Expected value (EV) is one of the most important concepts in gambling to understand — especially if you have aspirations of being a professional gambler of any kind. When you have a positive expected value (+EV), you have “the best of it.”

On the other hand, when you have a negative expected value (-EV), you have “the worst of it.”

Professional gamblers try to only put their money into action when they have the best of it. In some cases, they could be wrong about whether the bet they’re making is positive or negative expectation. But if they’re right often enough, they’ll make a profit in the long run.

Other gamblers — those with the worst of it — will eventually lose all their money if they keep playing. That’s just how the math behind the probability works.

A concept related to expected values is risk and the tolerance of risk. A situation can have a positive expectation, and you’ll still lose that bet most of the time.

The more often this is likely to happen, the more risk you’re required to take when placing that bet. Your “risk tolerance” describes how much risk you’re willing to take.

This post explains what expected value is, specifically, and it also explains how you should use this concept to inform your gambling decisions. It also looks at some of the implications involved with having a high and/or low risk tolerance.

## How to Calculate or Estimate the Expected Value (EV) of a Bet

You have 3 things to consider when estimating the expected value of a bet.

The first of these is how likely you are to win. You might have a 50% chance of winning a bet, a 75% chance of winning, or a 5% chance of winning. None of these by themselves tell you what the expected value of the bet is.

The 2^{nd} of these is how much you’ll lose if you lose the bet. Many bets are even money, which means you’ll lose as much as you’ll win. For example, if you bet $100 hoping to win $100, then you’re looking at an even-money loss and an even-money win.

Finally, you need to know how much you’ll win if you win. This is usually expressed as a ratio in odds format. For example, if you bet on a specific number in roulette, the payoff is 35 to 1.

If you bet $100 on that number and won, you’d win $3500.

None of these factors by themselves describe your expected value. You must look at them in aggregate. That’s probably easier than it sounds, too.

Finally, if you’re a professional gambler, you should avoid –EV bets and only make +EV bets — at least as much as possible. Most gambling games are games of incomplete information, so your ability to estimate which bets are +EV versus which bets are –EV has a huge effect on your success as a gambling pro.

The classic example is a bet on a coin toss. If you bet a buddy that a coin will land on heads, and you’re both putting up the same amount of money, your expected value for that bet is 0.

You’ll win half the time, he’ll win half the time, and your average win will be 0. This is also called “breaking even.”

Now let’s change one of the factors — how much you’re risking versus how much you’re winning. If you win $2 every time your opponent is wrong, but you only have to risk $1 if you’re wrong, then you have an average win of 50 cents per coin flip. That’s a positive expected value (+EV) situation. Over time, you’ll win all your buddy’s money.

This is how all casino games work, by the way. The casino makes you a wager where the payoff isn’t commensurate with the probability of winning. It’s rare that you can change a –EV casino game to a +EV proposition, although it is possible.

When you’re dealing with 2 parties, and the EV isn’t 0, then one of you must have a +EV, and one of you must have a –EV. If you add the 2 together, you should have a total of 0. If you have any other total, you did the math wrong.

## An EV Example from the World of No Limit Texas Hold’em

I recently read *Poker Winners Are Different* by Alan Schoonmaker. In that book, he has an example “story problem” from the world of no limit Texas hold’em. Here was the question.

You have $100 in a $2/$5 no limit Texas hold’em game, and you’re dealt pocket aces — the best possible starting hand in the game. You have 9 opponents. You push all-in preflop.

How many callers are you hoping for?

You can choose any number between 1 and 9.

I asked a couple of friends this question via text. One of them got it correct (9), and the other guessed 5.

But the only correct way to get the answer to this is to look at the expected value of the various outcomes.

I understand people who answer 1 or 2, by the way. I was taught that you raise preflop with a strong hand like that to narrow the field. You’re more likely to win with just 1 or 2 opponents. In fact, each player who calls lowers your probability of winning the pot. But expected value has more to consider.

If you only had a single caller in that situation, you’d win the pot 85% of the time.

If all 9 of your opponents called in that situation, you’d win the pot 30% of the time. (So yeah, you’d lose the pot about 2 out of 3 times.)

Let’s assume you play this hand 100 times and get mathematically average results on those 100 hands. In the case where you have 1 caller, you’d win $100 on 85 of those hands, for a gross win of $8500. You’d lose $100 on 15 hands, which would be a gross loss of $1500.

That’s a net win of $7000.

Averaged out over 100 hands, that’s an average of $70 in winnings per hand.

Now let’s assume you play this hand 100 times with 9 callers, again getting statistically average results. This time, you’re going to lose $100 on 70 hands, for total losses of $7000. But on 30 hands, you’re going to win $900, for total winnings of $27,000.

