Texas Hold'em Poker Probability - Pocket Pair vs. 2 Overcards(suited, connected) cont.

So how large is q, q is the winning rate when there is no 7, but any TJ combinations for the 5 cards on the table.

Let's check the possible 5 card combinations: (assuming pocket pair is 7s7c, player X has TdJd)

Case 1. You make a flush, the combination of the 5 card is
   a. five spade (including Ts or Js or both): C(11,4)+C(11,4)-C(10,3)=540
       less five spade (including Ts or Js or both, and all higher than 7): C(6,4)+C(6,4)-C(5,3)=20... tie
       five spade (8s9sTsJsKs, 8s9sTsJsAs): 2... you get straight flush

   b. four spade (including Ts or Js or both) with X, X is not in (7,T,J):
            (C(10,3)+C(10,3))*(52-4*3-10)+C(10,2)*(52-4*3-10)= 8,550
       four spade (with only Ts) with Tc/Th/Jc/Jh: C(10,3)*4= 480
       four spade (with only Js) with Tc/Th/Jc/Jh: C(10,3)*4= 480
       four spade (without Ts,Js) with Tc/Th/Jc/Jh: C(10,4)*4=840

   c. five club: same as 1.a: 540-20+2=522

   d. four club: same as 1.b: 8550+480+480+840=10,350

   The sum is 21,762, the probability is about 1.27%.

Case 2. You make a straight, the card is
   a. 3456T, 3456J: 4*4*4*4*3+4*4*4*4*3=1,536
       straight flush: 3s4s5s6sT, 3s4s5s6sJ, 3c4c5c6cT, 3c4c5c6cJ: 3*4=12

   b. 4568T, 4568J: 4*4*4*4*3+4*4*4*4*3=1,536
       straight flush: 4s5s6s8sT, 4s5s6s8sJ, 4c5c6c8cT, 4c5c6c8cJ: 3*4=12

   c. 5689T, 5689J: 4*4*4*4*3+4*4*4*4*3=1,536
       straight flush: 5s6s8s9sT, 5s6s8s9sJ, 5c6c8c9cT, 5c6c8c9cJ: 3*4=12

   d. 689Tx, x is not in (Q,5,6,7,8,9,T): 4*4*4*3*(52-4*7-1)=4,416
          straight flush: 6s8s9sTsx,6c8c9cTcx: (52-4*7-1)*2=46
       689T6, 6*4*4*3=288
          straight flush: 6s8s9sTs6,6c8c9cTc6: 3+3=6
       689T8, 4*6*4*3=288
          straight flush: 6s8s9sTs8,6c8c9cTc8: 3+3=6
       689TT, 4*4*4*3=192
          straight flush: 6s8s9sTsT, 6c8c9cTcT: 2+2=4

   e. 89TJx, x is not in (6,7,8,9,T,J,Q): 4*4*3*3*(52-4*7)=3,456
          straight flush: 8s9sTsJsX, 8c9cTcJcX: (52-4*7)*2=48
       89TJ8, 6*4*3*3=216
          straight flush: 8s9sTsJs8, 8c9cTcJc8: 3+3=6
       89TJ9, 4*6*3*3=216

          straight flush: 8s9sTsJs9, 8c9cTcJc9: 3+3=6

       89TJT, player X gets a full house.

          straight flush: 8s9sTsJsT, 8c9cTcJcT: 2+2=4

       89TJJ, player X gets a full house.

          straight flush: 8s9sTsJsJ, 8c9cTcJcJ: 2+2=4

   The sum is 13,680, the probability is about 0.8%

Combine the result of case 1 and case 2, the value of q is around 2%.

Texas Hold'em Poker Probability - Pocket Pair vs. 2 Overcards(suited, connected)

With continuing our previous discussion, say if you have pocket pair 77(no diamond), the other player X has TJ (suited-diamond).  Let's check the probability of player X getting a diamond flush.

