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.