Your net win is $20,000.

Averaged out over 100 hands, that’s an average of $200 in winnings per hand.

That’s 3 times the net win with 9 callers as opposed to 1 caller.

Your return on investment (ROI) is much better, too. You’ve wagered $10,000 in both those scenarios, but in one scenario, you see an ROI of 70% as opposed to an ROI of 200%.

## Expected Value in Probability

I write about gambling, so the explanations and examples I use are strictly gambling-related. In other words, I talk about expected value in terms of dollars.

Mathematicians who specialize in probability have a broader definition. When they talk about expected value, they’re talking largely about any long-term average of results in a random experiment.

For example, if you’re looking at the expected value of a single roll of a 6-sided die, the answer is 3.5. It’s just an average.

Here’s how you would calculate that.

You assume a perfect series of 6 rolls.

- You’ll get a 1, 2, 3, 4, 5, and 6 in that situation
- You’d add those together, getting a total of 1 + 2 + 3 + 4 + 5 + 6, or 21
- Divide that by 6, and the average is 3.5

If you read the Wikipedia article on expected value, it looks like it’s a really complicated subject.

But for our purposes, as gamblers and poker players, it’s not that complicated at all.

## What Expected Value Means for the Gambler in the Long Run

In the short term, expected value doesn’t mean much. On a single hand of no limit hold’em, you might make a highly +EV play and still lose 2 out of 3 times. When you factor variance in, you might get your pocket aces cracked a dozen times in a row.

This doesn’t mean much to the savvy gambler, though.

Savvy gamblers understand that if you consistently put money into +EV situations and bets, in the long run, you’ll eventually profit.

Casinos use this principle daily. They don’t care if you win a big jackpot on the slot machine games. They don’t mind if you have a good run at roulette. They don’t even mind if you have a winning streak at the blackjack table — unless you’re counting cards.

They will DEFINITELY care if you’re counting cards.

When you’re playing slots, you have a –EV and the casino has a +EV. When you’re playing roulette, you have a –EV and the casino has a +EV. When you’re playing blackjack, you have a –EV and the casino has a +EV.

But when you’re counting cards, you have a +EV and the casino has a –EV. That’s something the casino can’t allow. If the casino puts you in a +EV situation, they know that they’ll lose money in the long run. And businesses don’t like to lose money in the long run.

## Expected Value and Online Casino Bonuses

There was a time when you could get into some nice +EV situations on the internet by using a casino signup bonus. Those days are gone now, for reasons I’ll explain below.

When you sign up for an online casino bonus, the casino gives you extra money in your account to reward you for signing up. The most common offer in the early days of online casinos was a $200 bonus on your first deposit of $100.

You’d deposit $100, but you’d have $300 in your account to gamble with.

Of course, you were required to make a certain number of wagers before being allowed to cash any money out. But if you understand that the house edge in blackjack is very low if you use basic strategy, you could easily clear that wagering requirement with really low expected losses.

Let’s say they require you to wager that $300 15 times before cashing out. The total amount you must wager in that case is $4500.

If you assume that the blackjack game has a 1% house edge, your expected loss on that amount of action is $45.

- You deposit $100
- You get $200 as a bonus
- You lose $45
- You have $255 in your account
- You cash out with a $155 profit, on average

Online casinos grew wise to this advantage-play tactic pretty quickly, though. They started limiting the games and the action which counted toward those wagering requirements. They also increased the amount you were required to wager before cashing out.

Let’s assume you can only use wagers on slots toward your wagering requirement. You don’t know what the house edge is for slot machines at an online casino (or in a land-based casino, for that matter), but let’s assume that it hovers around 6%.

On $4500 worth of action, you’re now looking at $290 in expected losses. Since you started with $300, you’d only have $10 left after fulfilling your wagering requirements.

But then assume that the casinos increased the wagering requirements from 15 times to 30 times your deposit bonus.

Now you’d have to get really lucky to walk away with a profit. The expected value for the casino is to win all your money and then some. Even with a lot of variance, over time, the casino expects to come out way ahead.

I don’t know of any modern online casinos where you can get a positive expected value from a signup bonus anymore.

## Conclusion

Expected value is an important mathematical concept every gambler should understand, but it’s especially important for poker players and aspiring advantage gamblers.

It’s just the average amount you expect to win or lose on average on a particular bet over time.

The concept is useful because if you put yourself in positive expectation situations repeatedly, you’ll eventually show a profit. If you constantly put yourself in negative expectation situations, you’ll eventually lose money in the long run.

For recreational gamblers, that might not matter as much, but I’d suggest it still matters some. I don’t think you should be playing against a stacked deck unless you know and understand it’s stacked and how.

Understanding expected value is how you understand that.

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