A Closer Look at the AQo Hand
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.
- 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.
- 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.
- BB folds and UTG calls, and I’m ahead. 52% * 18% = 9.4% of the time. I profit $88.50.
- BB folds and UTG calls, I’m behind. 52% * 10% = 5.2%. I lose $55.50
- Both call, and I beat both. 30% * 18% = 5.4%. I win $64 + $40 + $40 + $40 = $184, for a profit of $122.50.
- Both call, and I lose to at least one. 11% * 18% + 30% * 10% + 11% * 10% = 6.1%. I lose $55.50
- 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.
- 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.
- 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.
- BB folds and UTG calls, and I’m ahead. 47% * 42% = 19.7% of the time. I profit $88.50.
- Both call, and I beat both. 36% * 42% = 15.1%. I win $64 + $40 + $40 + $40 = $184, for a profit of $122.50.
- Both call, and I lose to at least BB. 9% * 50% = 4.5%. I lose $55.50
- 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?