I understand what you mean now.
Easy way to proceed. (lets assume ties are possible?)
there are 8*8 possible combinations of hands.
Figure out how many of those 64 times player 1 will win.
_______________Player1_______Player 2
Straight Flush_______0____________0
4 of a kind_______1/741_________1/741
Full House________5/741_________5/741
Flush___________15/247__________0
Straight___________0___________7/741
3 of a kind_______43/741________43/741
2 pair__________170/741_______170/741
1 pair_____________1____________1
So, with a pair, player 1 can't win.
with 2 pair, he can win 1 way (player 2 has 1 pair.)
with 3 of a kind, he can win 2 ways
with a straight, 3 ways,
... 1 + 2 + 3 + 4 + 5 + 6 + 7 ways out of the 64 to win.
Find the probability of each of these occuring, add them together.
(Sorry, it seems like a pita to do so)
Also, shouldn't each person's probabilities sum to 741/741? And, what do you mean by the probability of a pair is 1? Did you mean that it's the complement of the sum of the other probabilities?