Where is the betting margin?

[VivaLaViking]

Player Hero acts first, he is certain player Villian has a 4-flush after the flop. Hero knows he will win unless the next suited card falls. Hero knows Villian has 9 outs and stands at 35% of making the flush in the next two cards, at ~19% on the tun and at ~19% on the river. How much should Hero bet on the flop to make Villian's call incorrect.

Hero considers that the bet, B, should be > 35% of the pot and feels that is where the margin is. But then he considers that Villian only has a ~19% chace of making the hand on the next card so the proper bet would be > 19% of the pot. What should hero bet, more than 35% of the pot, more than 19% of the pot, or some other percent of the pot ?

[jba]

[ QUOTE ]
Player Hero acts first, he is certain player Villian has a 4-flush after the flop. Hero knows he will win unless the next suited card falls. Hero knows Villian has 9 outs and stands at 35% of making the flush in the next two cards, at ~19% on the tun and at ~19% on the river. How much should Hero bet on the flop to make Villian's call incorrect.

Hero considers that the bet, B, should be > 35% of the pot and feels that is where the margin is. But then he considers that Villian only has a ~19% chace of making the hand on the next card so the proper bet would be > 19% of the pot. What should hero bet, more than 35% of the pot, more than 19% of the pot, or some other percent of the pot ?

[/ QUOTE ]

it need only be >19% of the pot on each street in order for your actions to turn a profit. In order to know the maximum profit possible, we must know when and how much villain will be willing to incorrectly call.

[Tom1975]

The key amounts aren't .35*pot and .19*pot, they're [.35/(1-.35)]*pot and [.19/(1-.19)]*pot. This is because to force your opponent to make an incorrect call, you must force him to call >35% or >19% of the pot AFTER you put your bet in. Example - the pot is $100. You bet $20 (which is > 19%). Now he only has to call $20 to win $120 which is 16 2/3%, which is less than 19%. The 'correct' amount would be .19/.81*100=$23.45 or greater.

[VivaLaViking]

Tom is absoluetely correct but the math may be simplified a little to:
.
HisProabilityOfWinning . . .19
----------------------- = --- ~ 23.46%
YouProbabilityOfWinning . . .81
.

[AaronBrown]

Not quite. If there is A in the pot already and you bet B, your opponent will analyze the call as having probability p of winning A+B and probability (1-p) of losing B.

p*(A+B)-(1-p)*B = 0

implies

B = p*A/(1-2p)

So on the river, with 9 outs and 44 unseen cards (assuming you have nothing of the suit), p = 9/44 and B = 9*A/26 or $34.62 for a $100 pot before your bet. On the turn, with 9 outs and 45 cards, p = 9/45 and B = 9*A/27 or $33.33 for a $100 pot. This makes your opponent indifferent between calling or folding.

Of course, these are precisely the bets you don't want to make, as they give your opponent no opportunity to make a mistake. If he folds, it doesn't matter what you bet. Although it's wrong for him to fold if you check, or bet 10% of the pot, that mistake doesn't help you. But if he calls a bet that's too large, that does help you. So the straightforward play is to bet more than the optimal amount. How much more depends on what you think your opponent might call.

Your advantage here is he may feel there is some chance of beating you without hitting the flush; or he may assign som equity value to making the flush and still getting you to call him. Making a large bet could encourage him in this belief, as it looks as if you are trying to force him out. That depends on how he reads your betting. Of course, if you make it too large, it could force him out anyway. Also, people are curious and like to see what they would have gotten, a lot of them will pay you money to see the hand drawn out.

Another gambit is to check, hoping he will raise. A lot of players will do this, figuring you must be too weak to beat him even without the flush. Of course, other players will figure you're slow-playing a full house; or will just say thank you and take the equity you've given away.

Personally, I might be inclined to bet slightly lower than optimal against most opponents. It does not cost you much equity and having called that bet, a lot of players will call a larger bet on the river. You have the advantage of knowing when you are beaten, so you never throw extra money away.

You also want to consider deception value. The next time you're in this situation, you might not have him beaten even without the flush, or you might be holding a suited Ace, so you're hoping he (and you) get the flush, or you may be holding a full house. Whatever you bet now, you'll want to think about how that changes your equity when the situation (from your opponent's point of view) recurs.

[VivaLaViking]

Aaron, you know I always appreciate your views. It is interesting to me the way the problem was stated you using the fact that 2 flush cards were known to exist in Villians hand. In real world problem things are not as cut and dry but maybe it is correct to calculate the math assuming Villian is suited. I'm dog tired but it has given me another thought process. The reason I was looking for the margin is it is useful for determining the most Hero should call. More tomorrow, thanks.

BTW: what B&M do you go to, Sun or NJ, I'll keep an eye out for you.?

[AaronBrown]

Thanks for the kind words.

Although you never know what the other guy is holding (except for the ones who wear mirror sunglasses), it often helps to figure out the best play against the most likely hand. At least you know your play is right against some cards, a lot of players do things that can't be right against any cards.

There are a lot of possible poker hands, and only three possible choices, fold, call or raise (or maybe a few more with spread limits and no limits, but still a lot less than the number of hands). If one of these actions is right against the most likely hand, it's not a bad guess that it's right against the majority of hands your opponent could have weighted by probability. Anyway, there are worse ways to play poker, like making it too complicated and doing something really stupid.

I usually figure the most likely case, then try to figure something that could make it worse, and something that could make it better. That's usually enough to make a decision at the table. At home with computer is another story.

I have not made it to either place this year yet, I've been playing only private games. Too busy. You wouldn't know that from 2+2 posts, but I do those mostly when I'm taking a five minute break from working.