Poker Predictor: a Free Texas Hold’em Odds and Probability Computer

The finest poker brains on your side!

In real life poker games many factors affects the game. A good poker player has to study  facial expression, breath frequency, hands movements etc. In online games you have no opponent to look  in the eyes,  the math behind poker takes a leading role to gain an edge against others players. To become a good online poker player you have  to know at least some basic poker theory.

First of all I want to say something about the relationship between math and poker. Poker is a cards game, this means luck counts, you wont win all the matches if you know all the odds and if you are a great bluffer.  To win, you also have to be a little luck and, fortunately, poker is still an human game, experience and skills are important too.

But what is important is to acquire a good gaming style that allows you to play correctly every time. It does not mind if you play correctly and you still loose. You may loose this game but you’ll see that on a large numbers of matches the edge you have against your opponent will turn into money.

Now how to acquire a mathematically correct good gaming style?

There are two options:

• You can study all the odds, read a thousand of poker game theory, make some quite complex calculations in your head each time a card falls on the table.
• You can Download Poker Predictor

What can Poker Predictor Do

Texas holdem poker is about:

• 2+5 cards each player
• 52 total number of cards
• 1326 starting hand configurations
• 207025 distinct head-to-head match ups
• 1712304 possible boards
• 690900 hands configurations with a single opponent
• over 6.22 e²⁶ with 9 opponents
• ……………..

All of them are in Poker Predictor

Poker Predictor is a tool to calculate poker games probabilities from whatever cards configuration. Probability are computed with random cards permutation, they are not the exact ones but the high simulations numbers ensure the error to be less than 1%.

Thanks to his strong computational engine, It can simulate 100′000 10 players Texas hold’em games in a matters of 0.3 seconds, so it is actually a real time tool, very useful for on line games, especially with high level players. It is possible to choose both players and opponents cards.

Although a little rough in the graphics (I am not a software developer, neither a programmer, just an engineer with poker passion) Poker predictor is a powerful tool, I am myself using it.

Parameters computed by poker predictor

OnLine Help for Poker Predictor

Downloads

Release History:

Poker Predictor v1.0 (24/02/2010) Download for Win 32
Poker Predictor v1.1 (09/03/2010) Download for Windows compatibility for win 64 systems 4 colors cards installer

This version is completely free. You can remove the ads on them with a 5 euros contribution to the project:

donations will be used to enlarge this utility

Poker Predictor was developed for Matlab at the beginning here you can find the matlab version: Matlab version of Poker Predictor

Poker predictor is always under development. Any question/suggestion? giaccariluigi@msn.com

Popularity: 5% [?]

Poker Predictor: How to Use it

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This page is some kind of poker predictor online help. Here you can find discussions about its game parameters. Suggestions are greatly appreciated, Poker Predictor is always under development.

Please go to Poker Predictor: How to Use it to view the survey

Profit

Each poker hand is just like an investment: you can loose the money you invest or you can make more money out of them. We need a parameter to know whether our cards are supposed to be a good investments or not.

Profit express the gain of money, or chips, you are expecting from the game. Basically are worth playing pre -flop cards with positive profit. This in the hypothesis you go to river all the hands. Obviously this is untrue. So you have to act like a “smart” player: if the game is very hard, full of rising players, you can play all hands with 10% (as example) profit, so you have more chance to “resist” to your opponents shoots. On the others side, if you meet soft players that always fold, even with low raises, you can play cards with negative profit (for example-10%)

Profit also take into account number of players,folds, Odds. For example folds decrease profit, since there will be less money on the pot. This may seems strange but it is true, folds increase your winning chance but actually decrease your capability to earn money.

Consider Profit is an instantaneous parameter which depends on the current cards configuration. There is also the profit of the entire game considering previous moves, but calculating this requires the history of the game and it is not supported.

Player and opponent higher rank

I suppose this need no explanation.

Winning chance

Probability to win after river. Notice that the probability suppose you go on until river card falls.

Tie chance

probability to tie after river.

Odds

the odds bookmakers would give to your game. When is odds useful? First of all, odds take into account both winning and tie chance. Ties are considered only partially because in that case the pot is divided among all the winners.

If the game is completely random, odds equal the number of players, this means every players has the same winning chance. So if odds overcomes players number this means your cards are worse than usual. On the other side if your odds is lower than the number of players your cards are good.

Odds are generally more useful than the winning chance. Let’s make an example: you have 30% of winning. Is that good?

The right answer id I don’t know, it depends on the number of players. If there are only two players 30% is bad winning chance but if we are 12 players of course is good. How do I find the limit? what is the number of players for which 30% winning is a good chance?

Well, you don’t have to get tired everytime on this computation, you can bypass them just by looking the odds number: is lower than the player number than is good, is higher than is bad. Odds and winning chance are just two different ways to estimate how  strong are your cards.

If you have zero winning chance your odds equals infinity, bookmakers will pay  a lot of money in case you win, but this just because you can not win. If odd equals one you are sure to win. The lower is the Odds the higher are you’re winning chance.

MaxBet/Pot

This is the answer to question call/fold when an opponent make raise. This is max bet, compared with the pot, you can do remaining in the mathematical correctness of the game. This means that if you can afford to bet the 30% of the pot you can call any raise which respect this condition. If this parameter overlaps 100% you can bet all the pot, you have more than 50% of winning and you are (theoretically) allowed to go all in, it is you or your opponent. Generally listening to this data carefully lead to run too much risk, the all in in poker may be without return !
What I suggest is to find to find a good feeling with the values of this parameter that fits your gaming style.

Consider that the pot you see when you call may increase during the game, this can not be calculated by the program, expert players must also consider this factor.

Any questions, suggestions?

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Popularity: 3% [?]

Poker Predictor: Matlab Version

The finest poker brains on your side!

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What is Texas Hold’em Poker:

• 2+5 cards each player
• 52 total number of cards
• 1326 starting hand configurations
• 207025 distinct head-to-head match ups
• 1712304 possible boards
• 690900 hands configurations with a single opponent
• over 6.22 e²⁶ with 9 opponents
• ……………..

All of them are in Poker Predictor: a Texas Hold’ em Probability and Odds Calculator.

Please go to Poker Predictor: Matlab Version to view the survey

What can Poker Predictor Do:

Poker Predictor is a Matlab tool to calculate poker games probabilities from whatever cards configuration. Probability are computed with random cards permutation, they are not the exact ones but the high simulations numbers ensure the error to be less than 1%. Thanks to his strong computational engine, It can simulate 100′000 10 players Texas hold’em games in a matters of 0.3 seconds, so it is actually a real time tool, very useful for on line games, especially with high level players. It is possible to choose both players and opponents cards.

Although a little rough in the graphics (I am not a software developer, neither a programmer, just an engineer with poker passion) Poker predictor is a powerful tool, I am myself using it. Here there is graphical explanation:

Collaborators Wanted:

I’d like to compile Poker predictor on different platforms. If you want to join the project your help will be greatly appreciated. I need people with the following requirements:

• Different platform than than Win32, with Matlab installed.
• Capable of building an external Matlab application.

Briefly I need to build an external Matlab application (MCR compiler included), I ‘ll give you all required files, if you are interested contact me at giaccariluigi@msn.com.

Extensions:

Maybe you need something different, look at this extensions or propose something yourself! Need a :

• different languages
• different cards design/colors
• different poker games
• poker tables
• others functionalities
• explanations on something unclear Propose !!!!!!!!!! your request may sounds very interesting, send everything you want on my e-mail.

This work is free thanks to users gratitude. If you find it useful please consider making a donation. Help me developing  future versions.

No need to tell you good luck if you have poker predictor on your side !

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Popularity: 57% [?]

Pre-Flop Texas Hold’em Poker Tables

This is quite a long post !!!

The first question that arise in a poker game is play/fold. There are many factors a good player should take into account like:

• playing order
• relative amount of chips of others players
• amount of chips on table
• psychology of the game, are my opponents currently happy, angry…

But one of the first important is sure the power of your cards. Every pro-player knows odds and winning chance for almost all pre flop cards. You can compare the odds of your first two cards and decide whether play or not. Generally are worth playing cards with odds lower than the number of opponents, but depending on your game style you can modify your strategy. Fortunately math can help, but poker is still an human game.

Here is shown a Matlab script that, using the Poker Predictor engine, builds html tables for pre-flop winning chances and odds from 2 to 10 players. Notice that to run this script you need prior download the Poker Predictor Matlab package. It also use a slightly modified version of Gus Brown Html Table Printer (this one is included).

Probability are referred to unknown opponents cards and to the river. Consider that often the game ends before the river due to folds. This is human factor no math can help!!! More they pretend all players go into the game and this is generally untruth, but we also have to consider that pre flop folds are often caused by bad cards.

These tables help you know a lot of little tips and tricks, for example did you that AA wins at 85% in 2 players game but only in 31% in 10 players game.

Downloads:

• Tables Printer
• Poker Predictor package (don’t forget to mex files!)

