What is Expected Value?

You have two choices when you gamble. You can do what most players do, and rely on luck. The other option is to understand how gambling games work, and make betting decisions knowing the long term outcome before you make your first wager.

Before you learn more about the second option, I want you to understand something. I’m not here to judge you, or even tell you which kind of gambler you should be.

Lots of people enjoy the chance to get lucky playing slots or blackjack and aren’t interested in the work involved in learning more about gambling games.

And it does take some work.

But the good news is that once you do the work you can benefit from it the rest of your life.

In addition, once you learn about expected value, you can use it to find opportunities to beat the casino on a consistent basis.

When a person learns how to look for opportunities t beat the casino, they become an advantage player.

An advantage player knows there are certain ways to beat the casino and is always looking for and exploiting an edge. Becoming an advantage player is even harder work than understanding expected value, but it can be worth the effort.

If you’re interested in learning how to improve your gambling results, this page will help you.

Even if you’re not sure you want to become an advantage player, if you take a few minutes to learn about expected value you can stretch your bankroll and understand which games offer the best value.

In the first section you’re going to learn exactly what expected value is and how to use it. Then I’m going to explain a little bit about how you can use it as an advantage player.

Gambling games are all mathematical based. If you know a few things about how math works and understand the mathematical concepts behind gambling games, you can determine how much you should win or lose on any wager.

I realize that many people are hesitant to dig into math too much. I used to teach math at a small college and many of my students were convinced that they couldn’t do math and / or were scared of it.

Please don’t stop reading just because I mentioned math. You’re going to see that expected value is easy to learn, and everything you need to know is on this page.

Everything you need to know is explained in simple terms and shown using examples so you can learn step by step.

Every bet you make has an expected value. The expected value of a bet is the average amount you win or lose in the long run.

The long run is tens of thousands of bets.

Because all bets have a mathematical base, if you know the correct numbers and formulas, you can determine exactly how much a bet is worth. On any individual bet, you either win or lose.

Here’s an example:

You place a $100 wager on black at the roulette wheel. The roulette wheel is one that has a zero and double zero space. This is usually called an American roulette wheel.

When the ball lands on black you get your $100 bet back and win $100. When the ball doesn’t land on black, you lose your $100 bet.

This is a fairly straightforward bet. You either win $100 or lose $100. But when you make this same bet thousands of times your results get closer to the expected value.

Expected value tells you the exact amount you lose on average on each individual wager. In the short run your results vary quite a bit from the expected value.

On the roulette wheel, there are 38 spaces. 18 spaces are black, 18 are red, and two are green.

You can use this information to determine the expected value of each bet. It’s not hard to work out the numbers, but the good news is you don’t need to work through it like this.

The hard work has already been done.

I’ll share the numbers for popular games below, but seeing how the numbers are determined will help you in the long run.

Because the roulette wheel has 38 spaces, let’s see what happens when you make 38 bets, with the results of the 38 spins landing on each of the 38 possibilities. In the long run, the ball lands on each space an equal number of times, because the results are random.

If you bet$100 on black 38 times your total cost of bets is $3,800. When the ball lands on red or green, you don’t get anything back. When it lands on black you get back your $100 bet and the $100 you win. It lands on black 18 times, so you get back $3,600.

To determine the expected value you use these numbers to determine the house edge for the game.

Here’s how you do it:

Divide the amount you win by the amount you bet. This means you divide $3,600 by $3,800. This gives you .9473684211. Now subtract this number from one. 1 – .9473684211 = .0526315789

This decimal is then converted to a percentage. You convert a decimal to a percentage by moving the decimal two places to the right. We then round the number down, so the percentage is 5.26% for roulette with 38 spaces.

The first decimal you computed represents the amount the casino returns on average on each bet and the second represents the amount the casino keeps on average from each bet. So the pay back percentage is 94.74% and the house edge is 5.26%.

Remember, I mentioned above that the hard work has already been done for you. You don’t have to work through the math in the example for every game.

You can simply see the house edge for popular casino games in the list below.

Take a look at the games and the house edge and you can see that some games have a relatively low house edge and some are much higher.

After you take a minute to look at the list, I’m going to show you how to use the house edge to determine the expected value of any bet you make.


