The draw: chase or push. Math approach.

[fiskebent]

This discussion on whether to chase or push with a draw seems to pop up a lot and I thought I'd do some math on it.

Let's say you have a draw after the flop. Your opponent has a made hand and is betting 1/2 the pot. If you draw, he'll bet 1/2 the pot until you hit. After you hit he'll pay off a 1/2 pot sized bet 50% of the time.

OK. Then let's set up some pot sizes.
After the flop, the pot is $1. He bets $0.50 and you call.
If you hit on the turn, you'll win $1.50 + an additional $1 50% of the time, so that's a $2 win.

If you miss on the turn, he'll bet another $1 that you call.
If you hit on the river, you'll win $2.50 + an additional $2 50% of the time, so that's a $3.50 win.
If you miss on the river, you lose $1.50.

Let's call the chance that you hit on the turn for HoT and the chance that you hit on the river for HoR. Then we get
EV = HoT*2 + (1-HoT)*((HoR*3.5)-(1-HoR)*1.5)

Let's compare this to pushing after the flop. Our chance to hit on the turn or river is 1-((50-outs)/50)*((49-outs)/49)

Let's assume that our opponent folds 20% of the time and that we push for 5 times the pot. Then we get
EV = 0.2*1 + 0.8*((hit%*6)-(1-hit%)*5). Hit% is the chance we hit on either the turn or the river.

That leads to the following numbers
Outs ChaseEV PushEV
1 -1.35 -3.45
2 -1.20 -3.10
3 -1.06 -2.77
4 -0.92 -2.44
5 -0.79 -2.11
6 -0.66 -1.80
7 -0.54 -1.49
8 -0.41 -1.19
9 -0.30 -0.89
10 -0.18 -0.60
11 -0.07 -0.32
12 0.03 -0.05
13 0.13 0.22
14 0.23 0.47
15 0.32 0.73
16 0.41 0.97
17 0.49 1.21
18 0.58 1.44
19 0.65 1.66
20 0.72 1.88
21 0.79 2.08

So in these scenarios the only spot where it's profitable to chase but not to push is with 12 outs. With 13 outs and better, it's more profitable to push. And with 11 outs or less you should fold after the flop.

These numbers and the conclusions will change with pot sizes, bet sizes, stack sizes, folding percentages, number of people in the pot and so on. So if you want to explore, you could make an Excel sheet where you can change the variables.

Comments are of course welcome.

[4_2_it]

This works great if you are HU with predictable opponents. Why is chase or push your only option? No bet or raise?

This looks an attempt to standardize play, which in NL will get you killed. I understand that having guidelines is important and understanding the math behind your concepts is VERY important, but trying to set-up a system of play based on this kind of stuff is just asking for disaster.

[fiskebent]

I'm not advocating always doing the same thing based on math on simplified scenarios. I'm advocating adding math based analysis to your play rather than always winging it because you have a gut feeling about what is the right play. I thought my last paragraph made that clear... Obviously it didn't...

[4_2_it]

[ QUOTE ]
I'm not advocating always doing the same thing based on math on simplified scenarios. I'm advocating adding math based analysis to your play rather than always winging it because you have a gut feeling about what is the right play. I thought my last paragraph made that clear... Obviously it didn't...

[/ QUOTE ]

All apologies, I misinterpreted the OP....With that clarification, I agree with what you are trying to do. Adding math as opposed to following your gut is definitely +EV.

[vulturesrow]

[ QUOTE ]
[ QUOTE ]
I'm not advocating always doing the same thing based on math on simplified scenarios. I'm advocating adding math based analysis to your play rather than always winging it because you have a gut feeling about what is the right play. I thought my last paragraph made that clear... Obviously it didn't...

[/ QUOTE ]

All apologies, I misinterpreted the OP....With that clarification, I agree with what you are trying to do. Adding math as opposed to following your gut is definitely +EV.

[/ QUOTE ]

Whats wrong with Push n Pray? Its a specialty of mine. [img]/images/graemlins/grin.gif[/img]

[theblitz]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I'm not advocating always doing the same thing based on math on simplified scenarios. I'm advocating adding math based analysis to your play rather than always winging it because you have a gut feeling about what is the right play. I thought my last paragraph made that clear... Obviously it didn't...

[/ QUOTE ]

All apologies, I misinterpreted the OP....With that clarification, I agree with what you are trying to do. Adding math as opposed to following your gut is definitely +EV.

[/ QUOTE ]

Whats wrong with Push n Pray? Its a specialty of mine. [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]
Mine is Push n Cry.

[Fallen Hero]

"Let's assume that our opponent folds 20% of the time and that we push for 5 times the pot."

wtf?