The remaining 5 cards has:

1. 3 diamond cards: C(11,3)*C(37,2)=109,890
    straight-flush: 4*C(37,2)=2,664
    a) 789TJ to TJQKA: 4

2. 4 diamond cards: C(11,4)*C(37,1)=12,210
    straight-flush: (7+7+7+8)*C(37,1)=1,073
    a) 789TJy, y is not Q: C(7,1)= 7
    b) 89TJQy, y is not K: C(7,1)= 7
    c) 9TJQKy, y is not A: C(7,1)= 7
    d) TJQKAy: C(8,1)= 8


3. 5 diamond cards: C(11,5)=462
    straight-flush: 4+21+21+21+28=95
    a) A2345TJ to 45678TJ: 4
    b) 789TJyz, yz is not Q: C(7,2)= 21
    c) 89TJQyz, yz is not K: C(7,2)= 21
    d) 9TJQKyz, yz is not A: C(7,2)= 21
    e) TJQKAyz: C(8,2)= 28

The probability of flush is about 7.16%, without straight-flush is 6.93%

If we roughly discount the previous winning rate against TJ(unsuited) by 7%, the probability of winning is about 48%.

=====================
Another approach...

The remaining 5 cards has:

1. 3 diamond cards:
    a. no 7,T,J: C(10,3)*C(30,2)=52,200
    b. with a 7 but no TJ:
          7 is diamond: C(10,2)*C(30,2)=19,575
          7 is not diamond: C(10,3)*C(30,1)=3,600
    c. with a 7 only one T/J:
          7 is diamond: C(10,2)*6*C(30,1)=8,100
          7 is not diamond: C(10,3)*6=720

    straight-flush: 1305+435+90+180+18=2028
    a. no 7,T,J: 89Q,9QK,QKA: 3*C(30,2)=1,305
    b. with a 7 but no TJ:
          7 is diamond: 789: 1*C(30,2)=435
          7 is not diamond: 89Q,9QK,QKA: 3*C(30,1)=90
    c. with a 7 only one T/J:
          7 is diamond: 789: 1*6*C(30,1)=180
          7 is not diamond: 89Q,9QK,QKA: 3*6=18

2. 4 diamond cards:
    a. no 7,T,J: C(10,4)*C(30,1)=6,300
    b. with a 7 but no TJ:
          7 is diamond: C(10,3)*C(30,1)=3,600
          7 is not diamond: C(10,4)=210
    c. with a 7 only one T/J:
          7 is diamond: C(10,3)*6=720

    straight-flush:
    a. no 7,T,J: (6+6+7)*C(30,1)= 570
             i) 89TJQy, y is not 7,K: C(6,1)= 6
             ii) 9TJQKy, y is not 7,A: C(6,1)= 6
             iii) TJQKAy, y is not 7: C(7,1)= 7

    b. with a 7 but no TJ:
          7 is diamond: (7+1+1+1)*30=300
             i) 789TJy, y is not Q: C(7,1)= 7
             ii) 89TJQy, y is not K, y is 7: 1
             iii) 9TJQKy, y is not A, y is 7: 1
             iv) TJQKAy, y is 7: 1

          7 is not diamond: (6+6+7)*30=570
             i) 89TJQy, y is not 7,K: C(6,1)= 6
             ii) 9TJQKy, y is not 7,A: C(6,1)= 6
             iii) TJQKAy, y is not 7: C(7,1)= 7
    c. with a 7 only one T/J:
          7 is diamond: (7+1+1+1)*6=60
            i) 789TJy, y is not Q: C(7,1)= 7
            ii) 89TJQy, y is not K, y is 7: 1
            iii) 9TJQKy, y is not A, y is 7: 1
            iv) TJQKAy, y is 7: 1

3. 5 diamond cards:
    a. no 7,T,J: C(10,5)=252
    b. with a 7 but no TJ:
          7 is diamond: C(10,4)=210

    straight-flush: 95

The total is 95,487(flush) less 3623(straight flush), the probability is 5.36%.  The winning rate is 51.73%-5.36%+q=46.37%+q.