Baccarat player bet	1.24%
Baccarat banker bet	1.06%
Baccarat tie bet	14.36%
Blackjack	.3% to 2% depending on house rules
Craps don’t pass bet	1.36%
Craps pass line bet	1.41%
Craps odds bet	0%
Keno	20% to 30%+
Let It Ride	3.51%
Pai Gow Poker	1.46%
Roulette 37 spaces	2.7%
Roulette 38 spaces	5.26%
Slot machines	2% to 20%
3 Card Poker	3.37%
Video Poker	.25% to 8%

You can see that a few games have a range of house edge percentages. Blackjack, keno, slot machines, and video poker all have a range of possible house edges. Keno and slot machines don’t offer good house edges, but blackjack and video poker both offer the chance to find games with a low house edge.

You can learn more about them in the advantage play applications section below.

In the next section you’re going to learn how to use the house edge percentages on the list to determine expected value for every bet you make, and how to use it to choose which games to bet on and how to determine how long your bankroll should last.

Using Expected Value

Now that you understand that expected value is the average amount each wager is worth, you can see it in action. Let’s continue using the roulette example from above.

Here’s an example:

You know the house edge for a bet on black on a wheel with 38 spaces is 5.26%. To determine the expected value of a bet you simply multiply the house edge times the amount of your wager.

A bet of $100 has an expected value of $5.26. $100 times 5.26% = $5.26

It’s important to understand that this is an expected loss of $5.26. In other words, the house edge creates a positive expected value for the casino and a negative expected value for you. On average, every time you bet $100 on black at the roulette table in the example you lose $5.26 and the casino makes $5.26.

You know that on any individual bet of $100 you either lose $100 or make $100, but in the long run you lose more than you win and the result is the long term expected value.

If you make a bet of $20, the expected value of the loss is $1.052. $20 times 5.26% is $1.052. You can plug any size wager into the formula to get the expected loss.

Here’s the formula:

Bet X house edge = Expected value

The expected value for the player is always negative, or an expected loss, as long as the house edge is positive.
Here are some examples for different games using different bet sizes:

Baccarat player bet:

$10 bet X 1.24% = $0.124
$50 bet X 1.24% = $0.62
$100 bet X 1.24% = $1.24

Baccarat banker bet:

$10 bet X 1.06% = $0.106
$50 bet X 1.06% = $0.53
$100 bet X 1.06% = $1.06

Craps pass line bet:

$10 bet X 1.41% = $0.141
$50 bet X 1.41% = $0.705
$100 bet X 1.41% = $1.41

Craps don’t pass line bet:

$10 bet X 1.36% = $0.136
$50 bet X 1.36% = $0.68
$100 bet X 1.36% = $1.36

Blackjack with .5% house edge bet:

$10 bet X .5% = $0.05
$50 bet X .5% = $0.25
$100 bet X .5% = $0.50

Roulette 37 spaces:

$10 bet X 2.7% = $0.27
$50 bet X 2.7% = $1.35
$100 bet X 2.7% = $2.70

It’s easy to look at the list of house edge percentages and decide that it’s always best to play the games with the lowest house edge.

While this is a good idea in general, the truth is that you also need to consider a few other things.

Some games are designed so you can use a certain amount of strategy to alter the house edge. When you play blackjack you can use something called basic strategy to lower the house edge.

If you don’t follow basic strategy the house edge goes up. Video poker games are the same way.

If you aren’t willing to learn and use the proper strategy, you’re better off playing a game that doesn’t require strategy that has a low house edge.

Here’s an example:

You find a blackjack games that has a house edge of 1% but you don’t know the proper strategy. You’re better off playing baccarat and always making the banker bet with a house edge of 1.06% because the house edge on the blackjack game will be higher when you make mistakes compared to using perfect strategy.

The other thing to consider is how many bets you make per hour at each game. Some games play faster than others, so if you’re concerned with extending your bankroll you’re better off playing a slower game with a higher house edge than a fast game with a low house edge.

Here’s an example:

You’re comparing a video poker game and Pai Gow Poker. The video poker game has a house edge of .5% and you have a strategy card so you can use the best strategy, and Pai Gow Poker has a house edge of 1.46%. At first, it looks like the video poker game is the best choice.

The difference is that you play 300 hands per hour on the video poker machine and the Pai Gow Poker games only deals 30 hands per hour.

