Hold’em or Fold’em? Expected Value for Calling ALL IN

We’ve all been there:

Our aggressive opponent just raised all in and now we have a choice to make: call or fold?

To figure it out we have to calculate our Expected Value (EV) of a call. If our EV is greater than zero, we should call because in the long run we’ll make money. If our EV is less than zero, we can safely fold because we’ll lose money in the long run.

You vs. Freddy the Fish

You’re playing your friend Freddy in a $0.50/$1 No Limit game. Freddy is a wild, aggressive player who constantly makes all in moves. You’re the small blind with $40, Freddy is the big blind with a measly $15. It is folded to you and you raise to $3 with Qc 8c. Freddy immediately goes all in. Should you call or fold?

The Short Answer

You can calculate your EV of calling using the following equation:

EV Call = equity * (pot + tocall) - tocall

Your Q8s has about 0.44 equity vs Freddy’s range. The pot is $18 and you have to call $12.

EV Call = 0.44 * ($18 + $12) - $12
EV Call = +$1.2

You should call because calling has a positive expected value; you will make money in the long run.

The Long Answer

1. Estimate your opponent’s range

Since Freddy is a maniac, he’d probably go all in with any pocket pair, any ace, any king, any two face cards, and any suited connector 54s or higher. Also, he’d probably go all in with Q8, J8, T7, 98 or better.

2. Estimate your equity vs that range

During a hand, you’ll rarely be able to calculate your equity exactly. Thats why practicing is so important. You’ll get a good feel for your equity by using an equity calculator like ALL IN Expert or PokerStove. ALL IN Expert display your equity for every hole card at once:

Against Freddy’s loose range, our equity is 0.4375.

3. Estimate your EV and make a decision

They key to calculating your EV is understanding that a certain percentage of the time you will win some amount, and the rest of the time you will lose some amount. Here’s how we do it.

e = Your equity aka your win %
(1-e) = How often you'll lose
p = The current pot size
c = The amount you have to call

So, sometimes we’ll win the pot:

e * p

And sometimes we’ll lose (we only lose the amount we have to call):

(1-e) * c

So our total EV is:

EV = e * p - (1 - e) * c

A little simplification gives us the original formula:

ep - (c - ce)
ep - c + ce
e * (p + c) - c

Effective Stacks

In this example, Freddy started with $15 and we started with $40. So when Freddy went all in, we had to call $12. If the situation had been reversed and we had $15 and Freddy had $40, our calculations would remain the same. Here’s why:

Freddy goes all in for $40, but since we only started with $15, we only have to call $12, same as before.

We call this our effective stack size. It is the smaller of the two starting stacks. In this case, our effective stacks are $15. Neither one of us can risk more than that.

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