ACES

Probability that one or more players will have an Ace dealt to them with 9, 8, 7, 6, 5, 4, 3, 2 players

9: 83%
8: 72%,
7: 67%,
6: 62%,
5: 55%,
4: 47%,
3: 38%,
2: 27%,

If you are dealt an Ace, what is the probability that one or more of the other 8, 7, 6, 5, 4, 3, 2, 1 players will be dealt an A

8: 47%,
7: 42%,
6: 38%,
5: 32%,
4: 27%,
3: 21%,
2: 15%,
1: 7%,

AX rule => Will it be beaten by remaining players

Probability of facing an ace with larger kicker when holding AX          
AX Against 1 Against 2 Against 3 Against 4 Against 5 Against 6 Against 7 Against 8 Against 9
A2 11.0% 20.8% 29.6% 37.3% 44.2% 50.4% 55.8% 60.7% 65.0%
A3 10.0% 19.1% 27.2% 34.5% 41.1% 47.0% 52.3% 57.1% 61.4%
A4 9.1% 17.3% 24.8% 31.6% 37.8% 43.4% 48.6% 53.2% 57.5%
A5 8.1% 15.5% 22.3% 28.6% 34.4% 39.7% 44.6% 49.0% 53.2%
A6 7.1% 13.7% 19.8% 25.5% 30.8% 35.7% 40.3% 44.5% 48.5%
A7 6.1% 11.9% 17.3% 22.3% 27.1% 31.6% 35.7% 39.7% 43.4%
A8 5.1% 10.0% 14.6% 19.0% 23.2% 27.2% 30.9% 34.5% 37.8%
A9 4.2% 8.2% 12.0% 15.6% 19.2% 22.5% 25.7% 28.8% 31.8%
AT 3.2% 6.3% 9.3% 12.1% 14.9% 17.6% 20.3% 22.8% 25.3%
AJ 2.2% 4.4% 6.5% 8.5% 10.5% 12.5% 14.4% 16.3% 18.2%
AQ 1.2% 2.4% 3.6% 4.8% 6.0% 7.1% 8.3% 9.4% 10.5%
AK 0.2% 0.5% 0.7% 1.0% 1.2% 1.5% 1.7% 1.9% 2.2%
Better than 1 in 3 chance no better AX in the hand (3 opponents is the key)

Outs and Probabilities

Type of draw Hand The flop Specific outs # Outs Flop to turn Turn to river Turn & River
Pocket pair to set 22 Q49 22 2 4.3%(22-1) 4.3% (22-1) 8.4% (11-1)
One overcard A8 952 AAA 3 6.4% (14-1) 6.5%(14-1) 12.5% (7-1)
Inside straight draw J9 Q84 10 4 8.5% (11-1) 8.7% (11-1) 16.5% (5-1)
Two pair to full house KQ KQ5 KKQQ 4 8.5% (11-1) 8.7% (11-1) 16.5% (5-1)
One pair to two pair or trips AQ A 103 AAQQQ 5 10.6% (8-1) 10.9% (8-1) 20.3% (4-1)
No pair to pair 97 23J 999777 6 12.8% (13-2) 13% (13-2) 24.1% (3-1)
Two overcards to overpair AJ 10 8 2 AAAJJJ 6 12.8% (13-2) 13% (13-2) 24.1% (3-1)
Set to full house/ four of a kind 66 67J 6777JJJ 7 14.9% (11-2) 15.2% (11-2) 27.8% (5-2)
Open-ended straight draw 98 7103 J6 8 17% (5-1) 17.4% (5-1) 31.5% (2-1)
Flush draw KJ A68 2345789107 9 19.1% (4-1) 19.6% (4-1) 35% (2-1)
Inside straight draw and two overcards AK 10 2 JAAAKKK 10 21.3% (4-1) 21.7% (4-1) 38.4% (3-2)
Inside straight and flush draw AK JQ3 102456789Q 12 25.5% (3-1) 26.1% (3-1) 45% (5-4)
Open-ended Straight and Flush Draw KQ 10 J 4 9A235678J 15 31.9% (2-1) 32.6% (2-1) 54.1% (5-6)

