Top Two On The Flop Continued...A Combinatorial Perspective

So to continue the discussion about the 5-10-J flop and me holding 10-J. Remember there are two other players in the hand. The question this discussion asks is this, "What are the odds I am beat, or if both players call, what are the odds I will get drawn out on?"

I think this topic is an advanced poker topic that warrants further discussion after this initial post.

Please read the previous post first to understand the hand in question in more detail.

What hands you can likely put your opponents on in this scenario:

1) A big overpair (QQ or KK or AA)
2) A flush draw
3) A straight draw (something like QK or 9Q or 89) [NOTE: The 89 is much less likely, as is the 9Q for the big re-raise.]
4) A set
5) Another two pair, like J-5 or 10-5 (most likely for UTG here)


As a side point, it is more likely for UTG check-raiser to have a made hand like (1) or (4).


First question -- What are the odds I am beat?
Answer: There is only one hand that beats my top two, and that's a set. As in the previous blog entry, the odds of someone having a set are about 0.37% here. As an aside, the odds of the other hands are:

Odds of (1) = 4/47 X 3/46 X 3 = 0.01665 = about 1.7% (of having any of the three overpairs)
Odds of (2) = 11/47 X 10/46 = about 5%
Odds of (3) = 4/47 X 4/46 X 3 = about 0.74% (of having any of the three draws)
Odds of (4) = See last post. About 0.37%
Odds of (5) = 2/47 X 3/46 X 2 = about 0.56% (of having any of the two choices)

And as it turned out, the most likely hands (1) and (2) were the hands that were being held by the other two players.

Second question -- What are the odds I could get drawn out on, if both players call?
Answer: We need to look at the winning percentages of each likely hand.

Re-draw winning percentages with two cards to come:
Odds of (1) = (1/45 + 1/44) = about 4.5% (any of the three overpairs, given a flush draw) to make a set + make a better two pair (can only make a 5) is 3/45 + 3/44 = 13.5% thus 18%
Odds of (2) = (8/45 + 8/44) = about 36% to make the flush (cannot make a 10 clubs) + make a better two-pair = (3/45 X 3/44 X 2) = 0.0091 or 0.91% + make a set of aces = (3/45 X 2/44) = 0.003 or 0.3%, thus total is 36 + .91 + .3 = about 37%
Odds of (3) = (6/45 + 6/44) = about 27% (each of the three straight draws, given the flush draw) to make the straight
Odds of (4) = Re-draw on river to a boat or quads = [if J-J - no quads possible, needs 10, 5, or turn card for a boat = 2/44 + 3/44 + 3/44 = 0.18 or 18%] [if 10-10 - no quads possible, needs 5 or turn card for a boat = 3/44 + 3/44 = 13.6%] [if 5-5, can only get quads to win, 1/44 = 2.3%]
Odds of (5) = Re-draw to a boat = [can only get a 5 to win, 2/44 = 4.6%]


Based on my reads and the most likely hand combinations of (1) and (2) being present, 37 + 18 = 55% (I am only 45% to win) if both players call my all-in. Since player2 was likely to call an allin, this would make UTG player more likely to call another $80. Thus my odds of getting drawn out on are what we might say is a coin flip, which sucks for having only a $6 investment (actually $6 + $4) in the hand. This is another reason I folded. You can still argue against the fold as in my previous post.

But check out the re-draw winning percentages if the hands present were (2) and (3) - the flush draw and a straight draw. If both players call my allin, and these hands are present, I am now only 36% to win!! Let me repeat, this sucks for having a $10 investment in the hand, and another reason why I folded, since I was unsure if could get at least one player off the hand. And again, you can still argue against the fold. Of course in this case I was sure I was not up against both of these hands.

Any other combination I am quite the favorite, unless at least one player has the flush draw, then it really is close to even money.


CONCLUSION
In this type of betting action on the flop, when you're up against at least two other players and faced with an all-in decision on a pot in which you have a small investment, unless you are stone cold or have great reads in which you are quite sure you have a decided winning percentage with two cards to come, it is ok to fold what you are pretty sure is currently the best hand!