Join the Poker Predictor Facebook Page

2 Players Game

Pre Flop Winning Chance Percentage UnSuited Cards 2 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	50.46	32.41	33.30	34.28	34.05	34.80	36.81	39.28	41.76	44.34	47.23	50.55	54.80
3	32.41	53.53	35.30	36.45	36.08	36.65	37.45	39.97	42.54	45.39	48.16	51.58	55.82
4	33.30	35.30	57.28	38.17	38.00	38.58	39.56	40.79	43.49	46.32	49.17	52.38	56.79
5	34.28	36.45	38.17	60.31	39.89	40.42	41.51	42.69	44.19	47.00	49.99	53.23	57.62
6	34.05	36.08	38.00	39.89	63.25	42.20	43.34	44.67	46.10	47.93	51.03	54.27	57.67
7	34.80	36.65	38.58	40.42	42.20	66.17	44.93	46.38	47.89	49.84	51.83	55.31	58.89
8	36.81	37.45	39.56	41.51	43.34	44.93	69.10	48.08	49.61	51.79	53.63	56.02	59.82
9	39.28	39.97	40.79	42.69	44.67	46.38	48.08	72.14	51.45	53.24	55.32	57.86	60.60
10	41.76	42.54	43.49	44.19	46.10	47.89	49.61	51.45	75.07	55.28	57.35	59.86	62.55
J	44.34	45.39	46.32	47.00	47.93	49.84	51.79	53.24	55.28	77.41	58.24	60.61	63.44
Q	47.23	48.16	49.17	49.99	51.03	51.83	53.63	55.32	57.35	58.24	79.91	61.51	64.42
K	50.55	51.58	52.38	53.23	54.27	55.31	56.02	57.86	59.86	60.61	61.51	82.28	65.24
A	54.80	55.82	56.79	57.62	57.67	58.89	59.82	60.60	62.55	63.44	64.42	65.24	85.15

Pre Flop winning Chance Percentage Suited Cards 2 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	36.06	36.98	38.01	37.53	38.11	40.18	42.40	45.10	47.75	50.07	53.32	57.61
3	36.06	NaN	38.60	39.73	39.68	40.07	41.04	43.45	45.83	48.37	51.19	54.03	58.26
4	36.98	38.60	NaN	41.20	40.98	41.66	42.38	43.70	46.27	49.10	51.93	54.92	59.13
5	38.01	39.73	41.20	NaN	43.25	43.72	44.53	45.64	47.34	50.01	52.81	55.98	60.02
6	37.53	39.68	40.98	43.25	NaN	45.40	46.21	47.56	49.18	50.91	53.76	56.88	60.08
7	38.11	40.07	41.66	43.72	45.40	NaN	47.94	49.23	50.77	52.61	54.37	57.75	61.04
8	40.18	41.04	42.38	44.53	46.21	47.94	NaN	50.69	52.39	54.12	56.12	58.34	62.07
9	42.40	43.45	43.70	45.64	47.56	49.23	50.69	NaN	53.96	55.53	57.41	59.81	62.59
10	45.10	45.83	46.27	47.34	49.18	50.77	52.39	53.96	NaN	57.23	59.31	61.72	64.63
J	47.75	48.37	49.10	50.01	50.91	52.61	54.12	55.53	57.23	NaN	60.37	62.78	65.69
Q	50.07	51.19	51.93	52.81	53.76	54.37	56.12	57.41	59.31	60.37	NaN	63.46	66.36
K	53.32	54.03	54.92	55.98	56.88	57.75	58.34	59.81	61.72	62.78	63.46	NaN	67.22
A	57.61	58.26	59.13	60.02	60.08	61.04	62.07	62.59	64.63	65.69	66.36	67.22	NaN

Pre Flop winning Odds Suited Cards 2 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	2.77	2.70	2.63	2.66	2.62	2.49	2.36	2.22	2.09	2.00	1.88	1.74
3	2.77	NaN	2.59	2.52	2.52	2.50	2.44	2.30	2.18	2.07	1.95	1.85	1.72
4	2.70	2.59	NaN	2.43	2.44	2.40	2.36	2.29	2.16	2.04	1.93	1.82	1.69
5	2.63	2.52	2.43	NaN	2.31	2.29	2.25	2.19	2.11	2.00	1.89	1.79	1.67
6	2.66	2.52	2.44	2.31	NaN	2.20	2.16	2.10	2.03	1.96	1.86	1.76	1.66
7	2.62	2.50	2.40	2.29	2.20	NaN	2.09	2.03	1.97	1.90	1.84	1.73	1.64
8	2.49	2.44	2.36	2.25	2.16	2.09	NaN	1.97	1.91	1.85	1.78	1.71	1.61
9	2.36	2.30	2.29	2.19	2.10	2.03	1.97	NaN	1.85	1.80	1.74	1.67	1.60
10	2.22	2.18	2.16	2.11	2.03	1.97	1.91	1.85	NaN	1.75	1.69	1.62	1.55
J	2.09	2.07	2.04	2.00	1.96	1.90	1.85	1.80	1.75	NaN	1.66	1.59	1.52
Q	2.00	1.95	1.93	1.89	1.86	1.84	1.78	1.74	1.69	1.66	NaN	1.58	1.51
K	1.88	1.85	1.82	1.79	1.76	1.73	1.71	1.67	1.62	1.59	1.58	NaN	1.49
A	1.74	1.72	1.69	1.67	1.66	1.64	1.61	1.60	1.55	1.52	1.51	1.49	NaN

Pre Flop winning Odds UnSuited Cards 2 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	1.98	3.09	3.00	2.92	2.94	2.87	2.72	2.55	2.39	2.26	2.12	1.98	1.82
3	3.09	1.87	2.83	2.74	2.77	2.73	2.67	2.50	2.35	2.20	2.08	1.94	1.79
4	3.00	2.83	1.75	2.62	2.63	2.59	2.53	2.45	2.30	2.16	2.03	1.91	1.76
5	2.92	2.74	2.62	1.66	2.51	2.47	2.41	2.34	2.26	2.13	2.00	1.88	1.74
6	2.94	2.77	2.63	2.51	1.58	2.37	2.31	2.24	2.17	2.09	1.96	1.84	1.73
7	2.87	2.73	2.59	2.47	2.37	1.51	2.23	2.16	2.09	2.01	1.93	1.81	1.70
8	2.72	2.67	2.53	2.41	2.31	2.23	1.45	2.08	2.02	1.93	1.86	1.78	1.67
9	2.55	2.50	2.45	2.34	2.24	2.16	2.08	1.39	1.94	1.88	1.81	1.73	1.65
10	2.39	2.35	2.30	2.26	2.17	2.09	2.02	1.94	1.33	1.81	1.74	1.67	1.60
J	2.26	2.20	2.16	2.13	2.09	2.01	1.93	1.88	1.81	1.29	1.72	1.65	1.58
Q	2.12	2.08	2.03	2.00	1.96	1.93	1.86	1.81	1.74	1.72	1.25	1.63	1.55
K	1.98	1.94	1.91	1.88	1.84	1.81	1.78	1.73	1.67	1.65	1.63	1.22	1.53
A	1.82	1.79	1.76	1.74	1.73	1.70	1.67	1.65	1.60	1.58	1.55	1.53	1.17

3 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 3 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	30.65	19.66	20.65	21.44	20.63	20.57	21.63	23.07	24.81	26.55	28.69	31.20	35.14
3	19.66	33.71	22.52	23.45	22.70	22.55	22.48	23.76	25.48	27.47	29.44	31.99	35.84
4	20.65	22.52	37.02	25.50	25.07	24.67	24.46	24.44	26.59	28.22	30.35	33.09	37.39
5	21.44	23.45	25.50	40.16	26.67	26.43	26.28	26.54	27.26	29.27	31.16	34.05	38.49
6	20.63	22.70	25.07	26.67	43.46	28.65	28.44	28.93	29.29	29.76	32.11	34.84	38.10
7	20.57	22.55	24.67	26.43	28.65	46.25	30.33	30.38	31.22	31.62	32.78	35.93	39.21
8	21.63	22.48	24.46	26.28	28.44	30.33	50.13	32.60	33.32	33.90	35.40	37.22	40.60
9	23.07	23.76	24.44	26.54	28.93	30.38	32.60	53.45	35.69	36.34	37.75	39.07	41.52
10	24.81	25.48	26.59	27.26	29.29	31.22	33.32	35.69	57.46	39.05	40.33	41.89	44.25
J	26.55	27.47	28.22	29.27	29.76	31.62	33.90	36.34	39.05	61.04	41.60	43.16	45.37
Q	28.69	29.44	30.35	31.16	32.11	32.78	35.40	37.75	40.33	41.60	64.75	44.40	46.76
K	31.20	31.99	33.09	34.05	34.84	35.93	37.22	39.07	41.89	43.16	44.40	68.64	48.10
A	35.14	35.84	37.39	38.49	38.10	39.21	40.60	41.52	44.25	45.37	46.76	48.10	73.43

Pre Flop winning Chance Percentage Suited Cards 3 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	23.94	24.62	25.38	24.86	24.56	25.98	27.28	28.71	30.51	32.54	35.04	38.76
3	23.94	NaN	26.62	27.21	26.62	26.28	26.43	27.82	29.44	31.20	33.02	35.64	39.67
4	24.62	26.62	NaN	29.17	28.69	28.26	28.18	28.32	29.93	31.92	33.91	36.12	40.64
5	25.38	27.21	29.17	NaN	29.96	30.05	30.21	30.24	30.79	32.51	34.93	37.30	41.29
6	24.86	26.62	28.69	29.96	NaN	31.80	31.71	32.03	32.66	33.30	35.51	38.32	41.08
7	24.56	26.28	28.26	30.05	31.80	NaN	33.82	33.99	34.79	35.13	36.37	39.12	42.14
8	25.98	26.43	28.18	30.21	31.71	33.82	NaN	36.06	36.79	37.34	38.59	40.35	43.52
9	27.28	27.82	28.32	30.24	32.03	33.99	36.06	NaN	38.72	39.65	40.76	42.48	44.57
10	28.71	29.44	29.93	30.79	32.66	34.79	36.79	38.72	NaN	41.87	43.26	44.84	47.14
J	30.51	31.20	31.92	32.51	33.30	35.13	37.34	39.65	41.87	NaN	44.23	45.92	48.19
Q	32.54	33.02	33.91	34.93	35.51	36.37	38.59	40.76	43.26	44.23	NaN	47.15	49.25
K	35.04	35.64	36.12	37.30	38.32	39.12	40.35	42.48	44.84	45.92	47.15	NaN	50.62
A	38.76	39.67	40.64	41.29	41.08	42.14	43.52	44.57	47.14	48.19	49.25	50.62	NaN