Rule of Thumb

The “ win probavility formula” looks like this:
(OUTS X 2) + 1 = % of getting a card you need
That formula works anytime you have between three and eleven outs.
If you have FEWER than three outs, it doesn’t really matter… since you should fold anyway.
For example, let’s say you pick up an open-ended straight draw and have eight outs.
(8 outs x 2) + 1 = 17%

"Rule of Four and Two". With two cards to come, the percent chance of hitting x outs is about (Outs X 4 (%); With one car to come its Outs X 2 (%)
Outs Make on turn or river Make on river
1 4.2% 2.2%
2 8.4% 4.3%
3 12.5% 6.5%
4 16.5% 8.7%
5 20.3% 10.9%
6 24.1% 13.0%
7 27.8% 15.2%
8 31.4% 17.4%
9 34.9% 19.5%
10 38.4% 21.7%
11 41.7% 23.9%
12 44.9% 26.1%
13 48.1% 28.2%
14 51.1% 30.4%
15 54.1% 32.6%
16 56.9% 34.8%
17 59.8% 36.9%
18 62.4% 39.1%
19 65.0% 41.3%
20 67.5% 43.4%
21 69.9% 45.6%
22 72.2% 47.8%
23 74.5% 50.0%
Better than 30%

Win rate for Suited and Unsuited 4 players

.. A K Q J T 9 8 7 6 5 4 3 2
As 64 40 38 37 36 33 32 31 30 30 29 28 27
Ks 43 59 36 35 34 31 29 28 27        
Qs 41 39 54 34 33 30 28            
Js 40 38 37 50 32 29 27            
Ts 39 37 36 35 46 29 27            
9s 36 35 33 33 33 42              
8s 35 33 31 31 31 30 38            
7s 34 32 29 29 29 28 28 35          
6s 33 31 29 27         32        
5s 34 30 28             29      
4s 33 29 27               27    
3s 32 29                   27  
2s 31 28                     27
                           
  >7                        
  >30%                        
  Suited A-2 K-5 Q-8 J-8 T-8              
  Unsuited A-2 K-9 Q-9 J-T TT              

Common Odds When Holding Three Gapped Connectors From A5 To AT:  
flopping a straight (including the slight chance of a straight flush in some cases) 0.3%
flopping an 8 out straight draw* 2.6%
Common Odds When Holding One Gapped Connectors From 53 To QT:  
flopping a straight (including the slight chance of a straight flush in some cases) 1.0%
flopping an 8 out straight draw* 8.0%
Common Odds When Holding Connectors From 54 To JT:  
flopping a straight (including the slight chance of a straight flush in some cases) 1.0%
flopping an 8 out straight draw* 10.0%
Common Odds When Holding Two Suited Cards:  
flopping a flush (including the slight chance of a straight flush in some cases) 0.8%
flopping a four flush 10.9%
Common Odds When Holding Two Unsuited Cards:  
flopping a four flush 2.2%
Common Odds When Holding Paired Hole Cards:  
flopping EXACTLY two pair by pairing the board 16.1%
flopping EXACTLY trips by flopping a set for your pocket pair 10.8%
flopping EXACTLY a full house, a set to your hole pair and pairing the board 0.7%
flopping EXACTLY a full house, by the board tripping up 0.3%
flopping EXACTLY four of a kind, two cards to your hole pair 0.3%
Common Odds When Holding Unpaired Hole Cards:  
flopping EXACTLY one pair by pairing a hole card 26.9%
flopping EXACTLY two pair by pairing a hole card AND pairing on the board 2.0%
flopping EXACTLY two pair by pairing EACH of your hole cards 2.0%
flopping EXACTLY trips by flopping two cards to one hole card 1.3%
flopping EXACTLY a full house, trips of one hole card and pairing the other 0.1%
flopping EXACTLY four of a kind, three cards to one of your hole cards 0.1%

Simplified Odds

With 4 cards of the same suit after the flop there is a 38.3% chance of completing the flush by the river.

With 3 cards of the same suit after the flop, there is a 32.6% chance of completing the flush by the river.

With 2 cards of the same suit after the flop, there is a 29.8% chance of completing the flush by the river.

