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StrategyPoker Basics

Probabilities in Texas Hold'em

Introduction

An understanding of basic probabilities will give your poker game a stronger foundation, for all game types. This article discusses all the important, and interesting, probabilities that you should be aware of.

Probabilities in poker

Probability means the degree of certainty that a possible event will occur. The classic definition is: "The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible." The probability of flipping heads in a coin toss is therefore 1 to 2, or 50%.

For poker players, stochastics is the most interesting part of studying probability. Stochatics deals with frequence-based probabilities. Combinatorics (card combinations), statistics (sample size) and other subdisciplines are all a part of stochastics.

Probabilities are always a number between 0 and 1 and are usually expressed as a percentage. Probability can also be given as odds, which tells how often one event will occur in relation to the number of times another event will take place (1:2).

In the following sections you will get an overview of many useful probabilities that will help you improve your game. You can see the calculations used to arrive at these results on the second page of this article.

Probability of being dealt a given starting hand

The following chart tells you exactly how likely you are to be dealt a given starting hand. Knowing these values can be very helpful when estimating the strength of your starting hand.

Starting hands
Prob. in % Odds
A specific pocket pair (AA, KK, etc.)
0.453 219.75:1
Pocket pair QQ+ 1.36 72.53:1
Pocket pair JJ+ 1.81 54.25:1
Pocket pair TT+ 2.24 43.24:1
Any pocket pair 5.88 16:1
A specific non-paired hand (AKo, AKs…) 1.21 82.64:1
Two specific suited cards (AKs, AQs…) 0.302 330.12:1
Suited cards 23.53 3.25:1
Suited connectors 3.92 24.5:1
Suited cards T or better 3.02 32.11:1
Connected cards 15.7 5.37:1
Connected cards T or better 4.83 19.7:1
Any two Q+ (AQ, KQ...) 4.98 19.08:1
Any Two J+ (AQ, AJ, KJ...) 9.05 10.04:1
Any Two T+ (AT, AQ, KT...) 14.3 5.99:1

Probability of facing a higher pocket pair when you have a pocket pair

The following two charts show you how likely you are to be behind with a given pocket pair. The first chart shows the probability of exactly one opponent having a higher pocket pair.

Your starting hand
Probability of facing a higher pocket pair (in %)
  1 Player 2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players
KK 0.49 0.98 1.47 1.96 2.44 2.93 3.42 3.91 4.39
QQ 0.98 1.95 2.92 3.88 4.84 5.79 6.73 7.66 8.59
JJ 1.47 2.92 4.36 5.77 7.17 8.56 9.92 11.27 12.59
TT 1.96 3.89 5.78 7.64 9.46 11.24 12.99 14.7 16.37
99 2.45 4.84 7.18 9.46 11.68 13.84 15.93 17.95 19.9
88 2.94 5.8 8.57 11.25 13.84 16.34 18.73 21.01 23.18
77 3.43 6.74 9.94 13.01 15.95 18.74 21.38 23.87 26.19
66 3.92 7.69 11.3 14.73 17.99 21.04 23.89 26.51 28.9
55 4.41 8.62 12.63 16.42 19.96 23.24 26.23 28.92 31.29
44 4.9 9.56 13.95 18.06 21.86 25.32 28.41 31.09 33.34
33 5.39 10.48 15.26 19.67 23.7 27.29 30.4 33 35.03
22
5.88
11.41 16.54 21.24 25.46 29.14 32.22 34.64 36.33

Probability of facing more than one higher pocket pair when you have a pocket pair

These values show how likely you are to be behind against more than one opponent before the flop when you have a pocket pair.