Pre Flop winning Odds Suited Cards 3 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	4.18	4.06	3.94	4.02	4.07	3.85	3.67	3.48	3.28	3.07	2.85	2.58
3	4.18	NaN	3.76	3.67	3.76	3.81	3.78	3.59	3.40	3.21	3.03	2.81	2.52
4	4.06	3.76	NaN	3.43	3.49	3.54	3.55	3.53	3.34	3.13	2.95	2.77	2.46
5	3.94	3.67	3.43	NaN	3.34	3.33	3.31	3.31	3.25	3.08	2.86	2.68	2.42
6	4.02	3.76	3.49	3.34	NaN	3.14	3.15	3.12	3.06	3.00	2.82	2.61	2.43
7	4.07	3.81	3.54	3.33	3.14	NaN	2.96	2.94	2.87	2.85	2.75	2.56	2.37
8	3.85	3.78	3.55	3.31	3.15	2.96	NaN	2.77	2.72	2.68	2.59	2.48	2.30
9	3.67	3.59	3.53	3.31	3.12	2.94	2.77	NaN	2.58	2.52	2.45	2.35	2.24
10	3.48	3.40	3.34	3.25	3.06	2.87	2.72	2.58	NaN	2.39	2.31	2.23	2.12
J	3.28	3.21	3.13	3.08	3.00	2.85	2.68	2.52	2.39	NaN	2.26	2.18	2.07
Q	3.07	3.03	2.95	2.86	2.82	2.75	2.59	2.45	2.31	2.26	NaN	2.12	2.03
K	2.85	2.81	2.77	2.68	2.61	2.56	2.48	2.35	2.23	2.18	2.12	NaN	1.98
A	2.58	2.52	2.46	2.42	2.43	2.37	2.30	2.24	2.12	2.07	2.03	1.98	NaN

Pre Flop winning Odds UnSuited Cards 3 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	3.26	5.09	4.84	4.66	4.85	4.86	4.62	4.33	4.03	3.77	3.49	3.20	2.85
3	5.09	2.97	4.44	4.26	4.41	4.43	4.45	4.21	3.92	3.64	3.40	3.13	2.79
4	4.84	4.44	2.70	3.92	3.99	4.05	4.09	4.09	3.76	3.54	3.30	3.02	2.67
5	4.66	4.26	3.92	2.49	3.75	3.78	3.80	3.77	3.67	3.42	3.21	2.94	2.60
6	4.85	4.41	3.99	3.75	2.30	3.49	3.52	3.46	3.41	3.36	3.11	2.87	2.62
7	4.86	4.43	4.05	3.78	3.49	2.16	3.30	3.29	3.20	3.16	3.05	2.78	2.55
8	4.62	4.45	4.09	3.80	3.52	3.30	2.00	3.07	3.00	2.95	2.83	2.69	2.46
9	4.33	4.21	4.09	3.77	3.46	3.29	3.07	1.87	2.80	2.75	2.65	2.56	2.41
10	4.03	3.92	3.76	3.67	3.41	3.20	3.00	2.80	1.74	2.56	2.48	2.39	2.26
J	3.77	3.64	3.54	3.42	3.36	3.16	2.95	2.75	2.56	1.64	2.40	2.32	2.20
Q	3.49	3.40	3.30	3.21	3.11	3.05	2.83	2.65	2.48	2.40	1.54	2.25	2.14
K	3.20	3.13	3.02	2.94	2.87	2.78	2.69	2.56	2.39	2.32	2.25	1.46	2.08
A	2.85	2.79	2.67	2.60	2.62	2.55	2.46	2.41	2.26	2.20	2.14	2.08	1.36

4 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 4 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	21.86	14.02	14.89	15.40	14.67	14.25	15.14	15.99	17.31	18.61	20.04	22.30	25.36
3	14.02	23.91	16.43	17.27	16.25	16.02	15.70	16.91	17.81	19.19	20.77	22.83	26.28
4	14.89	16.43	26.26	18.92	18.18	17.99	17.50	17.42	18.89	19.87	21.61	23.48	27.02
5	15.40	17.27	18.92	28.82	19.83	19.63	19.38	19.19	19.19	20.71	22.52	24.34	28.27
6	14.67	16.25	18.18	19.83	31.53	21.45	21.32	21.08	21.36	21.68	23.23	25.20	27.42
7	14.25	16.02	17.99	19.63	21.45	34.33	23.17	23.27	23.33	23.11	23.90	26.15	28.55
8	15.14	15.70	17.50	19.38	21.32	23.17	37.61	24.99	25.42	25.43	26.18	27.21	30.02
9	15.99	16.91	17.42	19.19	21.08	23.27	24.99	41.11	27.44	27.73	28.46	29.47	31.04
10	17.31	17.81	18.89	19.19	21.36	23.33	25.42	27.44	45.19	30.78	31.38	32.42	33.89
J	18.61	19.19	19.87	20.71	21.68	23.11	25.43	27.73	30.78	49.26	32.75	33.80	35.48
Q	20.04	20.77	21.61	22.52	23.23	23.90	26.18	28.46	31.38	32.75	53.70	35.32	37.13
K	22.30	22.83	23.48	24.34	25.20	26.15	27.21	29.47	32.42	33.80	35.32	58.37	38.63
A	25.36	26.28	27.02	28.27	27.42	28.55	30.02	31.04	33.89	35.48	37.13	38.63	63.95

Pre Flop winning Chance Percentage Suited Cards 4 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	18.06	18.85	19.47	18.68	18.19	19.41	20.18	21.51	22.66	24.23	26.19	29.25
3	18.06	NaN	20.46	21.18	20.36	20.00	19.70	20.68	21.95	23.05	24.83	26.63	29.90
4	18.85	20.46	NaN	22.68	22.20	21.63	21.37	21.42	22.84	24.02	25.50	27.42	30.88
5	19.47	21.18	22.68	NaN	23.55	23.26	23.04	23.07	23.13	24.75	26.07	28.37	31.82
6	18.68	20.36	22.20	23.55	NaN	25.18	25.11	24.99	25.14	25.38	27.36	29.22	31.40
7	18.19	20.00	21.63	23.26	25.18	NaN	26.68	26.59	26.88	27.16	27.86	29.78	32.61
8	19.41	19.70	21.37	23.04	25.11	26.68	NaN	28.42	28.78	29.10	29.78	30.93	33.27
9	20.18	20.68	21.42	23.07	24.99	26.59	28.42	NaN	31.03	31.38	31.96	33.16	34.50
10	21.51	21.95	22.84	23.13	25.14	26.88	28.78	31.03	NaN	33.80	34.48	35.50	37.19
J	22.66	23.05	24.02	24.75	25.38	27.16	29.10	31.38	33.80	NaN	35.61	36.81	38.62
Q	24.23	24.83	25.50	26.07	27.36	27.86	29.78	31.96	34.48	35.61	NaN	37.99	39.87
K	26.19	26.63	27.42	28.37	29.22	29.78	30.93	33.16	35.50	36.81	37.99	NaN	41.27
A	29.25	29.90	30.88	31.82	31.40	32.61	33.27	34.50	37.19	38.62	39.87	41.27	NaN

Pre Flop winning Odds Suited Cards 4 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	5.54	5.31	5.14	5.35	5.50	5.15	4.96	4.65	4.41	4.13	3.82	3.42
3	5.54	NaN	4.89	4.72	4.91	5.00	5.08	4.84	4.56	4.34	4.03	3.76	3.34
4	5.31	4.89	NaN	4.41	4.50	4.62	4.68	4.67	4.38	4.16	3.92	3.65	3.24
5	5.14	4.72	4.41	NaN	4.25	4.30	4.34	4.33	4.32	4.04	3.84	3.52	3.14
6	5.35	4.91	4.50	4.25	NaN	3.97	3.98	4.00	3.98	3.94	3.66	3.42	3.18
7	5.50	5.00	4.62	4.30	3.97	NaN	3.75	3.76	3.72	3.68	3.59	3.36	3.07
8	5.15	5.08	4.68	4.34	3.98	3.75	NaN	3.52	3.47	3.44	3.36	3.23	3.01
9	4.96	4.84	4.67	4.33	4.00	3.76	3.52	NaN	3.22	3.19	3.13	3.02	2.90
10	4.65	4.56	4.38	4.32	3.98	3.72	3.47	3.22	NaN	2.96	2.90	2.82	2.69
J	4.41	4.34	4.16	4.04	3.94	3.68	3.44	3.19	2.96	NaN	2.81	2.72	2.59
Q	4.13	4.03	3.92	3.84	3.66	3.59	3.36	3.13	2.90	2.81	NaN	2.63	2.51
K	3.82	3.76	3.65	3.52	3.42	3.36	3.23	3.02	2.82	2.72	2.63	NaN	2.42
A	3.42	3.34	3.24	3.14	3.18	3.07	3.01	2.90	2.69	2.59	2.51	2.42	NaN

Pre Flop winning Odds UnSuited Cards 4 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	4.58	7.13	6.72	6.50	6.82	7.02	6.61	6.26	5.78	5.37	4.99	4.49	3.94
3	7.13	4.18	6.09	5.79	6.15	6.24	6.37	5.91	5.61	5.21	4.81	4.38	3.81
4	6.72	6.09	3.81	5.29	5.50	5.56	5.71	5.74	5.29	5.03	4.63	4.26	3.70
5	6.50	5.79	5.29	3.47	5.04	5.09	5.16	5.21	5.21	4.83	4.44	4.11	3.54
6	6.82	6.15	5.50	5.04	3.17	4.66	4.69	4.74	4.68	4.61	4.31	3.97	3.65
7	7.02	6.24	5.56	5.09	4.66	2.91	4.32	4.30	4.29	4.33	4.18	3.82	3.50
8	6.61	6.37	5.71	5.16	4.69	4.32	2.66	4.00	3.93	3.93	3.82	3.68	3.33
9	6.26	5.91	5.74	5.21	4.74	4.30	4.00	2.43	3.64	3.61	3.51	3.39	3.22
10	5.78	5.61	5.29	5.21	4.68	4.29	3.93	3.64	2.21	3.25	3.19	3.08	2.95
J	5.37	5.21	5.03	4.83	4.61	4.33	3.93	3.61	3.25	2.03	3.05	2.96	2.82
Q	4.99	4.81	4.63	4.44	4.31	4.18	3.82	3.51	3.19	3.05	1.86	2.83	2.69
K	4.49	4.38	4.26	4.11	3.97	3.82	3.68	3.39	3.08	2.96	2.83	1.71	2.59
A	3.94	3.81	3.70	3.54	3.65	3.50	3.33	3.22	2.95	2.82	2.69	2.59	1.56

