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%.