Your starting hand
Probability of facing more than one higher pocket pair (in %)
  2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players
KK <0.001 0.001 0.003 0.004 0.007 0.009 0.012 0.016
QQ 0.006 0.018 0.037 0.061 0.091 0.128 0.171 0.22
JJ 0.017 0.051 0,102 0.171 0.257 0.36 0.482 0.621
TT 0.033 0.099 0.2 0.335 0.504 0.709 0.95 1.226
99 0.054 0.164 0.33 0.553 0.836 1.177 1.58 2.045
88 0.081 0.244 0.493 0.828 1.253 1.769 2.378 3.084
77 0.112 0.341 0.689 1.16 1.758 2.487 3.351 4.353
66 0.149 0.454 0.918 1.55 2.353 3.335 4.503 5.861
55 0.191 0.583 1.182 1.998 3.04 4.318 5.84 7.619
44 0.239 0.728 1.48 2.506 3.821 5.438 7.371 9.635
33 0.291 0.89 1.812 3.075 4.698 6.699 9.099 11.919
22 0.349 1.068 2.18 3.706 5.673 8.107 11.034 14.484

Probability of facing a better A when you have an Ax hand

The next chart shows you all possible Ax starting hands. You can then see the probability that an opponent will have an A with a better kicker to the right.

Your starting hand
Probability of facing a better A in (%)
  1 Player 2 Players 3 Players 4 Players 5 Players 6 Players 7 Players 8 Players 9 Players
AK 0.245 0.489 0.733 0.976 1.219 1.46 1.702 1.942 2.183
AQ 1.224
2.434
3.629 4.809 5.974 7.126 8.263 9.386 10.496
AJ 2.204 4.36 6.468 8.529 10.545 12.517 14.445 16.331 18.175
AT 3.184 6.266 9.25 12.139 14.937 17.645 20.267 22.805 25.263
A9 4.163 8.153 11.977 15.642 19.154 22.52 25.745 28.837 31.799
A8 5.143 10.021 14.649 19.038 23.202 27.152 30.898 34.452 37.823
A7 6.122 11.87 17.266 22.331 27.086 31.55 35.741 39.675 43.369
A6 7.102 13.7 19.829 25.523 30.812 35.726 40.291 44.531 48.471
A5 8.082 15.51 22.338 28.615 34.384 39.687 44.561 49.041 53.16
A4 9.061 17.301 24.795 31.609 37.806 43.442 48.567 53.227 57.465
A3 10.041 19.073 27.199 34.509
41.085 47 52.322 57.109 61.416
A2 11.02 20.826 29.552 37.315 44.223 50.37 55.84 60.706 65.037

Probability that no overcard will show up on the flop

You have pocket pair and want to know how likely you are to see an overcard on the flop? The following chart gives you the answer.

Your starting hand
No overcard on flop
No overcard on turn No overcard river
  Prob. in % Odds Prob. in % Odds Prob. in % Odds
KK 77.45 0.29:1 70.86 0.41:1 64.7 0.55:1
QQ 58.57 0.71:1 48.6 1.06:1 40.15 1.49:1
JJ 43.04 1.32:1 32.05 2.12:1 23.69 3.22:1
TT 30.53 2.28:1 20.14 3.97:1 13.13 6.61:1
99 20.71 3.83:1 11.9 7.40:1 6.73 13.87:1
88 13.27 6.54:1 6.49 14.40:1 3.1 31.26:1
77 7.86 11.73:1 3.18 30.44:1 1.24 79.64:1
66 4.16 23.04:1 1.33 74.18:1 0.4 249:1
55 1.86 52.76:1 0.43 231.56:1 0.09 1110.12:1
44 0.61 162.93:1 0.09 1110.12:1 0.01 9999:1
33 0.1 999.00:1 0.01 15352.33:1
<0.01 353125.67:1

Probability of making a specific hand (5 out of 52)

The following chart shows the probability of making a specific hand. You'll have the pleasure of holding a pair often enough; a straight or even royal flush is a lot less likely. 5 out of 52 means that you build your hand with using 5 cards.

