We don’t need to be mathematical boffins to understand how to calculate EV. Quite simply, we’re looking to make sure that we pick up enough chips/money via the pots we take down to more than compensate for the cumulative investments we lose when we don’t win.

Here is a formula that helps us do just that:

EV = (%W * $W) – (%L * $L)

This might initially look a bit intimidating but in fact it’s not at all difficult to get to grips with.

%W is how often we’ll win a given hand.

$W is the amount we’ll collect when we win.

%L is how often we’ll lose the hand.

$L is the amount we’ll lose when we lose the hand.

A simple way to illustrate this formula in practice is to use the very simple Heads/Tails Coin Toss game. Let’s say that a kind soul offers us the following very favourable odds: If the coin lands on Heads we win $2, while if it lands on Tails we lose $1.

How does this fit into the EV formula?

First, the value for each of %W and %L is obviously 50% because a coin toss is a classic example of a 50-50 outcome.

$W in this case is $2 (the amount we win if the coin lands on Heads).

$L is $1 (the amount we lose if the coin lands on Tails).

Filling in these values gives us:

EV = (0.50 x 2) – (0.50 x 1) = (1) – (0.50) = $0.50

This means that in the long-term we can expect to make an average profit of $0.50 per toss of the coin. The terms of the wager, then, by giving us an EV of (+)$0.50, are clearly very favourable.

If we translate the mechanics of this example from tossing a coin to poker, then the idea is that we seek out positive EV in order to maximise our potential for profit, while at the same time endeavouring to avoid negative EV. Poker is a game that on the one hand rewards us for performing well, but on the other allows us to pick up and even ‘perfect’ bad habits. Consequently, the trick is to find consistently optimal strategies and plays that afford us +EV, but to steer clear of poor play and habitually unsound tactics that amount to -EV. There’s no point securing well deserved long-term profit by working on various aspects of our game if this ends up being cancelled out by the cumulative losses caused by reckless, indulgent and ultimately avoidable play.

Calculating EV allows us to determine how best to continue and thus strengthens our foundations in terms of long-term positive expectation and, in turn, overall profit. Switching from the Heads/Tails game, here’s an example of how it works in poker:

We’ve reached the Turn and the pot is $200, and our lone opponent throws in a bet of $100. Given our opponent’s range, we give ourselves a 45% chance of winning. Let’s fill in the values.

EV = (%W * $W) – (%L * $L)

%W is 45%

%L is 55%

$W is $300 (the $200 Pot + the opponent’s $100 bet).

$L is $100 (the price of our call).

This gives us:

EV = (0.45 x 300) – (0.55 x 100) = (135) – (55) = $80

So, if we were to make this call our average profit would be $80. Of course, short-term outcomes can throw up any result, but the point of determining +EV plays is that we can continue playing optimally in the knowledge that over time such strategies will prove profitable.

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About the Author

AngusD

AngusD switched from pro chess to poker two decades ago and has been professionally involved in the game on numerous levels since the very beginning of online poker, including playing as a poker ambassador both online and at major festivals around the globe. He has written much about the game over the years, and brings to YPD a wealth of experience in all aspects of the poker industry. Meanwhile, his many years on the pro chess circuit (he’s an International Master and prolific author) afford him an interesting perspective on the psychology of poker.

Latest changes

The last changes of the page “The Expected Value Formula” was made by AngusD on May 10, 2021