I believe the correct set up for the general solution is:
p = C(52 - Ranks * 4, Opp * 2) / C(49, Opp * 2)
Where p is our percentage chance that the flop missed everyone,
Ranks is the number of ranks on the flop, and
Opp is the number of opponents you are facing
btw this is assuming we have no cards - that is, we're a neutral observer, but we can easily adjust for the times we do / don't have a pair. Also note that this formula does NOT account for the times villain holds a PP.
Here's an example - on a board with 3 ranks and a HUHU opponent:
p = C(52 - 3 * 4, 1 * 2) / C(49, Opp * 2)
Which after a little work comes out to ~66%.
Which meshes with the oft cited stat that AK will flop a pair ~1/3rd of the time.
right. but we know what the flop is in this case
its either xyz or xyy
so we have to take out those 3 cards from the deck. .. as we're working backwards from the AK will flop a pair 1/3 of the time. we have to think about it as the board is AK8, how often does he have a pair.
and we also know our cards (which is significant, i think.. especially in a case where we flop a full house, for example)