Hand Number of possibilities
Probability in % Odds
Royal flush 4 0.0001539077 649737:1
Straight flush 36 0.0013851695 72193.5:1
Four-of-a-kind 624 0.0240096038 4163.99:1
Full house 3744 0.144057623 693.17:1
Flush 5108 0.1965401545 507.8:1
Straight 10200 0.3924646782 253.8:1
Three-of-a-kind 54912 2.1128451381 46.3:1
Two pair
123552 4.7539015606 20:1
Pair
1098240 42.2569027611 1.366:1
High card 1202540 50.1177394035 0.995:1

Probability of making a specific hand (7 out of 52)

The following chart shows the probability of getting a certain hand. Whereas a pair floats by often enough, getting a straight or royal flush is less likely.

7 out of 52 means, that although you build your hand using 5 cards, you still have 7 cards from which to choose these 5. In the case of Texas Hold'em, there's the 2 pocket cards and 5 on the board. This way of working out the probabilities would be the more accurate way - however, just to note, the probabilities worked out on 5 of 52 are practically the same, and far easier to calculate.

Hand Number of possibilities
Probability in % Odds
Royal Flush 4324 0.003232062 30939:1
Straight Flush 37260 0.027850748 3589.57:1
Four-of-a-kind 224848 0.168067227 594:1
Full House 3473184 2.596102271 37.52:1
Flush 4047644 3.025494123 32.05:1
Straight 6180020 4.619382087 20.65:1
Three-of-a-kind 6461620 4.829869755 19.7:1
Two pair
31433400 23.49553641 3.26:1
Pair 58627800 43.82254574 1.28:1
High card 23294460 17.41191958 4.74:1

Probability of improving on the flop

Once you've picked up a promising starting hand, this chart will come in handy. You can see how likely you are to improve on the flop with a given starting hand.

Starting hand Improvement on flop
Probability in % Odds
Pocket pair Three-of-a-kind or better
12.7 6.9:1
Pocket pair Three-of-a-kind 11.8 7.5:1
Pocket pair Full house 0.73 136:1
Pocket pair Four-of-a-kind 0.24 415.67:1
2 unpaired cards
Pair
32.4 2.1:1
2 unpaired cards
Two pair
2 48.5:1
Suited cards Flush 0.842 118:1
Suited cards Flush draw 10.9 8.17:1
Suited cards Backdoor flush draw
41.6 1.4:1
Connectors 45o-JTo OESD 9.6 9.42:1
Connectors 45s-JTs Straight draw / flush draw
19.1 4.21:1
Connectors 45o-JTo Straight 1.31 75:1

Probability of improving on the turn

After the flop comes the turn - this chart shows how likely you are to improve on the turn.

Your hand
Improvement on turn
Probability in % Odds
Flush draw Flush 19.1 4.24:1
OESD Straight 17 4.9:1
Gutshot straight draw Straight 8.5 10.76:1
Three-of-a-kind Four-of-a-kind 2.1 46.61:1
Two pair
Full house 8.5 10.76:1
Pair Three-of-a-kind 4.3 22.26:1
Two unpaired cards
Pair (with hole card)
12.8 6.8:1

Probability of improving on the river

The following chart shows how likely you are to improve with the final community card.

Your hand Improvement on river
Probability in % Odds
Flush draw Flush 19.6 4.1:1
OESD Straight 17.4 4.74:1
Gutshot straight draw Straight 8.7 10.5:1
Three-of-a-kind Four-of-a-kind 2.2 45.46:1
Two pair
Full house 8.7 10.5:1
Pair Three-of-a-kind 4.3 22.26:1
Two unpaired cards
Pair (with hole card)
13 6.7:1

 

Probability of improving from flop to river

This chart shows how likely you are to improve your hand from flop to river. In other words, the turn and river are combined. These values can be very helpful for planning your post-flop play.

