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Calculating Expected Value

Let’s be honest—math is a pain. If you went on vacation next week, I’d bet you wouldn’t spend it doing calculus problems. It’s okay to not like math, but as a poker player, you do need to know some of the basics. After all, poker revolves around probabilities, and knowing the odds can help you win.

One of the most important poker math concepts to understand is expected value (EV). “What’s that?” you ask. Well, it’s pretty much what it sounds like: the average amount of units you can expect to win when you make a given play.

To clear up the concept, imagine flipping a coin. Pretend you’re going to call heads every single toss for 100 tosses. You’re a high roller, so you’re betting $1 per toss. What’s your EV when you make this bet?

Let’s break it down. There are 2 outcomes to tossing a coin: heads or tails. Assuming the coin is not weighted or tampered with in any way, both outcomes should be equally likely to occur. That is, there is a 50% chance of the coin landing on a given side each toss. To calculate EV, use the following formula:

((% chance to win) * (net win)) – ((% chance to lose) * (net loss)) = EV

Plug in your coin toss values:

(.50 * 1) – (.50 * 1) = 0

And you’ve got your expected value: $0.

So what does this really mean, in plain English? It means that in the long run (this is important), you can expect to neither win or lose any money by betting on a coin toss. If you tossed the coin 1000 times, or 10,000 times, or 100,000 times, chances are increasingly likely that you’d break even.

This does not mean that you’re guaranteed to break even in the short term. All chance games are governed by variance: short-term deviations from expected value. If you were to toss a coin 10 times, it’s certainly possible you might lose 7 or 8 times. Heck, it’s even possible that you’d lose every time.

EV is a calculation that looks to the long-term. Yes, you can lose over a small sample of bets, no matter how good your EV is. However as you run more and more iterations of the same bet, the results will approach the EV with closer accuracy. Expected value is not the same as guaranteed profit (or loss). It’s an indicator of how much you can expect to gain or forfeit from a proposition over time.

Pocket Aces Preflop

Looking to a relevant example, let’s calculate the EV of pushing all-in with pocket aces. Assume the following:

1. Both you and your opponent are 200BB deep.

2. Your know that your opponent will call when you raise all-in.

3. You’ve got the ace of spades and the ace of diamonds.

4. You know that your opponent has KQ of spades.

Let’s figure out your EV. Your first step is to run the hand matchup through a poker odds calculator. If you’re looking for a good one, I recommend PokerStove—It’s free, and easy to use. Upon running the odds, you’ll see that your aces are good roughly 82% of the time. Meaning, 82% of the time you’ll win, and 18% of the time you’ll lose.

Since both you and your opponent are 200BB deep, your possible winnings are (surprise) 200BB. Your possible losses are the same. Knowing all of this, you can calculate your EV. Just plug your data into the formula we saw above:

(.82 * 200BB) – (.18 * 200BB) = EV

164BB – 36BB = +128BB

Getting it all in preflop with your pocket aces against your opponent’s KQ of spades leaves you with an EV of +128BB. In other words, you will win an average of 128BB each time you make this bet over the long run.

You might lose 200BB 3 times in a row. You might win 200BB 8 times in a row. The short-term doesn’t really matter. What matters is that the bet is profitable in the long run, and you should make it every chance you get!

 

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