5 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 5 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	17.89	10.78	11.60	12.02	11.03	10.64	11.37	12.02	12.95	14.08	15.27	17.02	19.93
3	10.78	18.90	13.02	13.73	12.91	12.30	11.89	12.78	13.76	14.88	16.14	17.98	20.78
4	11.60	13.02	20.62	15.21	14.44	13.93	13.64	13.17	14.34	15.21	16.60	18.42	21.05
5	12.02	13.73	15.21	22.34	15.96	15.74	14.97	14.78	14.66	15.93	17.06	18.87	21.92
6	11.03	12.91	14.44	15.96	24.66	17.03	16.82	16.53	16.60	16.41	17.76	19.43	21.34
7	10.64	12.30	13.93	15.74	17.03	26.70	18.27	18.39	18.27	18.28	18.57	20.45	22.47
8	11.37	11.89	13.64	14.97	16.82	18.27	29.68	20.15	20.45	20.60	20.76	21.35	23.52
9	12.02	12.78	13.17	14.78	16.53	18.39	20.15	32.99	22.84	22.76	23.08	23.61	24.27
10	12.95	13.76	14.34	14.66	16.60	18.27	20.45	22.84	36.29	25.08	25.81	26.58	27.47
J	14.08	14.88	15.21	15.93	16.41	18.28	20.60	22.76	25.08	40.28	27.02	27.74	28.82
Q	15.27	16.14	16.60	17.06	17.76	18.57	20.76	23.08	25.81	27.02	44.83	29.22	30.42
K	17.02	17.98	18.42	18.87	19.43	20.45	21.35	23.61	26.58	27.74	29.22	49.59	32.44
A	19.93	20.78	21.05	21.92	21.34	22.47	23.52	24.27	27.47	28.82	30.42	32.44	55.83

Pre Flop winning Chance Percentage Suited Cards 5 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	14.98	15.58	16.17	15.46	15.09	15.61	16.36	17.49	18.35	19.90	21.61	24.39
3	14.98	NaN	16.82	17.46	16.58	16.32	16.11	16.93	18.04	19.11	20.26	21.80	24.74
4	15.58	16.82	NaN	18.80	18.47	17.83	17.45	17.30	18.60	19.49	20.84	22.59	25.43
5	16.17	17.46	18.80	NaN	19.69	19.46	18.97	18.72	18.95	20.16	21.45	22.80	25.89
6	15.46	16.58	18.47	19.69	NaN	20.81	20.64	20.43	20.60	20.82	21.87	23.68	25.44
7	15.09	16.32	17.83	19.46	20.81	NaN	22.25	22.13	22.18	22.25	22.38	24.24	26.30
8	15.61	16.11	17.45	18.97	20.64	22.25	NaN	23.95	23.87	23.82	24.21	25.06	27.17
9	16.36	16.93	17.30	18.72	20.43	22.13	23.95	NaN	25.92	25.74	26.17	27.11	28.00
10	17.49	18.04	18.60	18.95	20.60	22.18	23.87	25.92	NaN	28.46	28.99	29.69	30.79
J	18.35	19.11	19.49	20.16	20.82	22.25	23.82	25.74	28.46	NaN	30.10	30.86	32.21
Q	19.90	20.26	20.84	21.45	21.87	22.38	24.21	26.17	28.99	30.10	NaN	32.50	33.87
K	21.61	21.80	22.59	22.80	23.68	24.24	25.06	27.11	29.69	30.86	32.50	NaN	35.50
A	24.39	24.74	25.43	25.89	25.44	26.30	27.17	28.00	30.79	32.21	33.87	35.50	NaN

Pre Flop winning Odds Suited Cards 5 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	6.67	6.42	6.18	6.47	6.63	6.40	6.11	5.72	5.45	5.03	4.63	4.10
3	6.67	NaN	5.94	5.73	6.03	6.13	6.21	5.91	5.54	5.23	4.94	4.59	4.04
4	6.42	5.94	NaN	5.32	5.41	5.61	5.73	5.78	5.38	5.13	4.80	4.43	3.93
5	6.18	5.73	5.32	NaN	5.08	5.14	5.27	5.34	5.28	4.96	4.66	4.39	3.86
6	6.47	6.03	5.41	5.08	NaN	4.81	4.84	4.89	4.85	4.80	4.57	4.22	3.93
7	6.63	6.13	5.61	5.14	4.81	NaN	4.49	4.52	4.51	4.50	4.47	4.13	3.80
8	6.40	6.21	5.73	5.27	4.84	4.49	NaN	4.17	4.19	4.20	4.13	3.99	3.68
9	6.11	5.91	5.78	5.34	4.89	4.52	4.17	NaN	3.86	3.89	3.82	3.69	3.57
10	5.72	5.54	5.38	5.28	4.85	4.51	4.19	3.86	NaN	3.51	3.45	3.37	3.25
J	5.45	5.23	5.13	4.96	4.80	4.50	4.20	3.89	3.51	NaN	3.32	3.24	3.10
Q	5.03	4.94	4.80	4.66	4.57	4.47	4.13	3.82	3.45	3.32	NaN	3.08	2.95
K	4.63	4.59	4.43	4.39	4.22	4.13	3.99	3.69	3.37	3.24	3.08	NaN	2.82
A	4.10	4.04	3.93	3.86	3.93	3.80	3.68	3.57	3.25	3.10	2.95	2.82	NaN

Pre Flop winning Odds UnSuited Cards 5 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	5.59	9.28	8.62	8.32	9.07	9.40	8.80	8.32	7.72	7.10	6.55	5.87	5.02
3	9.28	5.29	7.68	7.28	7.75	8.13	8.41	7.82	7.27	6.72	6.20	5.56	4.81
4	8.62	7.68	4.85	6.58	6.93	7.18	7.33	7.59	6.97	6.58	6.02	5.43	4.75
5	8.32	7.28	6.58	4.48	6.27	6.35	6.68	6.76	6.82	6.28	5.86	5.30	4.56
6	9.07	7.75	6.93	6.27	4.06	5.87	5.95	6.05	6.02	6.09	5.63	5.15	4.69
7	9.40	8.13	7.18	6.35	5.87	3.75	5.47	5.44	5.47	5.47	5.39	4.89	4.45
8	8.80	8.41	7.33	6.68	5.95	5.47	3.37	4.96	4.89	4.86	4.82	4.68	4.25
9	8.32	7.82	7.59	6.76	6.05	5.44	4.96	3.03	4.38	4.39	4.33	4.24	4.12
10	7.72	7.27	6.97	6.82	6.02	5.47	4.89	4.38	2.76	3.99	3.87	3.76	3.64
J	7.10	6.72	6.58	6.28	6.09	5.47	4.86	4.39	3.99	2.48	3.70	3.60	3.47
Q	6.55	6.20	6.02	5.86	5.63	5.39	4.82	4.33	3.87	3.70	2.23	3.42	3.29
K	5.87	5.56	5.43	5.30	5.15	4.89	4.68	4.24	3.76	3.60	3.42	2.02	3.08
A	5.02	4.81	4.75	4.56	4.69	4.45	4.25	4.12	3.64	3.47	3.29	3.08	1.79

6 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 6 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	15.45	8.77	9.42	9.88	9.20	8.45	9.03	9.81	10.66	11.61	12.39	13.82	16.24
3	8.77	16.13	10.72	11.27	10.52	10.00	9.46	10.14	11.06	11.73	12.64	14.20	16.99
4	9.42	10.72	17.17	12.57	11.99	11.42	10.83	10.40	11.42	12.03	13.16	14.72	17.29
5	9.88	11.27	12.57	18.28	13.35	12.69	12.45	11.95	12.02	12.73	13.66	15.14	17.90
6	9.20	10.52	11.99	13.35	20.32	14.16	13.94	13.50	13.34	13.29	14.38	15.73	17.13
7	8.45	10.00	11.42	12.69	14.16	22.03	15.32	15.17	15.12	14.91	15.13	16.64	18.37
8	9.03	9.46	10.83	12.45	13.94	15.32	23.99	16.32	16.91	16.60	16.63	17.32	19.23
9	9.81	10.14	10.40	11.95	13.50	15.17	16.32	26.69	19.14	18.81	18.79	19.33	20.05
10	10.66	11.06	11.42	12.02	13.34	15.12	16.91	19.14	30.15	21.57	21.80	22.32	23.07
J	11.61	11.73	12.03	12.73	13.29	14.91	16.60	18.81	21.57	33.63	23.00	23.57	24.27
Q	12.39	12.64	13.16	13.66	14.38	15.13	16.63	18.79	21.80	23.00	38.17	25.05	25.88
K	13.82	14.20	14.72	15.14	15.73	16.64	17.32	19.33	22.32	23.57	25.05	43.08	27.97
A	16.24	16.99	17.29	17.90	17.13	18.37	19.23	20.05	23.07	24.27	25.88	27.97	49.07