Texas Hold'em Poker Probability - Pocket Pair vs. 2 Overcards(not suited, connected)

With continuing our previous discussion, say if you have pocket pair 77, the other player X has TJ (not suited).

1.Player X could get a 789TJ straight:

In case b, with a 7, but no TJ:

  b.1) 789yz: 2*4*4*C(28,2)  yz not in (7,8,9,T,J,Q)

Sum of (b.1) is 12,096, the probability is about 0.71%. 

In case c,

  c.1) 789Ty, 789Jy: 2*4*4*3*C(28,1)*2=5,376, the probability is about 0.31%. 

2.Player X could get a 89TJQ straight:

In case a, without 7,T,J, but 

  a.1) 89Qyz: 4*4*4*C(24,2)  yz not in (7,8,9,T,J,Q,K)

  a.2) 889Qy,899Qy,89QQy:  6*4*4*C(24,1)*3

  a.3) 8899Q,889QQ,899QQ: 6*6*4*3

Sum of (a.1,a.2,a.3) is 25,008, the probability is about 1.46%. 

In case b,

  b.1) 789Qy: 2*4*4*4*C(24,1)  y not in (7,8,9,T,J,Q,K)

Sum of (b.1) is 3,072, the probability is about 0.18%.

In case c,

  c.1) 789QT, 789QJ: 2*4*4*4*3*2=768, the probability is about 0.04%. 

3.Player X could get a 9TJQK straight:

In case a.
  a.1) 9QKyz: 4*4*4*C(24,2)  yz not in (7,9,T,J,Q,K,A)

  a.2) 99QKy,9QQKy 9QKKy:  6*4*4*C(24,1)*3

  a.3) 99QQK,99QKK,9QQKK: 6*6*4*3

Sum of (a.1,a.2,a.3) is 25,008, the probability is about 1.46%. 

In case b,

  b.1) 79QKy: 2*4*4*4*C(24,1)  y not in (7,9,T,J,Q,K,A)

Sum of (b.1) is 3,072, the probability is about 0.18%. 

In case c,

  c.1) 79QKT, 79QKJ: 2*4*4*4*3*2=768, the probability is about 0.04%. 

4.Player X could get a TJQKA straight:

In case a,
  a.1) QKAyz: 4*4*4*C(28,2)  yz not in (7,T,J,Q,K,A)

  a.2) QKAy,QKAy QKAy:  6*4*4*C(28,1)*3

  a.3) QQKKA,QQKAA,QKKAA: 6*6*4*3

Sum of (a.1,a.2,a.3) is 32,688, the probability is about 1.91%. 

In case b,

  b.1) 7QKAy: 2*4*4*4*C(28,1)  y not in (7,T,J,Q,K,A)

Sum of (b.1) is 3,584, the probability is about 0.21%. 

In case c,

  c.1) 7QKAT, 7QKAJ: 2*4*4*4*3*2=768, the probability is about 0.04%. 

The probability for player X to get a straight (without you getting a full house or 4-of-a-kind) is about 6.55%.  Your probability of winning is lowered to about 51.73%+q, anyway you could still win with a flush.

Texas Hold'em Poker Probability - Pocket Pair vs. 2 Overcards(not suited, not connected)

Say if you have pocket pair 77, the other player X has AJ (not suited).  What's your probability of winning before the flop?

Possible remaining 5 card combinations: 1,712,304=48*47*46*45*44/5!=C(48,5)

Let's try to calculate the upper bound of the probability by listing out the possible cases.

