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 …

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## Getting Value From Flush Draw on the Flop

Submitted by on February 21, 2010 – 10:28 pm17 Comments

This is a topic I’ve been wondering about, so I thought I’d do some calculations to figure out the best line of play here. So, imagine a situation like this.

Poker Stars \$0.01/\$0.02 No Limit Hold’em – 8 players

CO: \$1.92
BTN: \$2.93
SB: \$1.11
Hero (BB): \$2.58
UTG: \$3.00
UTG+1: \$4.83
MP1: \$2.64
MP2: \$3.33

Pre Flop: (\$0.03) Hero is BB with Q J
UTG calls \$0.02, 3 folds, CO calls \$0.02, 1 fold, SB calls \$0.01, Hero raises to \$0.10, 2 folds, SB calls \$0.08

Flop: (\$0.24) 2 2 8 (4 players)
SB checks, Hero checks

Ignore the preflop bet sizing, I know there’s already beena lot of debate over that. But there were a lot of people saying that I should have bet the flop, to get value from my draw. Basically the premise is that I won’t get paid off much if I hit the draw. Sounds logical, but let’s try and see quantitatively how true that is, and how much that bet should be.

Let’s assume that my opponent would fold if I hit the flush. On this flop, I have 9 outs to the flush, giving me a 9/47 = 19.1% chance of turning a flush, and 35.0% of hitting a flush by the river. However, we’ll take the odds of the flush hitting on the turn, because to take the case of the flush coming on the river, we have to consider if the opponent will bet a blank on the turn. Maybe I’ll consider that in a later post.

So let’s say a diamond comes on the turn, and my opponent shuts down here. If I had bet x cents on the flop, that gives me an EV of 0.191 * (24 + 2x) – x = 4.596 + 0.383x – x = 4.596 – 0.617x. For it to be a good play, that gives

EV > 0
4.596 – 0.617x > 0
4.596 > 0.617x
x < 4.596 / 0.617
x < 7.45

So I’d have to bet about 7c for it to be a good play. That’s a bit too small a bet, though, to make. If I bet that, he’s likely to raise (I would if I was the SB facing a 7c bet into a 24c pot), so that kind of doesn’t make sense. If I had to bet the flop, I’d have to bet something like 14c, at least. Let’s take a bet of 14c, that gives me an EV of 0.191 * (24 + 28) – 14 =-4.04. So it’s slightly negative.

I do have some sort of fold equity here. How much fold equity do I need to make this a positive play? Assuming fold equity f, that gives

EV = f * 24 + (1-f) * -4.04 > 0
24f – 4.04 + 4.04f > 0
28.04f > 4.04
f > 4.04 / 28.04
f > 0.144

Therefore, I need him to fold at least 14.4% of the time for that to be positive. I think that’s definitely a reasonable assumption, so a bet of 14 cents seems like it’s ok.

The question then becomes, what’s the relationship between the bet size, and the fold equity required to make it a positive EV play?

For the play to be +EV, you need the following to be satisfied, taking f as fold equity and b as the bet size.

EV > 0
f * 24 + (1 – f) * [0.191 * (24 + 2 b) – b] > 0
24 f + (1 – f) * (4.596 – 0.617 b) > 0
24 f + 4.596 – 0.617 b – 4.596 f + 0.617 * b * f > 0
19.404 f -0.617 b + 4.596 + 0.617 * b * f > 0
f (19.404 + 0.617 b) – 0.617 b + 4.596 > 0
f (19.404 + 0.617 b) > 0.617 b – 4.596
f > (0.617 b – 4.596) / (19.404 + 0.617 b)

I don’t think you’re really ever betting anything less than half the pot (12 cents) here – I know I wouldn’t. So, for bets of 12c to 24c, here are the required fold equities to make this a +EV play.

 Bet Size Required Fold Equity 12 0.104745 13 0.124886 14 0.144141 15 0.162567 16 0.180216 17 0.197136 18 0.213373 19 0.228965 20 0.243952 21 0.258367 22 0.272242 23 0.285608 24 0.298492

Let’s say your opponent is folding 30% of the time to your bet (which I think is a reasonable percentage). The more you bet, the lower the expected value is. The EVs for various bet sizes, assuming 30% fold equity, are as follows

 Bet size EV 12 5.2176 13 4.785 14 4.3524 15 3.9198 16 3.4872 17 3.0546 18 2.622 19 2.1894 20 1.7568 21 1.3242 22 0.8916 23 0.459 24 0.0264

Your EV from checking this street is simply 0.191 * 24 = 4.596. To make the bet better than checking, you require an EV of more than 4.596, which suggests a bet size of 12 or 13 cents. I’d probably include 14 (and maybe even 15 cents) in that range, because you will get paid off after hitting the flush every once in a while, so the implied odds would increase the EV. That depends on a lot of variables, of course – who the opponent is, your table image, stack sizes, etc.

But based on the analysis so far, I’d say that a bet of 12 cents to 15 cents, or half to about 60% of the pot, seems to be the best way to get value from a flush draw. Anything more is a bit too much, considering you’re only on a draw, and anything less is probably a bit too weak and likely to get raised.

What do you think? Do you bet your flush draws on the flop for value, in case you don’t get paid when the flush hits? If so, how much would you bet your draws?