Pre Flop winning Chance Percentage Suited Cards 6 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	13.26	13.71	14.15	13.24	12.82	13.41	13.87	14.96	15.57	16.79	18.13	20.81
3	13.26	NaN	14.69	15.43	14.53	13.85	13.64	14.25	15.12	16.10	17.09	18.67	21.21
4	13.71	14.69	NaN	16.24	15.85	15.18	14.74	14.59	15.55	16.38	17.45	18.98	21.64
5	14.15	15.43	16.24	NaN	17.14	16.78	16.17	15.92	16.18	17.05	17.95	19.44	22.25
6	13.24	14.53	15.85	17.14	NaN	17.91	17.64	17.52	17.44	17.42	18.51	20.03	21.54
7	12.82	13.85	15.18	16.78	17.91	NaN	19.00	18.90	18.84	18.92	19.10	20.95	22.31
8	13.41	13.64	14.74	16.17	17.64	19.00	NaN	20.28	20.57	20.69	20.80	21.26	23.22
9	13.87	14.25	14.59	15.92	17.52	18.90	20.28	NaN	22.38	22.29	22.50	23.17	23.84
10	14.96	15.12	15.55	16.18	17.44	18.84	20.57	22.38	NaN	24.77	25.02	25.88	26.59
J	15.57	16.10	16.38	17.05	17.42	18.92	20.69	22.29	24.77	NaN	26.25	26.81	27.88
Q	16.79	17.09	17.45	17.95	18.51	19.10	20.80	22.50	25.02	26.25	NaN	28.38	29.54
K	18.13	18.67	18.98	19.44	20.03	20.95	21.26	23.17	25.88	26.81	28.38	NaN	31.08
A	20.81	21.21	21.64	22.25	21.54	22.31	23.22	23.84	26.59	27.88	29.54	31.08	NaN

Pre Flop winning Odds Suited Cards 6 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	7.54	7.29	7.07	7.55	7.80	7.46	7.21	6.69	6.42	5.96	5.52	4.80
3	7.54	NaN	6.81	6.48	6.88	7.22	7.33	7.02	6.61	6.21	5.85	5.36	4.72
4	7.29	6.81	NaN	6.16	6.31	6.59	6.78	6.85	6.43	6.11	5.73	5.27	4.62
5	7.07	6.48	6.16	NaN	5.83	5.96	6.18	6.28	6.18	5.87	5.57	5.14	4.49
6	7.55	6.88	6.31	5.83	NaN	5.58	5.67	5.71	5.74	5.74	5.40	4.99	4.64
7	7.80	7.22	6.59	5.96	5.58	NaN	5.26	5.29	5.31	5.29	5.24	4.77	4.48
8	7.46	7.33	6.78	6.18	5.67	5.26	NaN	4.93	4.86	4.83	4.81	4.70	4.31
9	7.21	7.02	6.85	6.28	5.71	5.29	4.93	NaN	4.47	4.49	4.45	4.32	4.19
10	6.69	6.61	6.43	6.18	5.74	5.31	4.86	4.47	NaN	4.04	4.00	3.86	3.76
J	6.42	6.21	6.11	5.87	5.74	5.29	4.83	4.49	4.04	NaN	3.81	3.73	3.59
Q	5.96	5.85	5.73	5.57	5.40	5.24	4.81	4.45	4.00	3.81	NaN	3.52	3.39
K	5.52	5.36	5.27	5.14	4.99	4.77	4.70	4.32	3.86	3.73	3.52	NaN	3.22
A	4.80	4.72	4.62	4.49	4.64	4.48	4.31	4.19	3.76	3.59	3.39	3.22	NaN

Pre Flop winning Odds UnSuited Cards 6 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	6.47	11.40	10.62	10.12	10.87	11.83	11.07	10.20	9.38	8.61	8.07	7.24	6.16
3	11.40	6.20	9.33	8.88	9.51	10.00	10.57	9.86	9.04	8.53	7.91	7.04	5.88
4	10.62	9.33	5.82	7.95	8.34	8.76	9.23	9.61	8.76	8.31	7.60	6.80	5.78
5	10.12	8.88	7.95	5.47	7.49	7.88	8.03	8.37	8.32	7.86	7.32	6.60	5.59
6	10.87	9.51	8.34	7.49	4.92	7.06	7.17	7.41	7.50	7.53	6.95	6.36	5.84
7	11.83	10.00	8.76	7.88	7.06	4.54	6.53	6.59	6.61	6.71	6.61	6.01	5.44
8	11.07	10.57	9.23	8.03	7.17	6.53	4.17	6.13	5.92	6.03	6.01	5.77	5.20
9	10.20	9.86	9.61	8.37	7.41	6.59	6.13	3.75	5.22	5.32	5.32	5.17	4.99
10	9.38	9.04	8.76	8.32	7.50	6.61	5.92	5.22	3.32	4.64	4.59	4.48	4.33
J	8.61	8.53	8.31	7.86	7.53	6.71	6.03	5.32	4.64	2.97	4.35	4.24	4.12
Q	8.07	7.91	7.60	7.32	6.95	6.61	6.01	5.32	4.59	4.35	2.62	3.99	3.86
K	7.24	7.04	6.80	6.60	6.36	6.01	5.77	5.17	4.48	4.24	3.99	2.32	3.58
A	6.16	5.88	5.78	5.59	5.84	5.44	5.20	4.99	4.33	4.12	3.86	3.58	2.04

7 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 7 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	14.21	7.68	8.10	8.46	7.70	7.18	7.45	8.04	8.94	9.62	10.35	11.42	13.71
3	7.68	14.50	9.34	9.95	8.99	8.36	7.89	8.33	9.25	9.71	10.61	12.05	14.36
4	8.10	9.34	15.13	11.03	10.35	9.55	8.99	8.57	9.50	10.14	11.06	12.27	14.79
5	8.46	9.95	11.03	16.07	11.43	11.10	10.43	10.00	9.96	10.41	11.70	12.75	15.17
6	7.70	8.99	10.35	11.43	17.44	12.26	11.81	11.38	11.31	11.12	12.06	13.11	14.52
7	7.18	8.36	9.55	11.10	12.26	18.65	13.02	12.76	12.84	12.71	12.51	13.98	15.21
8	7.45	7.89	8.99	10.43	11.81	13.02	20.35	14.20	14.36	14.13	14.07	14.38	16.15
9	8.04	8.33	8.57	10.00	11.38	12.76	14.20	22.69	16.18	15.96	15.99	16.41	17.08
10	8.94	9.25	9.50	9.96	11.31	12.84	14.36	16.18	25.16	18.40	18.70	19.04	19.83
J	9.62	9.71	10.14	10.41	11.12	12.71	14.13	15.96	18.40	28.32	19.57	20.23	20.96
Q	10.35	10.61	11.06	11.70	12.06	12.51	14.07	15.99	18.70	19.57	32.63	21.68	22.53
K	11.42	12.05	12.27	12.75	13.11	13.98	14.38	16.41	19.04	20.23	21.68	37.59	24.66
A	13.71	14.36	14.79	15.17	14.52	15.21	16.15	17.08	19.83	20.96	22.53	24.66	43.33

Pre Flop winning Chance Percentage Suited Cards 7 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	11.91	12.15	12.57	11.82	11.28	11.65	12.32	13.11	13.94	14.65	16.15	18.38
3	11.91	NaN	13.25	13.79	12.83	12.28	11.94	12.45	13.19	14.09	15.10	16.43	18.69
4	12.15	13.25	NaN	14.62	14.04	13.62	13.00	12.65	13.51	14.29	15.25	16.59	18.99
5	12.57	13.79	14.62	NaN	15.15	14.77	14.29	13.83	13.91	14.74	15.53	16.85	19.34
6	11.82	12.83	14.04	15.15	NaN	15.84	15.64	15.04	14.88	14.98	16.06	17.20	18.75
7	11.28	12.28	13.62	14.77	15.84	NaN	16.64	16.67	16.42	16.13	16.42	17.99	19.49
8	11.65	11.94	13.00	14.29	15.64	16.64	NaN	17.87	18.21	18.00	18.19	18.70	20.24
9	12.32	12.45	12.65	13.83	15.04	16.67	17.87	NaN	19.86	19.90	20.07	20.58	21.21
10	13.11	13.19	13.51	13.91	14.88	16.42	18.21	19.86	NaN	21.85	22.10	22.73	23.55
J	13.94	14.09	14.29	14.74	14.98	16.13	18.00	19.90	21.85	NaN	23.10	23.67	24.54
Q	14.65	15.10	15.25	15.53	16.06	16.42	18.19	20.07	22.10	23.10	NaN	25.27	26.02
K	16.15	16.43	16.59	16.85	17.20	17.99	18.70	20.58	22.73	23.67	25.27	NaN	27.76
A	18.38	18.69	18.99	19.34	18.75	19.49	20.24	21.21	23.55	24.54	26.02	27.76	NaN

Pre Flop winning Odds Suited Cards 7 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	8.40	8.23	7.95	8.46	8.87	8.59	8.12	7.63	7.18	6.83	6.19	5.44
3	8.40	NaN	7.55	7.25	7.79	8.15	8.37	8.03	7.58	7.10	6.62	6.09	5.35
4	8.23	7.55	NaN	6.84	7.12	7.34	7.69	7.91	7.40	7.00	6.56	6.03	5.27
5	7.95	7.25	6.84	NaN	6.60	6.77	7.00	7.23	7.19	6.78	6.44	5.93	5.17
6	8.46	7.79	7.12	6.60	NaN	6.31	6.39	6.65	6.72	6.67	6.23	5.82	5.33
7	8.87	8.15	7.34	6.77	6.31	NaN	6.01	6.00	6.09	6.20	6.09	5.56	5.13
8	8.59	8.37	7.69	7.00	6.39	6.01	NaN	5.60	5.49	5.56	5.50	5.35	4.94
9	8.12	8.03	7.91	7.23	6.65	6.00	5.60	NaN	5.04	5.02	4.98	4.86	4.72
10	7.63	7.58	7.40	7.19	6.72	6.09	5.49	5.04	NaN	4.58	4.52	4.40	4.25
J	7.18	7.10	7.00	6.78	6.67	6.20	5.56	5.02	4.58	NaN	4.33	4.23	4.08
Q	6.83	6.62	6.56	6.44	6.23	6.09	5.50	4.98	4.52	4.33	NaN	3.96	3.84
K	6.19	6.09	6.03	5.93	5.82	5.56	5.35	4.86	4.40	4.23	3.96	NaN	3.60
A	5.44	5.35	5.27	5.17	5.33	5.13	4.94	4.72	4.25	4.08	3.84	3.60	NaN

