Probability of flopping top pair with X,X

[morello]

Hi,

My math is pretty rusty and I think I would make a mistake if I tried. Can anyone help me calculate the odds of flopping top pair with specific two card hands? I know that if AK flops a pair, it will be the top pair 100% of the time, and that it flops a pair approx 32~% of the time.

But I'm not sure how to do it for the other hands. KQ, KJ, KT, K9, QJ, QT, Q9, JT, J9, T9, 98, 97, 87, etc.

Anyone able to help, please?

[btmagnetw]

i remember a long time ago i asked this exact question. never got an answer.

[jay_shark]

It's easy actually . First lets omit all two pair or better hands and only work with single pairs .
Take KQ and work in cases .

case1) Find the number of flops with a top pair of kings but no aces or queens : 3*43c2 =2709.
case2) The number of flops with a top pair of queens but no kings or aces . As before it is 2709 .

The total number of single pairs using a king or a queen is just 2709*2/50c3 = 27.64%

Take q-10 .

case1) The number of flops with a top pair of queens but no k's or aces : 3*39c2=2223
case2) The number of flops with a top pair of 10's but no j's q's k's or aces : 3*32c2 = 1488

Your answer should be (2223+1488)/50c3 = 18.93%

Also note that AK will flop a "single" pair, 3*44c2*2/50c3 = 28.95% of the time .

[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.