Your hand Improvement by river
Probability in % Odds
Flush draw Flush 35 1.86:1
Backdoor flush draw Flush 4.2 22.8:1
OESD Straight 32 2.13:1
Gutshot straight draw Straight 17 4.88:1
Three-of-a-kind Four-of-a-kind
4.3 22.26:1
Two pair
Full house 17 4.88:1
Pair Four-of-a-kind 0.09 1100:1
Pair Three-of-a-kind 8.4 10.9:1

Probability of seeing a specific board on the flop

This chart can be very helpful when making your pre-flop decision. As you can see, a paired board occurs relatively often, whereas three-of-a-kind is much less likely. These values can help you make a better estimation of your actual hand strength before the flop.

On the flop Probability in % Odds
Three-of-a-kind 0.24 415.67:1
Pair 16.9 4.91:1
3 suited cards 5.17 18.34:1
2 suited cards 55 0.82:1
Rainbow 39.8 1.5:1
3 connected straight cards
3.45 27.99:1
2 connected straight cards
40 1.5:1
No connected cards
55.6 0.799:1

 

Deriving the probabilities

1. Probability of being dealt a specific starting hand

    a) Preliminary considerations

    Number of starting hands: 169
    Of them:
    - Pocket Pairs: 13
    - Suited Hands: 78
    - Offsuited Hands: 78 (excluding pockets)
    Number of all possible combinations: formel1

    b) Pocket pairs

    Number: 13
    Suit combinations per hand: formel
    (Example: 22, 22, 22, 22, 22, 22)

    Combinations (total): 13 x 6 = 78

    Probabilities
    - Specific pocket pair: formel3
    - In odds: 220:1

    - Any pocket pair: formel4
    - In odds: 16:1

    c) Suited Hands

    Number: 78
    Suit combinations per hand: formel5
    (Example: AK; AK; AK; AK)

    Combinations (total): 78 x 4 = 312

    Probabilities
    - Specific suited hand: formel6
    - In odds: 331:1

    - Any suited hand: formel7
    - In odds: 3.25:1

    d) Offsuited Hands

    Number: 78 (excluding pocket pairs)
    Suit combinations per hand: formel8
    (Example: AK; AK; AK; AK; AK; AK; AK; AK; AK; AK; AK; AK)

    Combinations (total): 78 x 12 = 936

    Probabilities
    - Specific offsuited hand: formel9
    - In odds: 110:1

    - Any offsuited hand: formel10
    - In odds: 0.417:1

    e) Ranges

    Ranges can be derived in nearly the same manner. You just divide the number of combinations in a given range by the number of total possible combinations.

Example 1:

Range: AKs, KQs, QJs, JTs
Number of combinations: 16 (4 per hand)
(Example: AK; AK; AK; AK; KQ; KQ; KQ; KQ; QJ; QJ; QJ; QJ; JT; JT; JT; JT)

Probability: formel11
In odds: 81.9:1

Example 2:

Range: AA, KK, QQ
Number of combinations: 18 (6 per hand)

Probability: formel12
In odds: 72.7:1

2. Probability of facing a higher pocket pair when you have a pocket pair

    a) Probability that a single opponent's pair will be higher

    formel13

    r = Rank of your pocket pair (2=2,... ,J=11, Q=12, K=13, A=14)

    There are (14 – r) x 4 higher cards. Your opponent can have any of the 50 remaining cards (you have the other 2). If his first card is higher than your pocket pair, 3 of the 49 remaining cards could give him a higher pair.

    b) Probability that one of several opponents' pairs will be higher
    Start by multiplying the probability of one player having a higher pair with the number of players remaining in the hand (n). Then subtract the probability that more than one opponent has a higher pocket pair. (formel14).

     

    formel15
    n = Number of players remaining in the hand
    formel 14 Probability that several opponents have a pocket pair, for which

    formel17
    formel18 Probability that exactly n players have a pocket pair, for whichformel19 .