Pre Flop winning Odds UnSuited Cards 7 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	7.04	13.01	12.34	11.82	12.98	13.93	13.42	12.44	11.18	10.40	9.66	8.76	7.29
3	13.01	6.90	10.71	10.05	11.12	11.96	12.67	12.01	10.81	10.30	9.42	8.30	6.96
4	12.34	10.71	6.61	9.06	9.66	10.47	11.12	11.67	10.53	9.87	9.04	8.15	6.76
5	11.82	10.05	9.06	6.22	8.75	9.01	9.59	10.00	10.04	9.61	8.55	7.84	6.59
6	12.98	11.12	9.66	8.75	5.73	8.16	8.47	8.78	8.84	8.99	8.29	7.63	6.89
7	13.93	11.96	10.47	9.01	8.16	5.36	7.68	7.83	7.79	7.87	8.00	7.15	6.57
8	13.42	12.67	11.12	9.59	8.47	7.68	4.91	7.04	6.97	7.08	7.11	6.96	6.19
9	12.44	12.01	11.67	10.00	8.78	7.83	7.04	4.41	6.18	6.27	6.25	6.09	5.85
10	11.18	10.81	10.53	10.04	8.84	7.79	6.97	6.18	3.97	5.44	5.35	5.25	5.04
J	10.40	10.30	9.87	9.61	8.99	7.87	7.08	6.27	5.44	3.53	5.11	4.94	4.77
Q	9.66	9.42	9.04	8.55	8.29	8.00	7.11	6.25	5.35	5.11	3.06	4.61	4.44
K	8.76	8.30	8.15	7.84	7.63	7.15	6.96	6.09	5.25	4.94	4.61	2.66	4.06
A	7.29	6.96	6.76	6.59	6.89	6.57	6.19	5.85	5.04	4.77	4.44	4.06	2.31

8 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 8 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	13.31	6.72	7.25	7.59	6.62	6.02	6.39	6.78	7.51	7.93	8.80	9.98	12.02
3	6.72	13.49	8.28	8.86	7.85	7.14	6.70	7.10	7.76	8.33	8.98	10.23	12.46
4	7.25	8.28	13.97	9.85	9.18	8.48	7.83	7.32	8.02	8.75	9.45	10.48	12.88
5	7.59	8.86	9.85	14.57	10.46	9.76	9.15	8.49	8.35	8.93	9.81	10.68	13.00
6	6.62	7.85	9.18	10.46	15.42	10.59	10.47	9.76	9.54	9.29	10.32	11.14	12.38
7	6.02	7.14	8.48	9.76	10.59	16.45	11.34	10.98	10.89	10.47	10.55	11.79	13.21
8	6.39	6.70	7.83	9.15	10.47	11.34	17.77	12.47	12.46	12.14	12.25	12.43	13.91
9	6.78	7.10	7.32	8.49	9.76	10.98	12.47	19.40	14.16	13.75	13.74	14.03	14.64
10	7.51	7.76	8.02	8.35	9.54	10.89	12.46	14.16	21.84	16.16	16.29	16.56	17.05
J	7.93	8.33	8.75	8.93	9.29	10.47	12.14	13.75	16.16	24.73	17.24	17.77	18.23
Q	8.80	8.98	9.45	9.81	10.32	10.55	12.25	13.74	16.29	17.24	28.36	19.25	19.78
K	9.98	10.23	10.48	10.68	11.14	11.79	12.43	14.03	16.56	17.77	19.25	33.09	21.45
A	12.02	12.46	12.88	13.00	12.38	13.21	13.91	14.64	17.05	18.23	19.78	21.45	38.72

Pre Flop winning Chance Percentage Suited Cards 8 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	10.72	11.24	11.46	10.65	10.15	10.55	10.97	11.66	12.13	13.08	14.24	16.26
3	10.72	NaN	12.21	12.62	11.77	11.28	10.64	11.14	11.89	12.41	13.34	14.76	16.70
4	11.24	12.21	NaN	13.38	12.75	12.21	11.69	11.19	12.18	12.69	13.74	14.76	17.11
5	11.46	12.62	13.38	NaN	13.48	13.46	12.84	12.34	12.45	12.98	13.96	15.08	17.44
6	10.65	11.77	12.75	13.48	NaN	14.21	14.10	13.62	13.41	13.57	14.49	15.54	16.80
7	10.15	11.28	12.21	13.46	14.21	NaN	14.82	14.94	14.70	14.53	14.46	15.78	17.13
8	10.55	10.64	11.69	12.84	14.10	14.82	NaN	15.80	16.04	15.76	16.15	16.54	18.01
9	10.97	11.14	11.19	12.34	13.62	14.94	15.80	NaN	17.85	17.68	17.72	18.05	18.63
10	11.66	11.89	12.18	12.45	13.41	14.70	16.04	17.85	NaN	19.92	20.14	20.58	21.16
J	12.13	12.41	12.69	12.98	13.57	14.53	15.76	17.68	19.92	NaN	20.95	21.41	22.25
Q	13.08	13.34	13.74	13.96	14.49	14.46	16.15	17.72	20.14	20.95	NaN	22.39	23.11
K	14.24	14.76	14.76	15.08	15.54	15.78	16.54	18.05	20.58	21.41	22.39	NaN	24.90
A	16.26	16.70	17.11	17.44	16.80	17.13	18.01	18.63	21.16	22.25	23.11	24.90	NaN

Pre Flop winning Odds Suited Cards 8 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	9.33	8.90	8.73	9.39	9.85	9.48	9.12	8.58	8.24	7.64	7.02	6.15
3	9.33	NaN	8.19	7.92	8.49	8.86	9.40	8.98	8.41	8.06	7.50	6.77	5.99
4	8.90	8.19	NaN	7.47	7.84	8.19	8.56	8.93	8.21	7.88	7.28	6.77	5.84
5	8.73	7.92	7.47	NaN	7.42	7.43	7.79	8.10	8.03	7.71	7.16	6.63	5.73
6	9.39	8.49	7.84	7.42	NaN	7.03	7.09	7.34	7.45	7.37	6.90	6.43	5.95
7	9.85	8.86	8.19	7.43	7.03	NaN	6.75	6.69	6.80	6.88	6.92	6.34	5.84
8	9.48	9.40	8.56	7.79	7.09	6.75	NaN	6.33	6.23	6.34	6.19	6.05	5.55
9	9.12	8.98	8.93	8.10	7.34	6.69	6.33	NaN	5.60	5.66	5.64	5.54	5.37
10	8.58	8.41	8.21	8.03	7.45	6.80	6.23	5.60	NaN	5.02	4.97	4.86	4.73
J	8.24	8.06	7.88	7.71	7.37	6.88	6.34	5.66	5.02	NaN	4.77	4.67	4.49
Q	7.64	7.50	7.28	7.16	6.90	6.92	6.19	5.64	4.97	4.77	NaN	4.47	4.33
K	7.02	6.77	6.77	6.63	6.43	6.34	6.05	5.54	4.86	4.67	4.47	NaN	4.02
A	6.15	5.99	5.84	5.73	5.95	5.84	5.55	5.37	4.73	4.49	4.33	4.02	NaN

Pre Flop winning Odds UnSuited Cards 8 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	7.52	14.88	13.80	13.17	15.10	16.62	15.64	14.76	13.32	12.61	11.36	10.02	8.32
3	14.88	7.41	12.07	11.29	12.74	14.00	14.91	14.09	12.88	12.01	11.13	9.78	8.03
4	13.80	12.07	7.16	10.15	10.90	11.80	12.78	13.66	12.47	11.43	10.58	9.54	7.77
5	13.17	11.29	10.15	6.87	9.56	10.25	10.93	11.78	11.97	11.20	10.20	9.36	7.69
6	15.10	12.74	10.90	9.56	6.48	9.44	9.55	10.24	10.48	10.76	9.69	8.98	8.08
7	16.62	14.00	11.80	10.25	9.44	6.08	8.82	9.11	9.18	9.55	9.48	8.49	7.57
8	15.64	14.91	12.78	10.93	9.55	8.82	5.63	8.02	8.02	8.24	8.16	8.04	7.19
9	14.76	14.09	13.66	11.78	10.24	9.11	8.02	5.16	7.06	7.27	7.28	7.13	6.83
10	13.32	12.88	12.47	11.97	10.48	9.18	8.02	7.06	4.58	6.19	6.14	6.04	5.86
J	12.61	12.01	11.43	11.20	10.76	9.55	8.24	7.27	6.19	4.04	5.80	5.63	5.49
Q	11.36	11.13	10.58	10.20	9.69	9.48	8.16	7.28	6.14	5.80	3.53	5.19	5.06
K	10.02	9.78	9.54	9.36	8.98	8.49	8.04	7.13	6.04	5.63	5.19	3.02	4.66
A	8.32	8.03	7.77	7.69	8.08	7.57	7.19	6.83	5.86	5.49	5.06	4.66	2.58

9 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 9 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	12.63	6.10	6.57	6.84	6.04	5.53	5.56	5.97	6.39	6.93	7.73	8.62	10.50
3	6.10	12.49	7.40	8.01	7.13	6.38	5.72	6.05	6.69	7.12	7.91	8.75	10.66
4	6.57	7.40	12.90	9.03	8.37	7.63	6.85	6.46	7.09	7.47	8.25	9.10	11.06
5	6.84	8.01	9.03	13.28	9.20	8.68	7.92	7.40	7.09	7.77	8.39	9.53	11.53
6	6.04	7.13	8.37	9.20	13.92	9.52	9.24	8.58	8.36	8.06	8.73	9.67	10.88
7	5.53	6.38	7.63	8.68	9.52	14.56	10.00	9.73	9.45	9.12	9.12	9.99	11.34
8	5.56	5.72	6.85	7.92	9.24	10.00	15.65	10.70	11.02	10.69	10.65	10.92	11.92
9	5.97	6.05	6.46	7.40	8.58	9.73	10.70	17.26	12.57	12.19	12.08	12.31	12.82
10	6.39	6.69	7.09	7.09	8.36	9.45	11.02	12.57	18.92	14.53	14.51	14.60	15.38
J	6.93	7.12	7.47	7.77	8.06	9.12	10.69	12.19	14.53	21.41	15.36	15.71	16.03
Q	7.73	7.91	8.25	8.39	8.73	9.12	10.65	12.08	14.51	15.36	25.09	17.05	17.42
K	8.62	8.75	9.10	9.53	9.67	9.99	10.92	12.31	14.60	15.71	17.05	29.28	19.05
A	10.50	10.66	11.06	11.53	10.88	11.34	11.92	12.82	15.38	16.03	17.42	19.05	34.55