The remaining 5 cards:

a) Without 7, A, J: 658,008=C(40,5)  the number of remaining cards without 7AJ is 40.

b) With a 7, but no AJ: 182,780=2*C(40,4)  two possible 7

c) With a 7, and only one A or J: 118,560=2*6*C(40,3)  two possible 7, six possible AJ

d) With a 7, and only (AA, JJ, AJ): 23,400=2*(3+3+9)*C(40,2)  3 possible AA, 3 JJ, 9 AJ

e) With 77, and but no AAA(JJJ): 15,178=C(46,3)-2  combination of the remaining 46 cards minus 2 for AAA/JJJ

f) Without 7, but with any AJ combinations: assuming your winning rate is q.

If we don't consider the other flush or straight possibilities for player X to win, the sum from a to f would be the upper bound for you to win.

The sum is 997,926+, the probability is about 58.28%+q, this is the upper bound of the probability for you to win.

Player X could get a TJQKA straight:

In case a, without 7,A,J, but 

  a.1) TQKyz: 4*4*4*C(28,2)  yz not in (7,A,J,T,Q,K)

  a.2) TTQKy,TQQKy TQKKy:  6*4*4*C(28,1)*3

  a.3) TTQQK, TTQKK,TQQKK: 6*6*4*3

Sum of (a.1,a.2,a.3) is 32,688, the probability is about 1.91%. 

In case b,

  b.1) 7TQKy: 2*4*4*4*C(28,1)  y not in (7,A,J,T,Q,K)

  b.2) 7TTQK,7TQQK,7TQKK:  2*6*4*4*3

Sum of (b.1,b.2) is 3,584, the probability is about 0.21%. 

  

In case c,

  c.1) 7TQKA, 7TQKJ: 2*4*4*4*3*2=768, the probability is about 0.04%. 

So, these cases will lower your winning probability by 2.16% to 56.12%+q without considering you win with a flush.

Texas Hold'em Poker Probability - Paired Flop cont..

From the previous discussion, we assumed that the players play any kind of starting hands.  Or we should say that the probability distribution is the same for each starting hand.

If a player is selective on the starting hands with considering each player's chips, strategy, style, and position,  the probability of each starting hand will be different.

So the estimated probability of the possible 5 card combination (after the flop) is based on the fact (the paired flop) and the probability distribution of the player's starting hand.

For example, if the flop is 99K, and from the past experience player X only plays starting hand 99.  Then we would know the probability of four of a kind is 100% for player X.

Texas Hold'em Poker Probability - Paired Flop cont.

With your two hole cards, the other players might have...

Possible starting hands: 1081=47*46/2=C(47,2)

Assuming the flop is 99K,

a) if you flop a three of a kind with 9x

Four of Kind: 0

Three of a Kind or Full House: 1*46+3=49  type 1: 9y, type 2: KK

Two Pair: 3*43+10*6+3=192  type 1: Ky, type 2: yy, type 3: xx

Others: 43*42/2-(10*6+3)=840  any combination without 9 and K without one x card(43*42/2), minus yy and xx pairs.

There is about 22.3% (1 in 4.5) having a two pair or better.

b) if you have a paired starting hand with QQ

Four of Kind: 1

Three of a Kind or Full House: 2*45+3=93  type 1: 9y, type 2: KK

Two Pair: 3*42+10*6+1=187  type 1: Ky (y is any card <> K9), type 2: yy (y<>Q), type 3: QQ

Others: 42*41/2-(10*6+1)=800  any combination without 9s and Ks and two Q cards(42*41/2), minus pairs.

There is about 26% (1 in 4) having a two pair or better.  It's slight better than the original 25% that doesn't consider your hole cards.

c) if you flop a two pair with Kx

Four of Kind: 1

Three of a Kind or Full House: 2*45+1=91  type 1: 9y, type 2: KK

Two Pair: 2*43+10*6+3=149  type 1: Ky, type 2: yy ( y not in (9,K,x)), type 3: xx

Others: 43*42/2-(10*6+3)=840

There is about 22.3% (1 in 4.5) having a two pair or better.

d) if your starting hand is XY, X<>Y and not in (9,K)

Four of Kind: 1

Three of a Kind or Full House: 2*45+3=93  type 1: 9y, type 2: KK

Two Pair: 3*42+9*6+2*3=186  type 1: Kz (z is any card not in (9,K)), type 2: zz (z not in (X,Y)), type 3: xx ( x is X or Y)

Others: 42*41/2-(9*6+2*3)=801

There is about 26% (1 in 4.5) having a two pair or better.

