Draws...

Lets say I'm heads up in a pot, I hold a set and my opponent holds a flush draw (and I know this). Is it mathematically correct for me to bet in such a way as to give my opponent the pot odds to call (so as to extract money), or to overbet to make him fold? If possible, please show the mathematics involved as well.

I think it's difficult to give a straight answer since we may factor in: our opponents style, our table image, stacks sizes/implied odds and blind level - but ideally, I'd say you'd want to make a bet whereby rather than shutting your opponent out of the hand - you are enticing him to make a mistake when he calls your bet.

In my mind that means making a bet which doesn't give him proper odds to call.

On the flop with 3-to-a-flush, your opponent is roughly 2 to 1 to hit his flush by the river(considering he sees both the turn and the river)
He is around 4 to 1 to hit his flush on the next card(turn)

If he misses on the turn and stays till the river he's roughly 4 to 1 to hit his flush on the river.

Let's say on the turn the pot is 1600. He has one river card to come and he's roughly 4 to 1 to hit it. You bet say 800. The pot is now 2400 and your opponent must call 800. He's getting exactly 3 to 1 to call.

From Sklanskys Fundemental Theory of Poker an opponent makes a mistake (and thus you capitalise) when an opponent would have played their cards differently had they been able to see yours.

Given the correct odds to call for a fush draw, your opponent would have made the correct play even if you showed your cards to him before he decided what to do. Yes he is behind but the odds justifiy the call.

Infact his only mistake would be to fold given those odds.

So the correct play here would seem to be to not give him sufficient odds to call. If he does then in the long run you will make money out of him.

i see. my thinking was that instead of overbetting, making the opponent fold and taking down the pot immediately, you could bet an ammount that barely gives them the pot odds to call, to extract more money out of them than what is already in the pot (although sometimes they would make their draw and you would lose the whole pot (which was why i was curious about the mathematics)). i think it depends on the size of the pot compared to the blinds. if both me and my opponent just called preflop, the pot is pretty small, so it would seem a better choice to try to extract more money. if the pot was raised preflop though, the opposite seems to be better.

I see what you are saying, but the closer to the correct odds you give him, the less of a mistake he makes. So when his draw does com in, the more he will make back.. ok not enough to put him in profit, but again not so much of loss.

When you know your opponent is on a draw, and you hold a set, you want to bet the largest possible bet that he will call that gives him incorrect odds to continue.

So for example (in this example, I'm using top pair, but its pretty much the same with a set):

You have , the flop is

Your opponent shows you

There is $10 in the pot currently.

Your bet must be bigger than his odds of hitting his flush on the turn. His odds of hitting the turn are 4.5:1. So our bet must be larger than $2.85 to have a positive $ expectation.

However, you want to bet as large as you think he will possibly call. So if our opponent will call a $100 bet, that is the bet we need to make.

We do not want our opponent to fold! Folding gets us no money. We want him to call incorrectly.

You know, I sometimes forget this. I will get a little scared, and try to overbet so I can take down the pot. This does hurt me in the long run. Thanks for the reminder

i see. that makes sense