# 4Betting Mathematics

28 Mar, 2011

This was inspired by a friend of mine who plays NL50. She wanted to work out whether she should **4bet/call** in a spot against a regular.

So here is an example of mine to explain it.

**Hero CO ($100)**

BTN ($100)

SB ($100)

BB ($100)

**Preflop:**Hero is the CO and is dealt

Hero raises to $3, BTN folds, SB re-raises $11, BB folds, Hero...

*considering whether we can 4bet/call*

## The basic mathematics.

I will try to keep the following mathematics as algebra free as I can.

Our expected value (EV) in this situation can be broken down into two parts:

- Our EV when we get all in. (i.e. we 4bet and then call any raises).
- Our EV when our opponent folds.

To calculate the EV of 4bet/calling, you first need to work out the EV for the following two situations:

```
EV when all in = ( [total pot - what we risk] * our equity) - (what we risk * opponent's equity)
```

```
EV when opponent folds = we win whatever is in the pot
```

We can then *combine* these two EV equations together to get the Total EV for our 4bet/call:

```
Total EV = (EV when all in * how often we get it in) + (EV when opponent folds * how often opponent folds)
```

## Ranges.

Let's think about our opponent's range, and assume they are going to be shoving *over* our 4bet with JJ+ and AK. PokerStove tells us that our equity in the hand against this range is `33.6%`

.

We also know that if we commit to 4bet/calling, we will be risking `97bb`

(we put in `3bb`

already) to win a total pot of `201bb`

(both of our stacks plus the big blind).

Therefore, the EV of getting the stacks in is:

```
EV when all in = (201-97 * 0.336) - (97 * 0.634)
EV when all in = 34.9 - 61.5
EV when all in = -26.6bb
```

So clearly we have a **negative expectation** when we get all the money in, and that is because we are getting the money in with `33.6%`

equity vs their range. However, "EV when all in" is only *part* of the equation – we also have to consider **how much we win when our opponent folds** too.

When our opponent folds after we 4bet, we win `15bb`

(their `11bb`

, plus the `1bb`

and our `3bb`

initial raise). Therefore:

```
EV when opponent folds = +15bb
```

## Fold equity.

There must be a cutoff point where the small blind folds to our 4bet so often that it doesn't matter if we lose money when we get all in, because we more than **make up for it the times they fold**. This is called fold equity.

We can construct a simple equation to find out whether we have any fold equity. All we need to do is replace *how often we need them to fold* with `x`

, and set the *answer* of the equation to `0`

. Then we can solve the equation for `x`

, and it will tell us **how often our opponent must fold in order to break even**.

It is basically a re-arrangement of the formula for Total EV of the play:

```
Fold Equity:
0 = [what we win when opponent folds * x] - [what we lose when opponent doesn't fold * (1-x)]
```

From the figures we've already calculated, we know that when our opponent folds we win `+15bb`

, and when they do not fold we lose `26.6bb`

(because we get all the money in).

Therefore we can *substitute* the numbers into the equation, solve it, and get a figure for `x`

:

```
Fold Equity:
0 = [15 * x] - [26.6 * (1-x)]
0 = 15x - 26.6 + 26.6x
0 = 41.6x - 26.6
26.6 = 41.6x
x = 26.6/41.6
x = 0.64, or 64%.
```

In other words, **our opponent needs to fold more than 64% of the time for our 4bet to be profitable**.

To put this into more useful terms:

- If our opponent is 3betting
`8%`

or less, then we make a loss. - If our opponent is 3betting
`9%`

of hands in this spot and shoving the top 3% of hands (`JJ+`

and`AK`

), we show a profit by 4bet/calling`TT`

here. - If our opponent is 3betting
`10%`

or more, then we are just printing money by 4betting.

It is impossible to model whether 4betting is better than calling, but you don't want to spend a ton of time focusing on these type of spots – once you know whether 4betting is +EV or not, you can just go from there.

## Conclusion.

You should spend a decent amount of time going back over this article and re-entering numbers based on different *shoving ranges* and different *starting hands* for yourself. For example:

- What happens when our opponent is 5bet bluffing sometimes?
- What if if they are only shoving
`QQ+`

? - What if we have
`AQ`

instead of`TT`

?

Jack.