3. Probability of facing several higher pocket pairs

This is derived by the same principle, however: formel20, for which formel21

4. Probability of facing a better A

    a) The probability than a specific opponent will have AA when you have an Ax hand

    formel22
    There are 50 cards remaining (you hold two, one of which is an ace), three of which are aces. If your opponent's first card is an ace, there are 2 of 49 remaining cards that could give him AA.

    b) Probability that an opponent has AA when you have an Ax hand

    formel23
    n = Number of opponents

    c) Probability that a single opponent will have a better A when you have an Ax hand

    formel24
    for which r represents the rank of your second (kicker) card (2=2,..., J=11, Q=12, K=13)

5. Probability that no overcard will come when you have a pocket pair

    a) Preliminary considerations

    Possible flops by any starting hand: formel25

    Possible turns by any starting hand: formel26

    Possible rivers by any starting hand: formel27

    b) No overcard on the flop

    formel28, for which r represents the rank of your pocket pair (2=2,..., J=11, Q=12, K=13)

    c) No overcard on turn

    formel29
    for which r represents the rank of your pocket pair (2=2,..., J=11, Q=12, K=13)

    d) No overcard on river

    formel30
    for which r represents the rank of your pocket pair (2=2,..., J=11, Q=12, K=13)

6. Probability of being dealt a specific hand

The probability of being dealt a specific hand is derived by dividing the number of possible card combinations for said hand by the number of all possible card combinations formel31

Royal Flush: Possible card combinations: formel32
Probability: formel33


Straight Flush: Possible card combinations: formel34
Probability: formel35


Four-of-a-kind: Possible card combinations: formel36
Probability: formel37


Full House: Possible card combinations: formel38
Probability: formel39


Flush: Possible card combinations: formel40
Probability: formel41


Straight: Possible card combinations: formel42
Probability: formel43


Three-of-a-kind: Possible card combinations: formel44
Probability: formel45


Two pair: Possible card combinations: formel46
Probability: formel47


Pair: Possible card combinations: formel48
Probability: formel49


High card: Possible card combinations: formel50
Probability: formel51

7. Probability of improving on the flop

formel52 x represents the number of outs your hand has before the flop; y represents how many of these outs you want to hit; a represents the number of remaining cards (50 - # of outs); and b represents how many of these cards (a) you want to hit.

If you don't want to hit any of these cards, the term formel53 is not necessary. formel54 stands for the number of possible flop combinations (19600).

Example 3:

A pocket pair should improve to (exactly) three-of-a-kind on the flop

formel55

Example 4:

2 suited cards should improve to a completed flush on the flop
formel56

8. Probability of improving on the turn

Simple odds and outs. On the turn:
formel57

Note: If you want to calculate the probability that the even will not take place, you must subtract the result from 1.

9. Probability of improving on the river

Again, simple odds and outs. On the river:
formel58

10. Probability of improving from flop to river

Once again, it's a matter of calculating odds and outs:
formel59

The following formula can be used to calculating runner runner outs:

formel60, for which Outs stands for runner runner outs. With a backdoor flush draw, for example, you have 10 outs for the flush draw and then 9 outs to complete.

Note: This cannot be used for straight/straight flush draws, since the outs depend on each other. In such a case the following formula can be used:

formel61, for which x represents the number of outs for the first runner and y the number of outs for the second.

11. Probability of seeing a specific flop

This calculation does not take your or your opponent's cards into consideration, but calculates the probability of seeing a specific flop with 52 cards in the deck.

Option 1: This can be calculated with the help of binomial coefficients. For the number of possible card combinations for 3 of 52 cards we get formel62. You then calculate the number of card combinations for a specific flop and divide by the number of all possible flops.

formel63

Here are a few examples to help illustrate this:

    a) Probability of seeing three-of-a-kind in the flop
    formel64
    b) Probability of seeing three legs of a straight in the flop (without the possibility of a straight flush)
    formel65

    You subtract the 48 possible combinations that would also be part of a straight flush, since you only want to know the probability of seeing three legs of a normal straight on the flop.

