How often will pocket aces lose to 4 flush or set after the flop?

Question

Having pocket aces in good position I make a reasonable bet so as not to blow everyone out of the pot, and get a lot of callers which I wanted. There is some betting on the flop but not enough to convince me that I was beat. I was low on chips since this was 1st good hand in about 3 hours so I went all in to drive out anyone with connected or suited. I was pretty much advertising my aces and everyone went out except someone with a pile of chips, who needed to complete a 4 flush or a set. As it turns out he completed the set and sent me packing. I'm kind of new to NLHE so I couldn't tell you what his pot odds were but the pot was probably 4 times my all in raise. So - what are the odds that pocket aces will lose to 4 flush or a set after the flop? IOW: was this a good bet on his part?

Answer 1

There are tools available that will compute this exactly for you. I used a freely available tool, PokerStove, to compute your so-called "hot/cold" equity assuming you had 2 black Aces, your opponent had suited 34 in Hearts, there were two Hearts and one diamond on board. You are about a 2:1 favorite in this scenario:

Text results appended to pokerstove.txt

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 990  games     0.048 secs    20,625  games/sec

Board: Kh 7h 2s
Dead:  

    equity  win     tie           pots won  pots tied   
Hand 0:     62.222%     62.22%  00.00%             616          0.00   { AcAs }
Hand 1:     37.778%     37.78%  00.00%             374          0.00   { 4h3h }


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But you don't need PokerStove to compute this for you, and you can't use PokerStove at the table. You should be able to do this yourself, at the table. Here's how.

First, figure out how many outs your opponent has. This will be an estimate, based on the range of hands he could hold. In the case above, there are 9 cards that can come that will make your opponent's flush -- so he has 9 outs.

Now, compute you're opponent's equity in the hand expressed in terms of the chance that he will win. Your equity will simply be 100-x where x is your opponent's chances of winning. You can compute your opponent's chance of winning by using the Rule of 2/4. On the flop, where there are 2 streets to come, multiply your opponents outs by 4. The result is the chance he will make his hand. On the turn, where there's only 1 card to come, multiply by 2 instead.

So, when the flop is Kh7h2d and you go all in, your opponent has 9 outs, and he has a (9x4) = 36% chance ow winning the hand. Now suppose you waited until the turn, when the board was Kh7h2dQc. He still has 9 outs, and he has a (9x2) = 18% chance of making his hand. So, conversely, on the flop you have a 67% chance of winning, and on the turn you improve to a 82% favorite.

These values are approximate, and don't account for backdoors (such as trips and straights) -- but the result is close enough to be useful at the table, and it's easy to remember and compute.

Answer 2

First of all, if someone has one 8 in hand and two 8s on the board, they are called TRIPs, not a set.

If he has a SET (of two 8s in hand and one on the board), your Aces are ALREADY beaten. But that's not the case here.

Here, you have A-A, and your opponent has an 8 of say, spades, which pairs with a non-spade 8 and matches three other spades.

The chances of a fifth spade coming by the river for the flush to beat your A-A are about 35%.

The chances of a third 8 coming by the river for the rips to beat your A-A are 7%-8%.

If he gets an 8 on the turn, you have a (4%) "redraw" to a SET of As on the river to beat his trip 8s. You also have a 4% draw to a third A on the turn to head off the trips he may make on the river.

This 4%-draw will not beat a spade flush unless you get runner-runner A-A for QUADS.

Answer 3

what are the odds that pocket aces will lose to 4 flush or a set after the flop

If I compute the odds of a hand like 8c 7c for your opponent on a 8d 3c 2c board vs your pocket Aces, I get basically 49/51, so from looking purely at the odds, it's a coinflip...

Now to know if it was a good call from your opponent, we have to take several things into account including: the pot odds he's getting and how his hand fares versus your range (because your opponent cannot be sure you have aces).

Not only is he getting very good odds (4:1) but also his hand will do very well versus your range of possible hands.

Seen that it's a tournament, we could also take into account ICM computation (Independent Chip Model) computation... But we'll leave that for another poker.stackexchange.com question :)

Was this a good bet on his part?

If I understand your question correctly, you were all-in and your opponent did call. So you question could be rephrased as: "Was it a good call on his part seen the odds?"

He's getting about 4:1 on his call (you wrote that the pot probably had about 4 times what you did bet all-in) and according to your description, he has both a flush draw and a pair. That's a very good hand.

So even if you're not giving very precise numbers, it's safe to say your opponent did make a good call.

EDIT As 'Jeffrey Blake' commented, improving to flush or a set is not the only way for your opponent to improve his hand. In the example I made up above, he could hit a seven for two pairs. On the real board, he may even have had another option: a straight draw or a backdoor straight draw. All this is taken into account by equity calculator like PokerStove.