[mykey1961]

Actually he asked 3 questions.

1) The title question.

It's not specific about who's AA wins or loses, so the answer is more like 4.34619%

2) How often will one opponent have AA if you have AA.
either 1/1225 or N/1225 depending on how you interpret "one opponent"

3) A combination of #2, and how often will the board show 4 of one suit given that both of you have AA.

if you assume "4 flush" means 4 on the board, and the A making a 5th.

(4*12*11*10*9*36*5)/(48*47*46*45*44*1225) = 0.003398%

or

(n*4*12*11*10*9*36*5)/(48*47*46*45*44*1225) = n*0.003398%

again depending on how you interpret "one opponent".

if you assume 4 flush to mean came up short of making a flush.

(12*11*10*36*35*20*4)/(48*47*46*45*44*1225) = 0.05286110%

or

(n*12*11*10*36*35*20*4)/(48*47*46*45*44*1225) = n*0.05286110%

again depending on how you interpret "one opponent".

#3 doesn't ask how often your AA will win or lose.