[FONT=NotoSans, Lato, arial, sans-serif]I made a mistake above. It is necessary to multiply by (48-12) instead of 48 for cases when there are 4 cards of the same suit on the board.
So, we get the following result for the probability of seeing a hand in which aces lose to aces:[/FONT]
P = 36*(4/52 * 3/51)*(2/50 * 1/49)*4*[C(5 of 12) + (48-12)*C(4 of 12)]/C(5 of 48) = 0.000133*4*[792+17820]/1712304 = 0.000133*0.043478 = 0.0000057826 or < 0.0006%
About 1 time on each 173.000 hands.
AA vs AA will happen once every 7520 hands at a 9-max table. And flash will happen exactly 1 time out of 23.
7520*23 ~ 173.000
And if we already have AA in our hands, then every 1225 hands someone will also have aces and one of the two players will get a flush 1 time out of 23. This will happen 1 time every 28175 hands. Thus, the probability of our losing with AA to another player with AA, calculated preflop, is approximately 1 of 56000.
If the cards are already turned over and both players have AA, then the probability of losing each of the players is exactly 1/46, if we do not take into account the discarded cards of other players, that is, if we do not know them.