• Mac says:

That god but are you forgetting the board is paired? What if he is checking 22,88, 28, or a8 and the turn is the Ad or something.

I haven't played that limit in awhile but I think if you bet the turn bet the flop because the fact that if you hit flush and bet big on the turn, he will only call you with two pair plus. And he will have trips some of the time, giving you less chance.

I am not sure this is all about feel, and how the opponents play. I play much higher so I somewhat forgot..

• Derrick Kwa says:

Firstly, let me clarify. This was intended as a general analysis of how much you should bet on flush draws to get value (when you believe you'll win if you hit your flush). The hand was just an example.

In this specific example, though, I think it still holds. If he's holding 2-8, then, so be it. It's hard to ever put him on 2-8 when he called the preflop raise. 22-99 (I think anything higher wouldn't have limped preflop), mid/high suited connectors, K2 to AT (again, I think higher would have open-raised), I think you're still ahead of enough of his range here, and if he has trips, you'll definitely get paid out even more if you hit the flush.

I think when placing a bet like that, I wouldn't really be afraid of a fold on the flop. I'm still on a drawing hand, so I'd be fine taking the pot down here. But by betting that somewhere between half to 60% of the pot, I'd be getting even more value from if I hit the flush, and still not losing too much if I miss. The math seems to show that the bet is a +EV play. The only thing I'd be worried about if he's on a higher flush draw, but really I don't think I can be too afraid of that too often.

• airconman says:

I almost managed to will myself to write a long-ass post about my feelings about the mathematical-centric approach you've been highlighting on this blog, but i think all my garbled thoughts would probably be much more coherent if i verbalized them to you in a conversation rather than a huge chunk of text. Basically, i think what you're doing is great and definitely legit, but it really feels that you're neglecting many of the concepts that *most* winning players have based their gameplay and strategies around. Hit me up on msn and hopefully you'll get a better idea of what im saying.

• Derrick Kwa says:

Lol, actually I'd prefer for you to write a long-ass post here. =). Share the knowledge, you know.

• Matt says:

You're making the assumption here that you need to hit your flush to win.However, if the guy doesn't have 88, KK, AA, or 2x, then you are a *favourite* in the hand. Suppose you knew the guy had AK and he check-raises you on the flop then you should reraise for value!

Secondly, suppose you bet, get check-raised, just call, and hit your flush. The opponent may bet again, or he may call a couple more bets with a hand like A2 just because he doesn't believe you have a flush.

I would tend to bet at least the full pot on this flop. Your main goal with bet-sizing here is to try and get as much of your stack in as possible if you do make your flush. It's true that if he check-raises all-in you may have to fold and then you've bet yourself off the draw. But there are a lot of other possibilities; I don't think you should let fear of the chance that he has the 2% or so of his range that beats you, dissuade you from getting maximum value or fold-equity in the more likely cases.

• Derrick Kwa says:

I think the thing for this specific hand is that the hand was meant more as an example for the general idea of betting flush draws. It probably wasn't the best example hand, but yeah. The calculations were meant for the general idea.

You're actually about 50-50 against AK and 46-54 behind to AQ. So, you aren't exactly favorite. I get what you mean, but yeah, just saying.

As for betting full pot, I wouldn't actually bet full pot, especially since you're just no a draw. The analysis does show that it's still +EV as long as the opponent is folding 30% of the time (which in this specific hand is probably true), but in general, with flush draws, I think 50-60% is safer.

• mathwannabe says:

I'm enjoying the dialog. I have a family member who is good in math…I was going to ask them to help me learn the level of algebraic equations that will help me solve these types of problems. What level do your recommend?

Standtall

• Derrick Kwa says:

Glad you enjoy it. =). I'm gaining a lot from the dialogue as well.

What level of math…well, what I'm doing so far is really just basic probability, I guess? Shouldn't require too advanced math.

• jason says:

you shouldn't assume fold equity to hold constant.
i generally bet flush draws to take down the pot, and it also looks like i'm trying to avoid getting drawn out on, so if i catch, i feel like i have better calling equity. my bet sizes overlap regardless of which side of a flush draw i'm on to help disguise my play, but i bet bigger generally without the draw.

• Titan says:

Tbh at 2NL I don't think fold equity is at 30%. If you intend to go really deep into the math for this, you've to take in consideration when he has the nutflush draw over you, the backdoor nut flush draw (a solitary ace diamond) where you might be able to further extract more value.

• Derrick Kwa says:

What do you mean by I shouldn't assume fold equity to hold constant?

I'm basically calculating the *required* fold equity for the appropriate bet.

• Derrick Kwa says:

Agreed, definitely a lot more cases here to look at, but this is just a simple look. If I consider all the possibilities, I'd probably have to write a book. ðŸ˜‰ Thanks for bringing it up, though, it is a really important thing to note.

As for whether fold equity is at 30%, it really depends on who you're playing. When I was playing 2NL, I was choosing the tighter/nittier tables, so I do think I could have had a 20-30% fold equity.

• guitarizt says:

I'd bet 16 cents on the flop and barrel any broadway turn card because that's what I'd do with all my air here. Same for the river. If you're not betting with QdJd here you're not going to be able to represent anything when you're bluffing here in the future.

• philip says:

who really cares its peeny poker!!

• philip says:

who really cares its penny poker duh!!!

• xocai says: