April 3, 2010 – 10:41 am | 7 Comments

I’ve compiled a short (just 7-pages) e-book, an introduction to the mathematics of poker. It’s basically covers how to calculate your expected value in a certain spot – starting with explaining what EV is, all …

Read the full story »
Theoretical

Discussions about the theory behind the game.

Opponent Modelling

Ideas about modelling opponent’s play, and how to best do that with mathematical tools.

Hand Analysis

Analyses of past hands (my own or otherwise), using math as much as possible to determine how well the hand was played.

Mental Processes

Thoughts about the mental/psychological side of poker – dealing with tilt, etc.

Miscellaneous

Other information – book reviews, contests, announcements, or random spew.

## A Closer Look at the AQo Hand

Submitted by on March 8, 2010 – 11:32 pm7 Comments

Since my last post, I’ve had some discussions on the 2+2 forums about that hand. And so I thought I’d take a closer look at the hand. I realized I should be considering the odds on the flop, and not the equity across the whole hand.

Quick recap. I had AQ offsuit, raised from the SB, BB 3-bet, and UTG limp-called. I was getting \$15.50 to call into a \$48.50 pot. In retrospect, even the ranges I put them on were probably wrong. I stand by the range of 77+, AT+, QJ+ for the BB. But considering the UTG limped, I don’t think he had a high pocket pair. So I’m thinking he’s likely to have 77-JJ, QJ+.

So let’s assume I called, and take look at the various cases of the flop. If I called, the pot on the flop would be \$64. Obviously I can’t consider every possibility, but let’s look at the main ones.

Case 1: A flops, with no K
The flop contains an A and no K. If this happens, I’m probably quite confident. This happens (3*43*42*3)/(50*49*48) = 13.82% of the time. I’m only afraid of a set if this happens. I’d be first to act, and I’d bet about \$40, for a total investment of \$55.50. I think JJ, QQ, KK, AT, AJ would at least call that bet. The other hands in the range are probably going to fold. Let’s assume that all the hands in the ranges are equally likely.

The player in the BB will then call when behind about 30% of the time. 11% of the time I’d be behind. 52% of the time, he’s going to fold, and the other 7% is when he has AQ as well and we tie. UTG will call when behind 18% the time, will be ahead of me 10% of the time, fold 64% of the time, and tie the last 8%.

So, betting action. There are a few cases.

1. BB calls and UTG folds, and I’m ahead. This happens 30%  * 64% = 19.2% of the time. At this point, I win \$64 + \$40 + \$40 = \$144. And I’ve invested \$55.50, giving me a profit of \$88.50.
2. BB calls and UTG folds, and I’m behind. This happens 11% * 64% = 7.0% of the time. I win nothing, and lose the \$55.50 I’ve invested.
3. BB folds and UTG calls, and I’m ahead. 52% * 18% = 9.4% of the time. I profit \$88.50.
4. BB folds and UTG calls, I’m behind. 52% * 10% = 5.2%. I lose \$55.50
5. Both call, and I beat both. 30% * 18% = 5.4%. I win \$64 + \$40 + \$40 + \$40 = \$184, for a profit of \$122.50.
6. Both call, and I lose to at least one. 11% * 18% + 30% * 10% + 11% * 10% = 6.1%. I lose \$55.50
7. Both fold. 64% * 52% =33.3%. I win \$104, for a profit of \$48.50.

For now, let’s ignore the cases when I tie with one.

So basically, if an A flops, and no K, I have an EV of 0.192 * 88.5 + 0.70 * -55.5 + 0.94 * 88.5 + 5.2 * -55.5 + 5.4 * 122.5 + 6.1 * -55.5 + 33.3 * 48.5 = \$37.86.

Case 2: Q flops, with no A or K.
This happens (3 * 42 * 41 * 3) / (50 * 49 * 48) = 13.17% of the time. This is similar to the above, except I’m behind to KK and QQ as well, and I’m probably being called by QJ/QK, instead of AJ/AK. KK, QQ, AA are not hands I put in UTG’s range (he would have open-raised with those hands, I think, and not limped/called), so I’m not behind to UTG for sure.

So, again, the same few cases of bets.

1. BB calls and UTG folds, and I’m ahead. This happens 36%  * 51% = 18.4% of the time. At this point, I win \$64 + \$40 + \$40 = \$144. And I’ve invested \$55.50, giving me a profit of \$88.50.
2. BB calls and UTG folds, and I’m behind. This happens 9% * 51% = 4.6% of the time. I win nothing, and lose the \$55.50 I’ve invested.
3. BB folds and UTG calls, and I’m ahead. 47% * 42% = 19.7% of the time. I profit \$88.50.
4. Both call, and I beat both. 36% * 42% = 15.1%. I win \$64 + \$40 + \$40 + \$40 = \$184, for a profit of \$122.50.
5. Both call, and I lose to at least BB. 9% * 50% = 4.5%. I lose \$55.50
6. Both fold. 47% * 51% = 24.0%. I win \$104, for a profit of \$48.50.

This gives me an EV of 0.184 * 88.5 + 0.046 * -55.50 + 0.197 * 88.5 + 15.1% * 122.5  + 4.5% * -55.5 + 0.24 * 48.5 = \$58.81.

Case 3: I miss the flop.
When no A or no Q hits. This happens close to 70% of the time, I think. Let’s take 70%, in general. When this happens, I’m probably going to get bet into, and fold, losing the \$15.50 I called preflop.

So therefore, my total EV = 0.7 * -15.50 + 0.1317 * 58.81 + 0.1382 * 37.86 = \$2.12.

Again, however, it seems like I should make the call here.

But let me place a disclaimer here. The actual EV would be lower if the ranges my opponents are on are tighter. A lot of the calculations made here are based on my read of what hands they’d call a bet with, etc. I understand that the ranges are probably a bit wide, but I feel like the game was quite loose, and those reads are what I genuinely felt.

What do you think, though? Am I thinking through this correctly – or am I missing something else again?

### 7 Comments »

• […] « Getting Value From Flush Draw on the Flop A Closer Look at the AQo Hand […]

• lovesuperkarma says:

Just asking if you can have what you say in video the numbers is freaking me out. but kudos on your very unique blog.

• Derrick Kwa says:

Thanks for the kudos. =). Appreciate it. As for video, good suggestion, I'll definitely be looking into that.

• vipeldeu says:

it certainly is a great blog. i will go through it carefully.

i like the approach. i like it very much. i would have folded online too, OOP being extremely important.

I think that live games are looser, checking through the ranges I'd may tighten BB a bit more, but I don´t think that would make much difference.

So after reading your math, something I am simply not capable of doing, your decision was wrong.

could we put being OOP in a math equation?

• jason says:

with those bets, if i were an opponent, i might not call an ace on the flop with a pair.

• Derrick Kwa says:

Hm…Well, how does being OOP affect the hand, though? It basically means that you have to act first, and you lose information. But I would say that the approach I'm trying to take kind of negates that factor. By predicting how your opponent would react to a bet with various cards, and how often you come out ahead, it gives you the “information” required. Of course, that means even more uncertainty to deal with.

As for the decision being wrong, I assume you mean that I should have folded, and that the call is wrong? I feel like the math shows it's kind of 50-50. It's a very small +EV, *given those assumptions*. But I think if the assumptions/reads I have are wrong, it could swing it either way, so it really becomes how confident you are of your reads of the players. I think a case for it can be made either way.

• Derrick Kwa says:

Yeah, I know. And a lot of you probably won't agree with me on this, but I feel like it was a very situation dependant read, and that's what I'm basing this analysis on (as I mentioned). It's so marginal, though, that a different read would probably swing the EV to below 0.