achievements and titles to brag about, he is a very good player. I guess he just played so much that he knows instinctively what the other players "could" have. It's still a matter of experience for sure.

Surely they don't memorize this in order to increase your chance of winning. I've watched some poker matches where the poorest hands are able to get the other player with better cards to fold with their bluffing skills. Even Kevin Hart did it succesfuly. The odds will be the same no matter how you memorize this table, it's a mix game of skill and luck as OP said, skills to bluff and luck for cards, that's what makes you win poker base on what I learn from watching videos and reading OPs post.

That's the best and important skill that you can have, the best poker face in the world. Making you hard to read and never reveal your true emotion. It's possible that they know it in a different way, like the possible hands that players could have.

That's correct. Professional poker players do not just automatically fold if they have very poor hands. As always mentioned, poker games are not just about luck. If you are given 3 of diamonds and 9 of hearts, it doesn't mean you no longer have to play the game. But they must also have rough ideas as to their opponent cards.

That's just the "Fundamentals Of" the game. Lol. It includes the bluffing part and making them know that you have an amazing hand. Do your best in risk-taking, right? Show it and sell it.

[Beparanf]

Do you think the professional poker players memorized this? Does this increase the chances of determining the correct hands of the other players?

Even they somehow know the probability to win, the key to win is proper execution. That's why even with good cards, it's not an assurance of an easy win.

That's correct. Professional poker players do not just automatically fold if they have very poor hands. As always mentioned, poker games are not just about luck. If you are given 3 of diamonds and 9 of hearts, it doesn't mean you no longer have to play the game. But they must also have rough ideas as to their opponent cards.

That's true but it varies in different situation. Just like on Poker All Star table. They all know each other that they are all Pro, so bluffing to a master that knows the probability like Daniel and Phil Ivy will not gonna work because this guy surely eat alive the bluffer if they really holding the strong hand.

I know many user above considering the bluffing in poker because it is what always see during the highlights, But in reality, the chance that you can see a bluff is very rare in real poker scene because once you caught bluffing, You never use that again on same table or else you will burn. Most of the poker player memorized the probability that's why they can easily assess whether they will still call or not.

Try to watch full poker tourney. Pro usually fold when they have a very weak hands. They didn't even consider to bluff.  

[Yogee]

....
That's correct. Professional poker players do not just automatically fold if they have very poor hands. As always mentioned, poker games are not just about luck. If you are given 3 of diamonds and 9 of hearts, it doesn't mean you no longer have to play the game. But they must also have rough ideas as to their opponent cards.

Hehe this is not really the case when the minimum bet to start the round are quite high. I would probably agree with your statement if they will just call or check and nobody else raises but that rarely happens in a professional tournament does it? I've watched some live actions before and cards like that are usually folded in an instant.

....
Try to watch full poker tourney. Pro usually fold when they have a very weak hands. They didn't even consider to bluff. 

This.