Option 2:
You could also work with probabilities, you just have to make sure you are using the right ones. The first card can basically be any given card and will be written as formel66, though it could be left out altogether. Then look at the events that should/should not take place and multiply the terms.

Here are a few examples to help illustrate this option better:

    c) The probability of seeing three cards of the same suit on the flop
    formel67
    The formel68 simply shows that we start with any given card and can be left out of the formula, as it does not change the result in any way. formel69 and formel70 are the probabilities of the second and third card being the same suit as the first.
    d) The probability of seeing a paired flop
    formel71

    Once again, the first card can be any given card. Once it has fallen, there are 3 of 51 remaining cards that could pair the board. Then one of the 48 cards that does not pair the board must come. You then multiply by 3 since the non-pair card could be any one of the three cards.

    e) Probability that no suited cards will be in the flop
    formel72

    Here you multiply the probabilities of no two cards of the same suit appearing in the flop.

Note

Converting probabilities to odds:

You have not yet learned to convert probabilities to odds. To do so, use this formula:

formel73

P represents the probability you wish to convert to odds. The ":" represents "to" as in '1 to 1' and does not mean to divide by 1.

You can also write formel75instead of formel74 .

 

Comments (41)

#1 theboydave, 03 Sep 09 13:08

What a handy article a download chart of these odds would be a good idea i think thanks.<br />

#2 oedipa, 04 Sep 09 08:52

What a wonderful article !<br /> I've been looking exactly for something like this.<br /> Thank you and keep up the good work :)

#3 ciasteczkoM, 04 Sep 09 09:12

Good job! I was making this equations or at least some of them for my own use from time to time (thanks to format C:) ). Now all i need is here. Thanks again.

#4 NiekamNeidomu, 04 Sep 09 09:47

math T_T

#5 JuiceQuadre, 04 Sep 09 09:58

Nice one!

#6 RahXephon1, 04 Sep 09 10:29

Very nice article. Specially the part with the probability that a better hand wakes up after you. should be usefull to corectly judge the edge an isolation raise should have to be profitable.<br /> <br /> The last part is interesting from a theorethic point of view, but it's probably easier to just remember the probabilities for the most important situations, than calculate them. Also the more marginal ones shouldn't be that important for calculating the profitability of the hand.<br /> <br /> And BTW, can someone please explain to my opponents they have less than 1% to flop a flush, so they won't take my money everytime I go all in on a suited flop.<br /> <br /> Just kidding obviously :)

#7 vladnz, 04 Sep 09 10:36

this is some serious math shit :D

#8 Pardel, 04 Sep 09 14:02

Thanks for this very nice and serious article!

#9 Sinnology, 04 Sep 09 14:02

GREat job.<br /> As someone already mentioned, a download chart of those would be great.<br /> Now I only need to read this like 20 times to absorb everytihng.<br /> Thx

#10 Bojim1, 04 Sep 09 16:26

nice article, thank you

#11 EagleStar88, 04 Sep 09 16:53

@#9, whilst not a download as such, you can of course use the new print function (icon at top right of article) to print off a copy for a little light bed time reading :o)<br /> Great work Indy.

#12 drawback, 04 Sep 09 17:32

Thank you a lot. Will have something really great to study ;)

#13 excelgeo, 05 Sep 09 02:15

what a great article.<br /> <br /> i was wondering about this recently.<br /> <br /> thanks!

#14 xLFC, 05 Sep 09 04:43

Great article, very useful!<br /> <br /> It would also be nice to have the probabilities of improvement from three-of-a-kind to full house (or better) in the charts.<br /> <br /> Thanks!

#15 fffgreek, 05 Sep 09 23:19

At least !!!<br /> You made my day guys ;)

#16 Koshburger, 16 Jan 10 10:48

ok

#17 1z2x3y, 21 Feb 10 17:42

Steven Hawkings eat your heart out ! =]

#18 Rap1d007, 06 Mar 10 10:44

interesting charts

#19 sindeon, 21 Feb 11 21:32

Thanks a lot! One of the best and most useful articles i´ve read here.

