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%.

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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.