Getting Value From Flush Draw on the Flop
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.
Hero (BB): $2.58
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|
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
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?