[UtzChips]

But what if you don't know what a "c" is, or want a flop that doesn't include an underpair to your AK, like K55?

All three "events" (the 3 cards on the flop) are dependent of each other, so we have to multiply them together. In addition, there are 3 ways to flop a pair. You can catch your card on the 1st, 2nd or 3rd card of the flop.

We start with the first card of the flop being an ace or king:

(6/50)*(44/49)no ace or king * (40/48)no ace/king or same rank as the 2nd card on the flop =

.12*.898*.8333 = .0898

We pair on the 2nd card of the flop:

(44/50)*(6/49)*(40/48) = .0898

We pair on the 3rd card:

(44/50)*(40/49)*(6/48) = .0898

.0898*3 = 26.94% chance

So, the chances of flopping top pair with KQ and no pair on board:

[(6/50)*(40/49)*(36/48)]+[(40/50)*(6/49)*(36/48)]+[(40/50)*(36/49)*(6/48)] = 15.23%

I seperated the 3 flops by brackets. The first flop you hit one of your 6 outs, so you have 5 cards left in the deck that are A/K and 4 aces, for a total of 9 cards that cannot come on the 2nd card. On the 3rd card you have 9 + the 3 of the rank of the card that did come on the 2nd card.

on the 2nd flop, you didn't hit your card or an ace (10 cards) on the 1st card. You did hit one of your 6 outs on the 2nd card. Now there are 12 outs that cannot come on 3rd card.

On the 3rd flop, you didn't hit an ace or one of your cards on the 1st card (10 outs), + 3 outs to the rank of the 1st card didn't hit on the 2nd card and then you hit one of your 6 outs on the 3rd.