Texas Hold'em Poker Probability - Paired Flop

As we might see a paired flop about 17% of time, let's take a look at what kind of hands players might have.

Possible starting hands: 1176=49*48/2=C(49,2)  we can see here the possible starting hands are updated based on the flop (based on the fact).

Four of a Kind: 1

Three of a Kind or Full House: 2*47 + 3 = 97  type 1: pick one card with the same number as the paired card(2) and pick anyone from the remaining card without the same number(47), type 2: pick two cards from with the same number as the unpaired card(3)

Two Pair: 198=3*44+11*6  type 1: pick one with the same number as the unpaired card(3) and pick anyone from the remaining card without the number in the flop(44), type 2: pick any paired card without the number in the flop (11*6)

Others: 880=44*40/2=C(11,2)*4^2   pick any unpaired card without the number in the flop

There is about 25% (1 in 4) having a two pair or better.

* * *
The previous 3 cards or 5 cards analysis give players an idea of what might happen during the game before you see the flop or give a general idea of the distribution of the combinations.

But after you see the flop, the situation is changed.  You will have to update the distribution of the combinations based on the flop or based on the fact.

By the way, you also have two cards at hand, so what the other players might have will be different from the previous calculated numbers.

Texas Hold'em Poker Probability - Flop (3 cards)

So what exactly we can see on the flop (3 cards).

There are 22100 combinations, 22100=C(52,3).

Straight Flush: 48=12*4  from A23 to QKA with 4 suits.

Three of a Kind: 52 = 13*4  pick one number from A to K with 4 different suit combinations.

Straight: 720=12*(4^3-4)  from A23 to QKA with 4^3 suit combinations minus flush(4).

Flush: 1096=C(13,3)*4-48  pick 3 different numbers from A to K with 4 suits minus straight flush(48).

Pair: 3744=13*6*48  pick one number from A to K with 6 suit combination, pick one from the remaining 48 cards.

Others: 16440= (C(13,3)-12)*(4^3-4)  pick 3 different numbers from A to K minus straight, without flush (4^3-4).

So it's about 17% ( 1 in 6 times) to see a pair on the flop.

It's about 26% ( 1in 4 times) to see a pair or better or flush/straight on the flop.

***
We may also want to know what about 2 card flush on the flop, or some kind of straight like patterns on the flop, such as 679, 689, or 579.

2 Card Flush: 12168=C(13,2)*13*3*4  pick two different numbers from A to K for the 2 card flush (C(13,2)), pick another one from A to K for the different suit (13), the suit combination is 4*3.

12168 includes one pair and straight.

So to see a 2 card flush or flush is about 60%, it's quite high, almost more than 1 in 2 games.

What about 2 Card Flush without pair or straight?

It's 9864=(C(13,3)-12)*P(4,2)*3  pick three different numbers from A to K and minus straight(12), the suit combination is P(4,2)*3 (pick two different suits, one for two card flush, one for the other)

***
For straight like combinations, let's consider 3 cards with different suits first

A24~JQA: 264=11*P(4,3)  3 cards all have different suits, P(4,3)

A34~JKA: 264=11*P(4,3)

A35~TQA: 240=10*P(4,3)

If we include 2 card flush but not 3 card flush,

A24~JQA: 660=11*(P(4,3)+P(4,2)*3) = 11*(4^3-4)

A34~JKA: 660=11*(4^3-4)

A35~TQA: 600=10*(4^3-4)

So, we might see straight or straight-like flops about 12% ( 1 in 8 times).