With 4 connected cards after the flop, there is a 34.8% chance of completing a 5-card straight by the river.

With 2 connected cards after the flop, there is an 8.7% chance of completing a 5-card straight by the river.

With 3 connected cards after the flop, there is a 17% chance of completing a 5-card straight by the river.

With A-K-Q after the flop, you have approximately a 17.0% chance of completing a 5-card straight by the river.

Probability of Completing Hand - Flop => River
No. of Outs Percent No. of Outs Percent
20 68 10 38
19 65 9 35
18 62 8 31
17 60 7 28
16 57 6 24
15 54 5 20
14 51 4 16
13 48 3 12
12 45 2 8
11 42 1 4
       
Straight Flush Draw
Inside Straight Draw => 4
Flush Draw => 9
two pair or gunshot draw
Pair to Set of 3 => 2
Two pair to Full House => 4
Open Ended Straight Draw => 8

Poker Odds Chart
Odds and outs
Outs Odds:
Flop-Turn / Turn-River
Odds:
Flop-River
Example
1 46:1 22.5:1 Backdoor flush draw
2 22.5:1 11:1 Pocket pair to set
4 11:1 5:1 Gunshot straight
5 8:1 4:1 Two pair/trips
8 5:1 2:1 Open-ended straight draw
9 4:1 2:1 Flush draw
12 3:1 1:1 Flush draw + gutshot
13 2.5:1 1:1 OESD + pair
14 2.5:1 1:1 Flush draw + pair
15 2:1 1:1 Flush draw + OESD

Flush

Straight

Odds and Sods - Probabilities and Odds for Best Poker Bets

Frequency of Both Start Cards being Dealt

[6,7,8,9,10,J,Q,K,A]

>5    1:2
>6    2:5 [7,8,9,10,J,Q,K,A] 8 of 13 cards
>7    1:3
>8    1:5
>9    1:6
>10    1:11 [J,Q,K,A]

Basic Probabilities

Probability of being dealt:

Odds of connecting with the Flop in Hold’em

Flush Statistics

Probability of there being a flop with 3 Suited cards

Your first card can be anything. So you have 52 choices out of 52 cards (because no matter what card you draw you can get a full hand of the same suite). Your second card, has to be the same suit as your first card, so probability of that is 1251 because there are 13 of each suite and you have to subtract 1 for the one card you have drawn. Your third card has to be the same suite as the first and the second, notice there are only 11 cards left of that suite, so selecting that specific card will be 11/50

Giving a total probability of:

(52/52)×(12/51)×(11/50) = 0.05 => 1 in 20

Completing the Flush

With 4 suited cards (various combos) => you have 9 outs to make your hand on the turn or river (There are 13 cards per suit and you have 4 of them). So you have 9 outs out of 47 total unknown cards (52 cards in the deck – your 2 cards and – 3 more on the flop) => 9/47 => 1 in 5 chance to hit your flush after the turn or the river. The odds can get better late in the game as fewer cards remain in the pack BUT some of your suited cards may have been buried.

With 3 suited cards after the flop, you have 10 outs to make your hand after the turn and river but you need to score twice (both times) => Your chance of hitting the flush on the turn is 9/47 = 19.15% (about 1 in 5). If you don't hit on the turn, your change of hitting the river is 9/46 = 19.57%. Your overall chances are 19.15% + (19.57% * (1-0.1915)) = almost exactly 35%.

Odds On the Flop in Texas Hold’em

Outs

This is especially where “outs” come into your line of thinking and how all of these Texas Hold’em odds are generated. For example, if you have 4 cards to a flush you have 9 outs to make your hand on the turn. There are 13 cards per suit and you have 4 of them.

There are 9 unknown cards left that could complete your flush so you have 9 outs out of 47 total unknown cards (52 cards in the deck – your 2 cards and – 3 more on the flop). This is how Texas Hold’em odds are calculated. 9/47 = 19.1, or a 19.1% chance to hit your flush on the turn.

Straight and Flush Draw Odds

Drawing to open-ended straights and flushes, or fear of your opponents doing so, is one of the most common scenarios in Hold’em.

On the flop, when you have:

Note that the figures above also apply on the turn to calculate odds for the river since you have the same 1 card to come.

