I wish to study 4 bet and 5 bet shove theory

My first example would be the following:

You run 1,000 simulations of 5 bet shoving A2s vs an aggro 4 bettor and using all in equity you know vs the ranges you construct/model, that A2s is +$2.50 EV

This is of course pre rake.

1.) Sometimes you shove pre and villain folds so you win the pot and you pay no rake - using a standard sb vs bb situation of

$1.25 open, 3bet to $4 and 5 bet to $10.50 let's say

2.) When you shove and get called and lose you lose a clean $50

3.) When you shove and get called and win you win $98 (with $2 max rake)

Using 50NL as the limit.

how the f*** do I begin to build a math equation to prove what the overall profit of this situation would be over 1,000 trials?

Please keep the math equation as simple as possible pls I really suck at math

I would like to be able to run the equation with different scenarios like different 4 bet sizings, shoving 99 with +$3 equity instead of $2.50 , etc.

thanks in advance!

Note: I'm not a math expert and I've probably made an error in this, but I don't have time to double-check everything.

For EV equations, I like to put it all in natural language to include all possibilities, and then add it all up.
EV = [How often you win with no flop * how much you win when villain folds pre] + [How often villain calls] * [How often you win when he calls * how much you can win when he calls] - [How often villain calls] * [How much you can lose when he calls]

EV = [frequency that villain folds to shove * size of pot before you shove] + [frequency that villain calls * your equity * (pot + size of villain's final call)] - [frequency that villain calls * villain's equity * the size of your final raise]

e.g. Let's suppose you 3-bet to $4, and the pot is $20 after villain's 4-bet to $10.50, and you're going to jam $50 total.
Your final raise amount (your "risk") is $46, because you already put in $4. If villain calls, he's putting another $39.50 in the pot.
Let's suppose that villain folds 50% of the time, and calls 50% of the time, and you have 30% equity when called.

EV = [fold freq * pot] + [calling freq * hero equity * (pot + villain's call)] - [calling freq * villain's equity * hero's final raise]
EV = [50% * $20] + [50% * 30% * (20 + 39.50)] - [50% * 70% * 46]

EV = [10] + [8.925] - [16.1] = $2.825

i.e. if you jammed with 50% fold equity, and 30% hand equity when called, you make $2.83.

I'd suggest to get rid of dollars and do it in big blinds.

**** me this is hurting my head lol

I'll review this more when I get some time.

In other news, I figured out that rake has not a big effect in these situations bc most of your profit is actually won pre flop when villain folds to the jam

and of course when you shove and get called, you lose a lot so you don't pay rake anyway

you only pay rake when you get called and win

So if you use "all in expert" you can safely just take out about .5bb from the total EV profit so if it's +$2.80, subtract .25cents = $2.50 profit about.

As a buffer I'd like to take out .50cents probably so assume $2.25 profit.

Basically in the end you are risking about 90 bb's to win 4.5 bb's - I suppose this is worth the variance, +4.5bb's per hand is better than you can hope for in almost any situation.

risking 90 bb's for 1bb , however, may not be worth the variance bc of tilt factor being - EV long term

I'm basically trying to see how far I wish to push range vs range in 5 bet shove spots to stomach whatever variance might ensue.

All expected value (EV) calculations are built up from the basic formulation below (in plain language):

EV for an event equals the probability the event occurred multiplied by the payout when the event occurs.

Translated to something more mathematical:

EV(event) = P(event)*Payout for event (note payout can be negative here).

Now sometimes you have multiple events that can occur and you want to know the total EV of all the events. In that case to can just add all of the EVs together like so:

EV (total) = EV (event1)+EV (event2)+EV (event3)

Where

EV (event1) = P (event1)*Payout for event1
EV (event2) = P (event2)*Payout for event2
EV (event3) = P (event3)*Payout for event3

So all Arty was trying to demonstrate is that it's easier to start identifying your events and their payouts first using normal language and translate them into math later.

Another situation that might arise is one EV equation can be a payout for a more complex equation, but let's concentrate on basic EV equations first.


I would deduct the rake from your payout in your EV equations and not your final EV total.

I doesn't quite work like that. The total EV of your actions for open,3bet,5bet, etc are cumulative. So if you choose to 5bet something it's because the combination of EV from opening and getting folds, opening and getting calls, 3betting and getting folds, 3betting and getting calls, 5 betting and getting folds, and 5 betting and getting calls is positive (note the individual terms don't necessarily have to be positive but at least 0 because you have the option to fold at any point which costs nothing).

FWIW, I knocked up a spreadsheet this afternoon that uses the equation I mentioned earlier. It looks like this:



I'm not in a position to upload the spreadsheet right now, but I think it's fairly straightforward. (You can see the equation for the final EV calc in the 'address bar' at the top. All I have to enter are the numbers in the yellow boxes, and the calculation is done instantly.)