Pre Flop winning Chance Percentage Suited Cards 9 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	9.77	10.27	10.49	9.78	9.31	9.49	9.95	10.69	11.25	11.92	13.03	14.96
3	9.77	NaN	11.06	11.62	10.90	10.13	9.68	10.11	10.65	11.13	12.16	13.04	15.14
4	10.27	11.06	NaN	12.58	11.98	11.27	10.72	10.35	11.06	11.70	12.43	13.52	15.61
5	10.49	11.62	12.58	NaN	12.77	12.38	11.73	11.26	11.12	11.81	12.68	13.85	15.89
6	9.78	10.90	11.98	12.77	NaN	13.22	12.81	12.47	12.16	12.19	12.92	14.03	15.32
7	9.31	10.13	11.27	12.38	13.22	NaN	13.77	13.37	13.33	13.01	13.26	14.24	15.52
8	9.49	9.68	10.72	11.73	12.81	13.77	NaN	14.40	14.72	14.37	14.27	14.79	16.32
9	9.95	10.11	10.35	11.26	12.47	13.37	14.40	NaN	16.12	16.01	16.02	16.29	17.06
10	10.69	10.65	11.06	11.12	12.16	13.33	14.72	16.12	NaN	17.90	18.02	18.32	18.95
J	11.25	11.13	11.70	11.81	12.19	13.01	14.37	16.01	17.90	NaN	18.71	19.29	19.91
Q	11.92	12.16	12.43	12.68	12.92	13.26	14.27	16.02	18.02	18.71	NaN	20.38	20.98
K	13.03	13.04	13.52	13.85	14.03	14.24	14.79	16.29	18.32	19.29	20.38	NaN	22.62
A	14.96	15.14	15.61	15.89	15.32	15.52	16.32	17.06	18.95	19.91	20.98	22.62	NaN

Pre Flop winning Odds Suited Cards 9 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	10.23	9.74	9.53	10.23	10.74	10.54	10.05	9.36	8.89	8.39	7.68	6.68
3	10.23	NaN	9.04	8.61	9.17	9.88	10.33	9.89	9.39	8.99	8.22	7.67	6.60
4	9.74	9.04	NaN	7.95	8.35	8.87	9.33	9.66	9.04	8.55	8.05	7.39	6.41
5	9.53	8.61	7.95	NaN	7.83	8.08	8.53	8.88	8.99	8.47	7.89	7.22	6.29
6	10.23	9.17	8.35	7.83	NaN	7.56	7.80	8.02	8.22	8.21	7.74	7.13	6.53
7	10.74	9.88	8.87	8.08	7.56	NaN	7.26	7.48	7.50	7.69	7.54	7.02	6.44
8	10.54	10.33	9.33	8.53	7.80	7.26	NaN	6.94	6.80	6.96	7.01	6.76	6.13
9	10.05	9.89	9.66	8.88	8.02	7.48	6.94	NaN	6.20	6.25	6.24	6.14	5.86
10	9.36	9.39	9.04	8.99	8.22	7.50	6.80	6.20	NaN	5.59	5.55	5.46	5.28
J	8.89	8.99	8.55	8.47	8.21	7.69	6.96	6.25	5.59	NaN	5.34	5.18	5.02
Q	8.39	8.22	8.05	7.89	7.74	7.54	7.01	6.24	5.55	5.34	NaN	4.91	4.77
K	7.68	7.67	7.39	7.22	7.13	7.02	6.76	6.14	5.46	5.18	4.91	NaN	4.42
A	6.68	6.60	6.41	6.29	6.53	6.44	6.13	5.86	5.28	5.02	4.77	4.42	NaN

Pre Flop winning Odds UnSuited Cards 9 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	7.92	16.38	15.22	14.61	16.57	18.10	17.98	16.76	15.65	14.42	12.94	11.60	9.52
3	16.38	8.00	13.52	12.48	14.02	15.69	17.47	16.53	14.94	14.05	12.64	11.43	9.38
4	15.22	13.52	7.75	11.07	11.95	13.10	14.61	15.49	14.10	13.39	12.13	10.99	9.04
5	14.61	12.48	11.07	7.53	10.87	11.52	12.62	13.52	14.10	12.87	11.92	10.49	8.67
6	16.57	14.02	11.95	10.87	7.19	10.50	10.82	11.65	11.96	12.41	11.46	10.34	9.20
7	18.10	15.69	13.10	11.52	10.50	6.87	10.00	10.28	10.58	10.96	10.96	10.01	8.82
8	17.98	17.47	14.61	12.62	10.82	10.00	6.39	9.34	9.07	9.36	9.39	9.16	8.39
9	16.76	16.53	15.49	13.52	11.65	10.28	9.34	5.79	7.95	8.20	8.28	8.12	7.80
10	15.65	14.94	14.10	14.10	11.96	10.58	9.07	7.95	5.29	6.88	6.89	6.85	6.50
J	14.42	14.05	13.39	12.87	12.41	10.96	9.36	8.20	6.88	4.67	6.51	6.36	6.24
Q	12.94	12.64	12.13	11.92	11.46	10.96	9.39	8.28	6.89	6.51	3.99	5.87	5.74
K	11.60	11.43	10.99	10.49	10.34	10.01	9.16	8.12	6.85	6.36	5.87	3.42	5.25
A	9.52	9.38	9.04	8.67	9.20	8.82	8.39	7.80	6.50	6.24	5.74	5.25	2.89

10 Players Game

Pre Flop winning Chance Percentage UnSuited Cards 10 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	11.99	5.52	6.08	6.11	5.49	4.93	4.98	5.31	5.69	6.19	6.78	7.73	9.26
3	5.52	12.22	7.00	7.37	6.29	5.71	5.12	5.35	5.88	6.27	6.90	7.91	9.51
4	6.08	7.00	12.16	8.14	7.68	6.80	6.07	5.59	6.09	6.57	7.07	8.09	9.70
5	6.11	7.37	8.14	12.22	8.44	7.89	7.11	6.48	6.43	6.92	7.33	8.20	10.31
6	5.49	6.29	7.68	8.44	13.15	8.70	8.35	7.87	7.37	7.01	7.78	8.63	9.63
7	4.93	5.71	6.80	7.89	8.70	13.54	9.38	8.82	8.53	8.15	8.09	8.83	10.08
8	4.98	5.12	6.07	7.11	8.35	9.38	14.46	9.84	9.88	9.47	9.17	9.56	10.53
9	5.31	5.35	5.59	6.48	7.87	8.82	9.84	15.66	11.39	10.88	10.55	10.90	11.22
10	5.69	5.88	6.09	6.43	7.37	8.53	9.88	11.39	17.03	13.12	12.96	13.07	13.43
J	6.19	6.27	6.57	6.92	7.01	8.15	9.47	10.88	13.12	19.33	13.78	13.97	14.41
Q	6.78	6.90	7.07	7.33	7.78	8.09	9.17	10.55	12.96	13.78	22.27	15.34	15.72
K	7.73	7.91	8.09	8.20	8.63	8.83	9.56	10.90	13.07	13.97	15.34	26.11	17.43
A	9.26	9.51	9.70	10.31	9.63	10.08	10.53	11.22	13.43	14.41	15.72	17.43	31.17

Pre Flop winning Chance Percentage Suited Cards 10 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	9.24	9.58	9.77	9.06	8.49	8.80	9.11	9.74	10.38	11.27	12.13	13.79
3	9.24	NaN	10.24	10.95	9.96	9.33	8.66	9.07	9.86	10.31	11.01	11.88	14.03
4	9.58	10.24	NaN	11.66	10.94	10.50	9.91	9.50	9.97	10.56	11.43	12.37	14.20
5	9.77	10.95	11.66	NaN	11.81	11.45	10.83	10.43	10.14	10.69	11.36	12.34	14.32
6	9.06	9.96	10.94	11.81	NaN	12.08	11.93	11.33	11.00	10.83	11.72	12.74	13.99
7	8.49	9.33	10.50	11.45	12.08	NaN	12.75	12.36	12.36	12.05	12.17	13.25	14.30
8	8.80	8.66	9.91	10.83	11.93	12.75	NaN	13.34	13.48	13.20	13.19	13.44	14.76
9	9.11	9.07	9.50	10.43	11.33	12.36	13.34	NaN	14.90	14.50	14.44	14.85	15.32
10	9.74	9.86	9.97	10.14	11.00	12.36	13.48	14.90	NaN	16.62	16.56	16.81	17.42
J	10.38	10.31	10.56	10.69	10.83	12.05	13.20	14.50	16.62	NaN	17.29	17.78	18.34
Q	11.27	11.01	11.43	11.36	11.72	12.17	13.19	14.44	16.56	17.29	NaN	18.93	19.63
K	12.13	11.88	12.37	12.34	12.74	13.25	13.44	14.85	16.81	17.78	18.93	NaN	20.75
A	13.79	14.03	14.20	14.32	13.99	14.30	14.76	15.32	17.42	18.34	19.63	20.75	NaN