Odds of hitting a hand by the river from the flop

The following set of odds is the likelihood to complete these hands by the river on the flop, so with 2 cards to come.

On the flop, when you have:

All-in One-on-One in Texas Hold’em

Preflop matchups when played to showdown:

Extra Texas Hold’em odds

EXTRA ODDS

Pairs - chance overcard in flop .
KK 22%
QQ 41%
JJ 56%
TT 69%
99 80%
88 86%
77 92%
66 96%
55 98%
higher pair beats lower pair 82%
lower pair beats higher pair 18%
KK same chance as 33 or 22 against AA 18%
Pair beats 2 higher cards (22 vs AJ) 55%
Pair beats 2 lower cards (QQ vs T6) 86%
Pair beats 2 lower cards (suited or connected) 70%
Pair beats cards 1 higher/1 lower (QQ vs A7) 71%
Pair flops a set or better 11.8%
Pair flops a set 11%
Pair flops a full house 1%
Pair flops 4 of a kind 1 0.25%
Matched Pair to trip 8%
Have 2 pairs after flop => full house 16%
Odds get an A 15%
Odds get an A or a pair 21%
Dealt an A, probability of no one else having one: .
2 players 88%
3 players 77%
4 players 67%
5 players 58%
6 players 50%
7 players 43%
8 players 36%
9 players 30%
That is if you have an A with 9 other players, .
there’s a 70% chance that at least one other player also has an A! .
. .
Straights - chance to complete .
3 card straight after flop the chance of completing 30%
2 connected cards to complete straight 2%
J-10 is the best of the connectors in one way – because these are the only cards that make 4 straights and each straight is the nuts! nuts. .
. .
Trips .
Trips on the flop chance of full house or better 33%
. .
Flushes .
If you start with any 2 suited cards: .
to flop a flush 1%
to flop a four card flush 11%
to complete a flush with four cards flushed 35%
with any 2 suited cards get a complet flush 4%
.
AK .
AK unsuited vs small suited connectors 7:5.
AK suited vs small unsuited connectors 9:5.
AK unsuited vs a pair 10:11.
AK suited vs a pair 1:1
. .
Other odds .
Any 2 cards vs any 2 lower cards (e.g. AK vs 7-2) 63%
Two higher cards vs a pair (e.g. KQ vs 9-9) 45%
Two suited higher cards vs a pair (e.g. KQs vs 9-9) 50%
One card higher and one lower than a pair (e.g. K-10 vs QQ) 40%
One card higher and one in between Villain’s 2 cards (e.g. AJ vs K9) 150%
One card higher and one lower than Villain’s 2 cards (e.g. A-2 vs J-4) 56%
One card the same as Villain’s, the other higher (e.g. A-10 vs A-9) 71%
No pair in card improving to at least a pair 27%
. .
Other insights .
Most flops miss pairing most hands! (Indeed, if you hold AK, you won’t pair either card on the flop 68% of the time). .
The probability that your opponent is bluffing is a least 10% .
Any time you have an A in your hand and you are getting a little over 2:1 pot odds to call, you pretty much have to call. It’s only against AA that your chances are markedly worse. .

OUTS & ODDS

  Flop to Turn   Turn to River   Turn and River  
Outs % Odds % Odds % Odds
20 42.6% 1.35-1 43.5% 1.30-1 67.5% 0.48-1
19 40.4% 1.47-1 41.3% 1.42-1 65.0% 0.54-1
18 38.3% 1.61-1 39.1% 1.56-1 62.4% 0.60-1
17 36.2% 1.77-1 37.0% 1.71-1 59.8% 0.67-1
16 34.0% 1.94-1 34.8% 1.88-1 57.0% 0.75-1
15 31.9% 2.13-1 32.6% 2.07-1 54.1% 0.85-1
14 29.8% 2.36-1 30.4% 2.29-1 51.2% 0.95-1
13 27.7% 2.62-1 28.3% 2.54-1 48.1% 1.08-1
12 25.5% 2.92-1 26.1% 2.83-1 45.0% 1.22-1
11 23.4% 3.27-1 23.9% 3.18-1 41.7% 1.40-1
10 21.3% 3.70-1 21.7% 3.60-1 38.4% 1.60-1
9 19.1% 4.22-1 19.6% 4.11-1 35.0% 1.86-1
8 17.0% 4.88-1 17.4% 4.75-1 31.5% 2.17-1
7 14.9% 5.71-1 15.2% 5.57-1 27.8% 2.60-1
6 12.8% 6.83-1 13.0% 6.67-1 24.1% 3.15-1
5 10.6% 8.40-1 10.9% 8.20-1 20.3% 3.93-1
4 8.5% 10.75-1 8.7% 10.50-1 16.5% 5.06-1
3 6.4% 14.67-1 6.5% 14.33-1 12.5% 7.00-1
2 4.3% 22.50-1 4.3% 22.00-1 8.4% 10.90-1
1 2.1% 46.00-1 2.2% 45.00-1 4.3% 22.26-1

