Originally Posted by oskar

Great post.
I personally find it easier to think about the probability in percentage of the deck and not of all card combinations. For example AA would be 4/50x3/49 - and if you did a couple of those calculations, you can estimate the number pretty exactly. For practical purposes you would round it down - 4/50 is about 1 in 12 - 3/49 is about 1 in 15 - 15x12 is 10x15+2x15 is 180. So it's about 1 in 180. I'm sure most people can do that without pen and paper in a reasonable amount of time.

So your example if the board is KKQ - say you hold 63.
Then the possibility of a single person having AA is:
4/47x3/46
or or KQ is:
ok now you got me... I mean there's nothing wrong with doing an approximation (5/47x2,5/46)... but the calculation isn't rocket science either.
1-[(1-3/47x2/46)x(1-2/47x3/46)]
This is first the probability of getting QK and then KQ, and the rest is the "or" part. Actually the probabilities are the same whether you draw the K or Q first, you can cut it short to
1-(1-(3/47x2/46)²)
It's only that complex when there's not an even chance for both of the cars. Explanation is on the first or second page of any elementary probability book of your choice. There's about a 60% chance I got something terribly wrong here... it's been a long time