Learning Advantage Gambling Math Can Improve Your Casino Skills

Neon Sign Displaying Poker, Sic Bo, Roulette, Blackjack, Math Equation
The main difference between most gamblers, who lose on a consistent basis, and the small group who’re able to beat the casinos is a lack of knowledge. Some gamblers don’t have any idea that they can win in the long run, and some simply don’t know how to win, though they might understand that it can be done.

If you need a recap on how advantage gamblers differ from casual gamblers, take a look at my previous blog for more details.

One thing is certain, in order to become a winning gambler, which I call an “advantage gambler,” you need to understand advantage gambling math.

It’s not too hard, and even if you don’t care for math, you can learn what you need to know if you’re willing to spend a little time working on it.

I’m not trying to convince you that being an advantage gambler is easy. The truth is that it’s hard work. But the truth is also that it can be done, and if you have any interest in learning how to be a long-term winner, it starts with advantage gambling math.

Expected Value

Expected value is the basic math that is the foundation of all advantage gambling. Until you understand what it is and how it works, you’re going to have a hard time beating the casinos. The good news is that you’re going to learn everything you need to know about expected value in this section.

The first time I read about expected value I didn’t understand it. It was in a book by David Sklansky, and it sounded like something important, but it was a bit difficult to wrap my head around. Eventually, I started to understand it and went back over the basics many times.

Expected Value, Explained

Every decision you make as a gambler has an average value over the long run. If you make the same decision or play hundreds or thousands of times, there’s an average value. In the short term, usually just a single decision or play, the value is either just a win or a loss.

But if you average the value over many times, you can determine if a play is profitable or unprofitable on average. Advantage gamblers look for profitable expected value and do everything they can to make these bets. This is called positive expectation.

If you make positive expectation wagers more often than negative expectation ones, you win more than you lose. This is why expected value is the most important mathematical concept for advantage gamblers.

Colorful Question Marks, Two Red Dice, Win Lose Dice

Here’s an example of determining expected value:

If you study casino games, you’re aware that almost all of the games and bets offered in the casino have a house edge. With a casino house edge, how are you supposed to find positive expected value?

The answer is that most gamblers never learn how to find and make positive expected value bets. But smart players learn how to find and exploit them. I cover some of the best places to find positive expectation in the next section, but before you move on, consider what happens when you make positive expected value wagers.

Making Money With an Edge

Let’s say you’ve found a blackjack game with good rules and have mastered card counting. Instead of playing a game with a house edge, you’re playing a game with a small edge in your favor. Overall, you’re playing with an edge of .5% over the casino and are betting $2,000 per hour.

.005 X $2,000 = $10

This doesn’t seem like much, but compare winning $10 an hour to losing hour after hour. All of a sudden, instead of gambling for fun and exchanging money for entertainment, you’re being entertained and making money while you do it.

Some games don’t offer a chance to play with positive expected value. Advantage gamblers avoid these games. Here’s a list of games that generally don’t offer advantage play opportunities:

The reason I used the word “generally” above is because there are a few specific instances where you can play some of these games with a positive expectation. Occasionally, a progressive slot machine will have the jackpot climb so high that it effectively eliminates the house edge. You can also play a few video poker games, combined with perfect strategy and comps, which eliminate the house edge.

You can even find special situations, involving comps, promotions, coupons, or tournaments, where some of the other games on the list offer a small positive expected value. But for the most part, these games should be avoided. Once you master some of the advantage play methods for the gambling activities in the next section, you can start looking for positive expected value elsewhere. Just don’t start with the games listed above.

Potentially Profitable Gambling Activities

The most common forms of advantage gambling include poker, sports betting, blackjack, horse racing, dog racing, daily fantasy sports, and esports betting. Most gamblers still lose while participating in these gambling activities, but they all offer the opportunity for advantage gamblers to play with long-term positive expectation.

Poker is my favorite advantage play game because, other than a small rake collected by the casino or poker room, you’re competing directly against the other players. By studying the game and improving your skills, you can overcome the rake and make money in the long run.

Advantage poker players understand how to use particular starting hand requirements, pot odds, position, and pot size control to take advantage of weak opponents.

Smart poker players put themselves in more positive expectation situations than their opponents, locking in a long-term profit.

Some gamblers that play real money blackjack have learned how to find games with favorable rules, use the best possible strategy, and have mastered a card-counting system that gives them a long-term edge against the casino.

Yankee Stadium, New York Yankee Baseball Game

Sports betting is somewhat like poker, in that the sportsbook charges a fee on losing wagers called the “vig” to make money. This creates a situation where smart sports bettors can handicap games well enough to overcome the vig, creating a positive long-term expectation.

Horse racing and dog racing works somewhat like poker and sports betting, because all of the money collected is put in a pool and the track keeps a percentage of the pool. The rest of the money in the pool is paid out to gamblers who place bets on the winning animals. This gives the best handicappers a chance to win more than they lose in the long run.

Daily fantasy sports and esports betting both work on similar models to poker and sports betting. This gives the best gamblers an opportunity to win more than they lose.

The step by step process of beating each of the gambling activities on this list can fill entire books, so it’s outside of the scope of this article. The important thing is to know that you can become a winning gambler, and the place to start is understanding what you need to do.

Understanding Advantage Gambling Math

Let’s say you’re playing Texas hold’em and have four cards to a flush after the turn. The pot has $500 in it and your opponent bets $100. From your experience and what you know about your opponent, you know that if you hit your flush on the river, you’re going to win the hand. And if you don’t hit your flush, you’re going to lose the hand.

The question you have to answer in this situation is if calling the $100 bet has a positive expectation. It turns out that this is fairly easy to do once you know how.

After the $100 bet is added to the pot, the total is $600. You have to risk $100 for the chance to win the $600. You know the value of the two cards in your hand and the four cards on the board, so this leaves 46 unseen cards that can land on the river. Nine of these cards complete your flush and 37 of them don’t.

Because there are 46 possible river cards, if you run the numbers for each possible card and average the results, you can learn if it’s profitable to call.

Stacks of Colored Casino Chips on Table

By making the $100 call 46 times, your total investment is $4,600. When you lose, you receive nothing back. The nine times that you win, you receive the $600 in the pot back as well as your $100 call. This is a total of $700 and when you multiply this with the nine times you win, the total is $6,300.

In this situation, you win more than you lose, so the expectation is positive. To get the average expected value, you divide the difference between $6,300 and $4,600 by 46. This gives you a positive expected value of $36.96 every time you make the call.

You’re still going to lose 37 out of every 46 times you’re in this situation, but the nine times you win makes up for the losses. In addition, when you hit your flush sometimes, you might be able to win another bet on the river. Because you know you lose when you don’t hit your flush, you always fold on the river when you miss, so you don’t lose more money.

Advantage poker players learn about dozens of different situations like this and put themselves in these situations as often as possible. This locks in long-term profits for smart players.


Now that you understand how expected value is the key to advantage gambling math and how it works, you can quickly build your skills to start winning on a consistent basis. It’s not going to be easy, but with the skills you build, you can be an advantage gambler for the rest of your life.