Pre Flop winning Odds Suited Cards 10 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	NaN	10.82	10.43	10.24	11.03	11.78	11.36	10.98	10.26	9.63	8.87	8.24	7.25
3	10.82	NaN	9.77	9.13	10.04	10.72	11.55	11.02	10.14	9.69	9.08	8.42	7.13
4	10.43	9.77	NaN	8.57	9.14	9.52	10.09	10.53	10.03	9.47	8.75	8.09	7.04
5	10.24	9.13	8.57	NaN	8.46	8.73	9.23	9.59	9.86	9.35	8.80	8.10	6.98
6	11.03	10.04	9.14	8.46	NaN	8.28	8.39	8.83	9.09	9.23	8.53	7.85	7.15
7	11.78	10.72	9.52	8.73	8.28	NaN	7.84	8.09	8.09	8.30	8.22	7.55	6.99
8	11.36	11.55	10.09	9.23	8.39	7.84	NaN	7.50	7.42	7.57	7.58	7.44	6.77
9	10.98	11.02	10.53	9.59	8.83	8.09	7.50	NaN	6.71	6.90	6.93	6.74	6.53
10	10.26	10.14	10.03	9.86	9.09	8.09	7.42	6.71	NaN	6.02	6.04	5.95	5.74
J	9.63	9.69	9.47	9.35	9.23	8.30	7.57	6.90	6.02	NaN	5.78	5.63	5.45
Q	8.87	9.08	8.75	8.80	8.53	8.22	7.58	6.93	6.04	5.78	NaN	5.28	5.09
K	8.24	8.42	8.09	8.10	7.85	7.55	7.44	6.74	5.95	5.63	5.28	NaN	4.82
A	7.25	7.13	7.04	6.98	7.15	6.99	6.77	6.53	5.74	5.45	5.09	4.82	NaN

Pre Flop winning Odds UnSuited Cards 10 Players
	2	3	4	5	6	7	8	9	10	J	Q	K	A
2	8.34	18.11	16.45	16.36	18.21	20.27	20.09	18.83	17.57	16.16	14.75	12.94	10.80
3	18.11	8.18	14.29	13.57	15.90	17.50	19.52	18.70	17.01	15.95	14.48	12.64	10.52
4	16.45	14.29	8.22	12.28	13.03	14.72	16.47	17.89	16.42	15.22	14.14	12.36	10.31
5	16.36	13.57	12.28	8.18	11.85	12.67	14.07	15.43	15.56	14.44	13.65	12.20	9.69
6	18.21	15.90	13.03	11.85	7.61	11.50	11.98	12.70	13.57	14.27	12.85	11.59	10.39
7	20.27	17.50	14.72	12.67	11.50	7.38	10.66	11.34	11.72	12.28	12.37	11.32	9.92
8	20.09	19.52	16.47	14.07	11.98	10.66	6.92	10.16	10.12	10.56	10.91	10.46	9.50
9	18.83	18.70	17.89	15.43	12.70	11.34	10.16	6.39	8.78	9.19	9.48	9.18	8.92
10	17.57	17.01	16.42	15.56	13.57	11.72	10.12	8.78	5.87	7.62	7.72	7.65	7.44
J	16.16	15.95	15.22	14.44	14.27	12.28	10.56	9.19	7.62	5.17	7.26	7.16	6.94
Q	14.75	14.48	14.14	13.65	12.85	12.37	10.91	9.48	7.72	7.26	4.49	6.52	6.36
K	12.94	12.64	12.36	12.20	11.59	11.32	10.46	9.18	7.65	7.16	6.52	3.83	5.74
A	10.80	10.52	10.31	9.69	10.39	9.92	9.50	8.92	7.44	6.94	6.36	5.74	3.21

Popularity: 2% [?]

Barnard’s unconditional exact test on 2×2 matrix

Barnard’s unconditional exact test on 2×2 matrix

There are two fundamentally different exact tests for comparing the equality of two binomial probabilities – Fisher’s exact test (Fisher, 1925), and Barnard’s exact test (Barnard, 1945). Fisher’s exact test (Fisher, 1925) is the more popular of the two. In fact, Fisher was bitterly critical of Barnard’s proposal for esoteric reasons that we will not go into here. For 2 × 2 tables, Barnard’s test is more powerful than Fisher’s, as Barnard noted in his 1945 paper, much to Fisher’s chagrin. Anyway, perhaps due to its computational difficulty the Barnard’s is not widely used.
Consider a general 2×2 matrix

	Category 1	Category 2	Row sum
Condition 1	A	B	RS1
Condition 2	C	D	RS2
Column sum	CS1	CS2	total = N

Suppose that the common probability of infection under the null hypothesis is πA = πC = π. Then the probability of observing any generic table X is a product of two binomials:

The exact p-value is then the sum of such probabilities for all tables, X, that could have been observed which are at least as extreme as the observed table, X₀ , under the null hypothesis. Specifically, suppose T (X) is a “discrepancy measure” or test statistic that measures how discrepant any table X is relative to the type of table one would expect under the null hypothesis. Then, for any given π, the exact p-value of the observed table X₀ is:

Now there exist many different test statistics for determining if a table X is more or less extreme than the observed table X₀ under the null hypothesis: we will use the popular Wald statistic:

Although the Wald test statistic T (X) is well defined, equation (2) is not of much practical use for computing an exact p-value because it depends on knowing the value of π, the common probability under the null hypothesis. One could of course use the data to estimate π and then substitute this estimate into equation (2). But then the p-value would no longer be exact. The main difference between the Fisher’s and Barnard’s tests is the manner in which they eliminate this nuisance parameter from (2) without sacrificing the exactness.

Fisher’s exact test

Fisher’s exact test gets rid of π by restricting attention only to generic 2 × 2 tables X in which A + B = Rs₁, the sum of responses that were actually observed in the data. Accordingly define:

Fisher noted that, under the null hypothesis, provided we confine our attention only to tables X ∈ Γ(Rs₁), each table has a hypergeometric probability that no longer depends on the nuisance parameter π (see my post: Fisher’s conditional exact test on 2×2 matrix).

Barnard’s exact test

Barnard’s exact test is an unconditional test. It generates the exact distribution of T(X) by considering all the tables X ∈ Γ, and rather than just the tables X ∈ Γ(Rs₁). What shall we do with the unknown nuisance parameter π in the above p-value (equation 1)? Barnard suggested that we calculate p(π) for all possible values of π ∈ (0, 1) and choose the value, π^∗ , say, that maximizes p(π). And this is computationally very very difficult….

The algorithm

Now, let we see the algorithm in deep.

Step 1: The Wald’s Statistic

The first step is to compute the Wald’s statistic for each possible value of A and B (from now on I and J).

I∈[0:1:Cs₁] and J∈[0:1:Cs₂]. One could write:

Notice that for (I=0 & J=0) or (I=Cs(1) & J=Cs(2)) T(I,J)=0/0=NaN. So there are two points where T(I,J) is not defined and NaN should be replaced with 0.
Two nested for…end loops are time consuming and are useless considering MatLab functions…

A more efficient code should be:

The ndgrid function creates two square matrix:

So TX will be a Cs(1)+1 x Cs(2)+1 matrix. Using .* and ./ all elements of the matrix will be correctly multiplied or divided.

To set up the observed matrix (TX_o) and the index of all TX≥TX_o

Step 2: Compute P(X|π)

Now, we must compute P(X|π) for each I, J and π∈(0, 1) . I and J were just defined by ndgrid function. Expanding the binomials we obtain:

As I explained in my previous post Fisher’s conditional exact test on 2×2 matrix, it is useful to use the logarithms of P(X|π): the factorials can be obtained using the MatLab Gammaln function; the multiplications became sums; the divisions became subtractions and the power became multiplications.

CF is a matrix with the same dimensions of TX because it is a linear combination of I and J.

The S matrix is a 3D matrix: two dimensions are Cs₁+1 and Cs₂+1; the third dimension is set by the user. In fact in the arguments of this function, the user can choose to set the length (the Tbx variable) of the vector π (by default it is 100). Let we see the code:

Now we have two 1xTbx arrays (LP and ALP) and three Cs₁+1 x Cs₂+1 x Tbx “boxes” (E, F and CF).
As Peter J. Acklam wrote in his “Matlab array manipulation tips and tricks” , there is a no for…loop solution to multiply each 2D slice with the corresponding
element of a vector. He wrote:
Assume X is an m-by-n-by-p array and v is a row vector with length p.
How does one write:

Y = zeros(m, n, p);
for i = 1:p
Y(:,:,i) = X(:,:,i) * v(i);
end

with no for-loop? One way is to use:
Y = X .* repmat(reshape(v, [1 1 p]), [m n]);

So we can construct the two others 3D boxes:
Box1=CF
Box2=E.*repmat(reshape(LP,[1 1 Tbx]),[Cs1+1 Cs2+1])=E.*repmat(reshape(LP,A),B)
Box3=F.*repmat(reshape(ALP,[1 1 Tbx]),[Cs1+1 Cs2+1])=F.*repmat(reshape(ALP,A),B)
Finally, we can obtain the S box that is the sum of these three boxes (remember that we are using logarithms, so we must convert them using the exp function)

Now we are at the last point: for each 2D slice of the S box we must sum the values corresponding to TX>=TXo, obtaining a new vector P.
To use the logical indexing tecnique, we must replicate the idx matrix:
S(repmat(idx,A)) is a column vector that has L*Tbx row;
L=sum(idx(idx==1)) is the number of 1 in the idx matrix.
Using the reshape function we can obtain a new LxTbx matrix: in each column we’ll have the P(X|π & TX>=TXo) and using the function sum we’ll, finally,
obtain the P vector.

Now we can find the p-value and the nuisance parameter:

Display the results…

An example

Supposing to have this matrix

	Vaccine	Placebo	Row sum
Flu infection	7	12	19
Healthy	8	3	11
Column sum	15	15	30

Using the Fisher’s conditional exact test the p-value (1-tailed) is 0.6407: you will conclude that the vaccine is not effective.
Using the Barnard’s unconditional exact test the results will be:

2×2 matrix Barnard’s exact test: 100 16×16 tables were evaluated
——————————————————————————–
Wald statistic = 1.8943 – Nuisance parameter = 0.6666
p-values 1-tailed = 0.034074 2-tailed = 0.068148
——————————————————————————–

So with the Barnard’s test you will conclude that the vaccine is effective.
Anyway, using the Fisher’s conditional exact test with the Lancaster’s correction (mid p) you will obtain a 2-tailed p-value=0.04774 ….

If any problems occurs in execution, or if you found a bug, have a suggestion or question just contact me at: giuseppe dot cardillo-edta at poste dot it

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