Two pair by the river

[kingstalker]

If you start with say A-K what are the odds that you will make two pair using your two hole cards by the river?

[Lost Wages]

Do you mean 2 pair or better or exactly two pair? If you mean exactly 2 pair you need to specify if it is AKs or AKo.

Lost Wages

[kingstalker]

Just two pairs,AKo.

[BruceZ]

[ 9*(44*40*36/3! - 4^3)-
(4 + 1*2 + 2)*(C(11,3)-1) ] / C(50,5)

= 4.4%.

There are C(50,5) total boards. The first line is for 9 AK combined with 3 other cards that don't pair the board and excluding the 4^3 QJT that make straights. The second line subtracts flushes, but this is a minor contribution. There are 4 AK which match exactly 1 of the hole cards, 1 AK that matches both hole cards and allows for flushes in 2 suits, and 2 suited AKs that match neither hole card and allow the board to flush. There are C(11,3)-1 ways to pick the 3 remaining flush cards, excluding QJT royal flushes which we already excluded in the first line.

[BruceZ]

I excluded all cases of the board pairing, so I literally counted "only 2 pair" and didn't even allow "3 pair"! Here I will allow the board to pair as long as it doesn't pair the A or K to give a full house, and no 3-of-a-kind on board. There are C(44,3) ways to pick the remaining 3 cards, minus 11*4 triplets, and minus the 4^3 QJT as before.

[ 9*( C(44,3) - 11*4 - 4^3)-
(4 + 1*2 + 2)*(C(11,3)-1) ] / C(50,5)

= 5.5%.