Here are the expected loss numbers for each game:

Video poker – $5 per hand X .5% X 300 hands per hour = $7.50 expected loss.
Pai Gow Poker – $10 per hand X 30 hands per hour X 1.46% = $4.38 expected loss.

I used a $5 bet for video poker because it’s representative of a $1 coin value at five coins per hand. I used a $10 bet for Pai Gow Poker because most tables have a minimum bet amount, and you can find $10 tables.

Notice that even though you bet twice as much per hand playing Pai Gow Poker, and it has a higher house edge, you still have a lower expected hourly loss than playing video poker.

Of course, you can see how the different variables change your expected loss. You could play fewer hands per hour playing video poker, or you could play a quarter machine, making the bet per hand $1.25 instead of $5. Or you could play Pai Gow Poker online with a minimum bet of $1 or $5.

You can use this simple formula to determine how long your bankroll should last, or to determine how much bankroll you need in order to play for a set amount of time.

Here’s the simple formula:

Bet size X house edge X hands or spins per hour = Expected hourly loss

It’s important to remember that the expected loss is a long term average. This means that in any individual hour of play your actual results may vary a great deal.

You might end the hour winning more than you lose, or you might lose quite a bit more than the expected loss. Keep this in mind when you’re predicting how long your bankroll will last or how much you need.

Advantage Play Applications

So far you’ve learned what the house edge is for many casino games and have seen that every bet has a positive expected value for the casino and an expected loss for you.

I mentioned the possibility of advantage play in the opening section, so in this section I’m going to explain how it may be possible to turn the tables and make the house edge negative. When the house edge is negative, it makes the expected value positive for you and negative for the casino.

From what you’ve learned so far, it might seem impossible to be an advantage player. The truth is that becoming an advantage player is hard, but it’s not impossible.

You use the same formulas you learned above, but the key is figuring out how to make the house edge negative, or switch it to your favor.

Here are the most common advantage play games and methods. Each method requires a great deal of study and work, so if one of them interests you, you need to do a great deal of additional research and work.

Blackjack

Blackjack has a house edge that varies based on the house rules and your ability to play using the best strategy. When you play a blackjack game with good rules and use the best strategy, you can start with a house edge of less than .5%.

This is a low house edge, but if you don’t do anything more, you still have a negative expected value.

Advantage players use a few different methods while playing blackjack to change the house edge. The first one is counting cards. The basics are that when the remaining deck has a higher percentage of high cards than low cards the house edge swings into your favor. When this happens you can bet more and swing the long term edge into your favor.

Counting cards isn’t as hard as it sounds, but it’s not easy. You don’t have to memorize each card that gets played, but you do have to be able to keep a running count in your head and add or subtract from it while still playing perfect strategy.

I recommend learning more about card counting first if you want to try to become an advantage player.

The other method involving blackjack that advantage players use is called hole carding. Some dealers are sloppy when they deal and flash the value of their hole card so players at certain seats can see them if they’re watching for it.

When you can see the value of the dealer’s hole card, you can use the information to swing the house edge in your favor. Hole carding can also be a valuable at some other casino table games.

Poker

I haven’t mentioned poker yet, but just like casino games, every decision you make while playing poker has an expected value. Some decisions have a positive expected value and some have a negative expected value.

Poker is a complex game with easy rules so anyone can learn how to play. But it also has advanced strategy and can take years to master. The big advantage it offers an advantage player is that you play against other players instead of against the house.

This means that you can develop your skills to the point where you can make more positive expectation plays than your opponents and make a long term profit.

Sports Betting

Sports betting also hasn’t been mentioned, but it also offers the opportunity to win in the long run. When you make a sports bet you pay a little extra than you can win. This is called vig, and it’s how sports books lock in a long term profit.

If you develop your skills to the point where you can find lines offered by the books that have value, you can win more than you lose. This takes a great deal of work, but some players are long term winners.

Video Poker

A few video poker machines offer pay tables that have a house edge of less than .5%. A few machines have even been made that offer a negative house edge if you use perfect strategy, but these machines are rare.

Some players use comps and other things like free play and coupons, to play a break even game against the casinos. Video poker is the last advantage play method I suggest considering, because it doesn’t offer as many opportunities as the other methods on this list.

Other advantage play methods are possible, but the best way to get started is with one or more of the games listed above. Once you start seeing positive results you can start looking for additional methods to create positive expected value.