Extra Probabilities

The Probability that....

Odds

%

 

Non-pairs will pair at least one card

2-to-1

32%

 

Non-pairs will pair both hold cards

50-to-1

2%

 

A pair will flop a set

8-to-1

12%

 

A pair will flop four of a kind

400-to-1

0.3%

 

Two suited cards will flop a flush

118-to-1

1%

 

Two suited cards flop a four flush (flush draw)

9-to-1

11%

 

Two suited cards will make a flush by the river

15-to-1

7%

 
       

Probability that ....

     

You Hold

Hope to Make

Outs

 

A Pair

Three of a Kind

2

 

Two Pair

Full House

4

 

Inside Straight

Straight

4

 

Overcards

Pair

6

 

Open-ended Straight

Straight

8

 

Four Flush

Flush

9

 

Straight & Flush Draw

Straight / Flush or Better

15

 
       

Probability of improving on the flop

     

Starting hand

Improvement on flop

Probability in %

Odds

Pocket pair

Three-of-a-kind or better

12.7

7:1

Pocket pair

Three-of-a-kind

11.8

8:1

Pocket pair

Full house

0.73

136:1

Pocket pair

Four-of-a-kind

0.24

415:1

2 unpaired cards

Pair

32.4

2:1

2 unpaired cards

Two pair

2

48:1

Suited cards

Flush

0.8

118:1

Suited cards

Flush draw

10.9

8:1

Suited cards

Backdoor flush draw

41.6

2:1

Connectors 45o-JTo

Open Ended Straight Draw

9.6

9.:1

Connectors 45s-JTs

Straight draw / flush draw

19.1

4:1

Connectors 45o-JTo

Straight

1.31

75:1

       

Probability of improving on the turn

     

Starting hand

Improvement on flop

Probability in %

Odds

Flush draw

Flush

19.1

4.:1

OESD

Straight

17

5:1

Gutshot straight draw

Straight

8.5

11:1

Three-of-a-kind

Four-of-a-kind

2.1

47:1

Two pair

Full house

8.5

11:1

Pair

Three-of-a-kind

4.3

22.:1

Two unpaired cards

Pair (with hole card)

12.8

7:1

       

Probability of improving on the river

     

Starting hand

Improvement on flop

Probability in %

Odds

Flush draw

Flush

19.6

4.1:1

Open Ended Straight Draw

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

     

Starting hand

Improvement on flop

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

     

On the flop

Probability in %

Odds

 

Three-of-a-kind

0.24

415:1

 

Pair

16.9

5:1

 

3 suited cards

5.17

18:1

 

2 suited cards

55

0.8:1

 

Rainbow

39.8

1.5:1

 

3 connected straight cards

3.45

28:1

 

2 connected straight cards

40

1.5:1

 

No connected cards

55.6

0.81

 
       

Rules for Starting Cards

     

Starting Pairs

Number (suited)

Odds of getting these Start Cards

 

AA to A7

124 (31)

   

KK to K7

108 (28)

   

QQ to Q10

44 (11)

   

JJ and J10

28 (7)

   

Additional Pairs & Adjacent Cards >6

136 (24)

   

Total for all above with 2652 combos

440 (101)

about 1 in 6 hands

 

Including A6 to A2 suited , K6 to K2 suited and Q9 to Q7 suited

500

about 1 in 5 hands