Texas Hold'em Poker Probability - Flop (5 cards) cont.

Rather than the previous 5 card combinations, players might also be interested in knowing how often we get open-ended straight or 4 card flush after the flop just shown to us.

4 Card Flush: 111540=C(13,4)*4*(52-13)  pick 4 different numbers from A to K with 4 suits, and pick one from the remaining card without the same suit(52-13).

Open Ended Straight: 60480=9*(13-6)*(4^5-4*3*5-4)  from 2345 to TJQK and pick one from the remaining number without making straight (13-6), the total suit combination 4^5 minus 4-card-flush(4*3*5) minus flush(4).

***
After the flop is shown, there is about 1 in 13 players to have something better than one pair.

But if we also include the 4 card flush and open-ended straight, then it's about 1 in 7.

So if there might be 7 players into a game to see the flop, you would better be prepared to fight down through the river.

Texas Hold'em Poker Probability - Flop (5 cards)

There are 3 cards on the flop.  Including your two cards, there are total 5 cards.

Let's look at the 5 card combination first.

The total combination of 5 cards is 2,598,960= 52*51*50*49*48/5!=C(52,5)

Royal Flush: 4

Straight Flush: 36=10*4-4  from A2345 to TJQKA minus Royal Flush

Four of a Kind: 624=13*12*4   from AAAA* to KKKK*, * could be any of the other 48 cards

Full House: 3744=P(13,2)*P(4,2)  pick two different number from A to K for the 3 cards and 2 cards and with 4*3 suit combinations.

Flush: 5108=C(13,5)*4-36-4  pick five different numbers from A to K, four different suits, minus straight.

Straight: 10200=10*(4^5-4)  from A2345 to TJQKA with any kind of suit combination minus flush.

Three of a Kind: 54912=13*4*C(48,2)-3744  from AAA to KKK with 4 different suit combination, and pick any two cards from the remaining 48 cards, then minus full house.

                            54912=13*4*C(12,2)*4^2  from AAA to KKK with 4 different suit combination, and pick any two different numbers from the remaining 12 numbers (each number has 4 suit choices).

Two Pairs: 123552=C(13,2)*6*6*11*4  pick two different numbers from A to K, each number has 6 suit combination, and pick one from the remaining card without the same number(11*4).

One Pair: 1098240=13*6*C(12,3)*4^3  pick one number from A to K with 6 suit combination, then pick 3 numbers from the remaining 12 numbers (each number has 4 suit choices).

Others: 1302540=(C(13,5)-10)*(4^5-4)  pick five different numbers from A to K minus the straight combination(10), there are 4^5 suit combination minus flush(4).

The probability to get one pair or better after the flop is about 50%.

How to VERIFY: Play online poker and keep the record of the hands with the flop.  If you fold before the flop, you simply take that record out.

Texas Hold'em Poker Probability - Starting Hand

Poker is an imperfect information game, and it's all about the theory of probability.  The key to win is based on how we can use the probability to gain an edge with some call/raise skills.

In Texas Hold'em, you get two cards first, then the flop, the turn, and the river card.  In such a process, we can find out the probability of the starting hand (2 cards), the flop(3/5 cards), the turn (4/6 cards), and the river (5/7 cards).

If we simplify it a little bit, we focus on the starting hand, the flop, and the river.  We would like to know the 2-card, 3-card, 5-card, and 7-card probability of the combination of the cards.

There are 52 cards.

For the starting hand, 

the number of the combination of the starting hand is 1,326 = 52*51/2.

pair - 78 = 52*3/2

suited - 312 = 52*12/2

others - 936 = 52*(51-3-12)/2

pair : suited : others = 1:4:12

You have about 30% probability to get a pair(6%) or suited(24%) starting hand.

How to VERIFY: You can go to play online poker and keep a record of your starting hands to verify the result.