#20 DukeXIII, 22 Jul 11 10:27

GUYS.... I LOVE U :)

#21 maheepsangari, 23 Apr 12 12:36

Could someone please explain the calculation under 7. Probability of improving on the flop Example 3 where it calculated probability for hitting a set. Shouldn't it be 2C1*48C1*44C1/50C3. Actually what I calculated is also incorrect since that would lead to 4.6:1 odds and not 7.5:1 odds so can someone please explain the calculation here.

#22 maheepsangari, 23 Apr 12 12:53

I mean to calculate set or full house it would definitely be 2C1*48C2/50C3=11.51% meaning 7.68:1 odds so I'm wondering how to calculate for a set only directly without using the subtraction method.

#23 maheepsangari, 23 Apr 12 13:05

2C1*39C1/50C3 doesn't correspond to 11.5% but istead its 0.397%

#24 genia2q, 18 Jul 13 17:09

Thank you that was very interesting and valuable information I have printed it out and I will try to memorise it thanks.

#25 coolpokername, 12 Dec 13 01:09

I know I'm not expected to memorize all this, but what am I supposed to do with this info?

#26 biggood, 07 Sep 14 18:52

good

#27 coperneeque, 12 Oct 14 17:52

Hi masheepsangari (@21)<br /> <br /> You're right, I believe it's a mistake and they used wrong formula. When calculated it does give 0.397...% that you mention.<br /> I calculated it myself and the probability of 11.5% is correct.<br /> <br /> We have:<br /> C1, C2 - cards of rank C that make the pocket pair<br /> C3, C4 - cards of the rank C left in the deck of remaining 50 cards<br /> E - the event of hitting a set only on the flop (while holding a pocket pair)<br /> <br /> To hit a set only on the flop it means that we have hit either C3 or C4 and the remaining 2 cards are random.<br /> <br /> To calculate number of events corresponding to E we will start from the back end:<br /> Let's pick those 2 cards. From 48 cards we pick 2:<br /> 48! / [2!(48-2)!]<br /> Now, those 2 cards come with the 3rd card - C3 or C4 so that makes twice that number:<br /> because 2! = 2<br /> we have:<br /> 48! / (48-2)! = 48! / 46! = 48*47 = 2256<br /> and that's the number of events corresponding to E.<br /> <br /> Total number of events (all possible flops): 19600<br /> <br /> Now it's easy:<br /> 2256/19600 = 0.115... = 11.5%<br /> <br /> Hope this helps!

#28 toske1, 13 Mar 15 18:33

vrh

#29 mirth, 27 Mar 15 21:10

thankfully this has been simplified in other charts!

#30 vinkojudi, 31 Jul 15 11:20

nice

#31 vinkojudi, 31 Jul 15 11:20

nice

#32 FlyingDutchm1n, 08 Oct 15 23:49

Great overview right here I knew most of them but I especially found the stats on the straightyness (connectedness of the ranks like 3 connected straight ranks) superhelpful thank you pokerstrategy

#33 Leonventer112, 17 Oct 15 03:32

I love stats

#34 minsa0123, 12 Nov 15 06:09

Thanks for this very nice and serious article!

#35 bubamarasr, 22 Jan 16 20:15

Read it. Thank you.

#36 hassux, 25 Jan 16 21:30

badi nik kiss emou

#37 Hesticus, 02 Feb 16 09:22

@coperneeque: Do you like solving math problems? :D

#38 forceking, 21 Mar 16 20:21

a download option of these tables would be good

#39 RedFoxyBam, 03 Apr 16 16:51

Thanks very much for this article very useful in the probability of having a kind of hand, and it helped me in a way that I better organize my self in a matter of ranges... Have a good one people

#40 sedinbsng, 04 Apr 16 11:14

ok

#41 CroZoZo, 26 Apr 16 13:23

97