The first hand of the WSOP ME where Farha and Hudson went heads up with AT vs TT and flop was AAT and Farha nutted Hudson? What are the odds of this happening, where it's a full house vs higher full house and the non pair hand shares a card of the pocket pair? And while we're at it, let's look at the odds if played to the river as well. The more work can be shown, the better. I want to improve my ability to count cards so I may be making more posts here more often, but I got a long ways to go.

Flop (there are only 3 ways to make AAT):

3/C(48,3) =~ 5764:1


River:

3*C(44,2)/C(48,5) =~ 602:1


These assume that the 2 players start with the proper hole cards.

So I gather that 5764:1 is a cold deck for sure. The rest I'm having a hard time understanding. Hard to admit I'm having a hard time with the rest given I consider myself a grinder, or did before I was banned with the rest of America from playing online. I don't even know what the C means. Or 48,3 or =~. Why is it =~ instead of just = ... ?

I'm kind of a math n00b. But we gotta start somewhere. Thanks a lot by the way, and nicely done. I'm not asking you to teach a man to fish but any suggestions for how I can get myself started are appreciated. I've read some poker books but I haven't quite discovered one that helps me with the conventional ways like what you're probably using.

C(x,y) is the number of ways to choose y cards out of x cards without regard to the order that they are chosen (combinations). In the first case we are just using it for the total number of flops. C(48,3) is the number of ways to choose 3 cards out of the 48 remaining cards for the flop. In the second case, C(48,5) is the total number of 5 card boards. Then C(44,2) is the number of ways to get the turn and river cards out of the 44 remaining cards that are not an A.

C(48,3) = 48*47*46/(3*2*1)

C(44,5) = 44*43*42*41*40/(5*4*3*2*1)

C(44,2) = 44*43/(2*1)

What the C means.

A mentioned in that post, you can evaluate these directly in Excel or even in google.

C(x,y) is convenient notation on this forum. In texbooks you usually see _xC_y, or ^xC_y or C_x,y or . These are all equivalent.

You can also do the first one just using fractions like this:

3/48 *2/47 * 1/46 * 3 =~ 5764:1

where the 3 fractions are the probabilities of getting an A on the first flop card, times the probability of getting an A on the second flop card, times the probability of getting a T on the third flop card, and then we multiply by 3 since the T can actually occur in any of the 3 positions, not just the third card. It's good to start with fractions, but you'll want to learn combinations pretty quickly if you want to do counting problems. For example, if you try to do the second problem with fractions, you may end up needing combinations anyway because after you multiply by 5 for the number of places the T can occur, you would then have to multiply by C(4,2) places that the 2 aces can occur out of the other 4 cards.

The =~ means approximately equal to since the number is rounded off.