### DQB, a Mathematical Approach

I have just arrived in Las Vegas for another two week stay. I want to tell you all about my setup, the trip, the games, and so on. I intend to post every day while I am here. Yeah, I know you will believe it when you see it. I will take that as a challenge. I will get to the beginning of that tomorrow.

Today, I wanted to discuss a hand I have been kicking around in my ahead for some time. It was a $0.25/$.0.50NL game on Tilt. A brief summary of the hand: I was small blind, there were a few limpers to me, I completed with huge odds and 89o, and the big blind checks. I flop top two on an uncoordinated board, get all-in against the big blind, he shows me bottom set, and I get runners to make quads to take it down. For your reference, the full hand history is here.

After the hand, I said something like ‘sorry for the beat.’ He went on to tell me that he would have gone away in my spot. I took it back my apology and called him a pussy. I got to wondering if my play was correct. Intuitively, I was sure it was. At the time, I had gone through all of his possible holdings in my head. I recognized that there was a significant possibility he had bottom set, and a very small chance he had middle or top set.

The thing that makes this hand interesting to me is that he and I both had essentially random hands. Paradoxically, this turns out to make the analysis compact and clean. Here is what I did on paper after the fact:

There are some very unlikely holdings he could have that I am beating, like an over pair, or a straight draw, or a pure bluff. I ignore these possibilities for this discussion. His more likely holdings are some combination of threes, eights, and nines. How many combinations are there? There are two remaining nines, two remaining eights, and three remaining fours. That makes a total of seven cards. He most probably has two of these seven cards. I think any of these possible combinations are equally likely. On to the math:

There are C(7,2) possible hands he likely has. This is 7!/(5!*2!) which works out to 21 possible hands. There are six ways for him to make bottom set, one way to make middle set, and one way to make top set. This makes a total of eight possible ways for him to make a hand that is beating me. This makes 13 ways (21-8) that he can make a hand that I am beating. The odds that I am ahead are 13-8. Of course, I am assuming he plays any of these hands the same way. With little information on this player, I think the stakes we are playing drive me to this conclusion.

Something else I find interesting about this hand can be seen from his point of view. With his 44, he can make the same basic assumptions I did. If he does, I have six ways of making top set, and six ways of making middle set, the only hands that are beating him. That makes 9 (21-12) ways that I am not beating him. His odds that he is ahead are 3-4 (9-12). This means that, on average, that he is behind! He told me I should have gone away, but did not consider going away himself.

Now, looking at it at the top level, of course it all goes in every time. But I am happy that my intuition was correct there. And that the math shows he is a pussy if he would have gone away in my spot.

Tomorrow: My setup, why I am here in Vegas, and a quick update on how my live games have been going.

Today, I wanted to discuss a hand I have been kicking around in my ahead for some time. It was a $0.25/$.0.50NL game on Tilt. A brief summary of the hand: I was small blind, there were a few limpers to me, I completed with huge odds and 89o, and the big blind checks. I flop top two on an uncoordinated board, get all-in against the big blind, he shows me bottom set, and I get runners to make quads to take it down. For your reference, the full hand history is here.

After the hand, I said something like ‘sorry for the beat.’ He went on to tell me that he would have gone away in my spot. I took it back my apology and called him a pussy. I got to wondering if my play was correct. Intuitively, I was sure it was. At the time, I had gone through all of his possible holdings in my head. I recognized that there was a significant possibility he had bottom set, and a very small chance he had middle or top set.

The thing that makes this hand interesting to me is that he and I both had essentially random hands. Paradoxically, this turns out to make the analysis compact and clean. Here is what I did on paper after the fact:

There are some very unlikely holdings he could have that I am beating, like an over pair, or a straight draw, or a pure bluff. I ignore these possibilities for this discussion. His more likely holdings are some combination of threes, eights, and nines. How many combinations are there? There are two remaining nines, two remaining eights, and three remaining fours. That makes a total of seven cards. He most probably has two of these seven cards. I think any of these possible combinations are equally likely. On to the math:

There are C(7,2) possible hands he likely has. This is 7!/(5!*2!) which works out to 21 possible hands. There are six ways for him to make bottom set, one way to make middle set, and one way to make top set. This makes a total of eight possible ways for him to make a hand that is beating me. This makes 13 ways (21-8) that he can make a hand that I am beating. The odds that I am ahead are 13-8. Of course, I am assuming he plays any of these hands the same way. With little information on this player, I think the stakes we are playing drive me to this conclusion.

Something else I find interesting about this hand can be seen from his point of view. With his 44, he can make the same basic assumptions I did. If he does, I have six ways of making top set, and six ways of making middle set, the only hands that are beating him. That makes 9 (21-12) ways that I am not beating him. His odds that he is ahead are 3-4 (9-12). This means that, on average, that he is behind! He told me I should have gone away, but did not consider going away himself.

Now, looking at it at the top level, of course it all goes in every time. But I am happy that my intuition was correct there. And that the math shows he is a pussy if he would have gone away in my spot.

Tomorrow: My setup, why I am here in Vegas, and a quick update on how my live games have been going.

## 6 Comments:

Who was the aggressor on the hand? I can't get to the hand history........

2 pair, even top 2, is always a scary situation but not so much on the flop. That being said, I will probably (actually most definitely) go broke on that hand.

But if you take the math away, think for a second and decide what he would call an allin or raise allin with in this spot...... assuming you were aggressive on the flop.

TPTK? Overpair? A Draw? A set?

If all of these are potential push hands, then you have to make the move.......but with a tighter player it's a little more complicated.

I guess the real question is what lesser hand would push or call a push on the flop?

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BTW, I'm enough of a tard to not anticipate sets......I lose money alot with 2 pair on the flop. Most of the biggest pots I've lost holding 2 pair, even top 2, are usually against flopped sets with the money going in on the flop. I know that's stating the obvious, but that underlines the reasons to tread cautiously when facing aggression from an opponent (assuming he's not a tard).

Very good post man.........and good seeing you back. I'll hold your word to the "everydaypost"

Good luck in Vegas bro.

wow u are seriously questioning whether you made the right play?

No, not questioning it at all. I was simply wondering if the kind of analysis that I applied here would agree with my intuition. I even said that "Now, looking at it at the top level, of course it all goes in every time."

I don't know how you could think that was my point.

I probably didn't read the post carefully enough, but to analyze the hand so thoroughly doesn't make any sense to me. That's what I was saying.

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Again, you are still not getting it,flux. It was interesting to me because the randomness of the two hands paradoxicaly led to the possiblility of a closed, compact analysis that is not possible in other situations.

I enjoyed writing it. If you did not enjoy reading it, then combintorics are not for you. Move on and stop being so negative.

Readers that agree with you will want to go read your brilliant blog and not my pointless blog. The rest are interested in my pointless blog, so your comments are irrelevant and pointless.

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