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Explored: The Mathematics of Gambling

Jane ShawBy Jane Shaw Senior Editor Updated: 10/20/2023
Jane Shaw Jane Shaw Senior Editor See Full Bio

We count on Jane to inform our readers about the latest slot games in the US market. With her passion for video games and a degree in engineering, she’s our gambling tech expert. Jane’s also active in our blog section, where she tackles the curiosities and changes in the industry.

Players at Blackjack Table The Mathematics of Gambling

Gambling is a game of math, at its core – and understanding the underlying mathematics and probabilities is key to maximizing your chances of winning and reducing the likelihood of you losing! Today’s blog breaks down the essential mathematical concepts behind popular gambling games, odds, and probabilities – so let’s get into the mathematics of gambling.

Gambling Mathematics: The Basics

Casino mathematics can seem complex at first – but, as we just noted, gambling is really nothing more than a game of numbers. While games of chance rely on luck, there is an underlying mathematical structure that determines probabilities and odds. By understanding the math involved, you can make informed decisions to better your chances as a player.

This blog will explore core concepts like odds, house advantage, and expected value to shed light on the mathematical principles governing casino games. We’ll look at probability calculations, analyze why the house always wins overall, and we’ll even show you some of the best (and worst) casino games to play, from a math perspective.

Whether you stumbled upon this page while searching for US casinos with online roulette or just want to learn more about betting mathematics – let’s get right into it!

The Mathematics of Games and Gambling: An Introduction

Gambling is fueled by math. Behind every bet, dice roll, and deal of the cards lies a complex backbone of probabilities, calculations, and statistics. While casino games rely heavily on chance, the mathematics of games and gambling determine the likelihoods and outcomes.

Understanding the numbers allows players to make better bets. Probabilities reveal the percentage chance of certain events occurring. These probabilities can be calculated by examining the possible outcomes. Of course, every casino game uses a slightly different calculation or has different probabilities, but the core concepts generally remain the same.

When you gamble, you’ll probably find yourself coming across something called the house edge. The house edge represents the mathematical advantage casinos build into games by paying less than the true probabilities would dictate. This ensures profitability over time as the house edge plays out.

Players can use math to look for games with lower house edges and better odds for the player. Blackjack, for example, offers opportunities for skillful play like card counting to shift the odds in your favor. Craps bets like Pass/Don’t Pass have low house edges as well. However, legal blackjack casino sites don’t always offer the best blackjack table rules, so always check in advance!

While short-term results rely on luck, mathematical principles govern the long-term outcomes in gambling – and that’s the sole reason why casinos are able to stay in business; they know they’ll make a profit in the long run.

Odds and Probabilities

Odds are fundamental to gambling mathematics as they represent the probability or likelihood of an event happening. In games of chance like roulette or craps, odds show the mathematical advantage the house has over players. Understanding odds allows you to calculate your expected return on a bet – although this only really applies in the long run, as short-term volatility means swings can go either way!

At their core, odds compare the number of desired outcomes to possible outcomes. For example, the odds of rolling a 6 on a standard die with 6 sides are 1 in 6. This is because there is 1 desired outcome (rolling a 6) and 6 total possible outcomes (rolling 1, 2, 3, 4, 5 or 6). The odds can be expressed as fractions, decimals, or ratios.

The amount a bet pays out is based on its odds. Bets that are less likely to win, statistically, generally pay out more to make up for the lower probability. For example, hitting a single number in roulette pays 35:1 whereas correctly guessing red or black pays 1:1 (even money).

Understanding odds lets you calculate your expected return – how much you expect to win or lose on average. Expected return is simply the probability of winning multiplied by the payout, minus the cost of the bet. At roulette, the expected return betting $1 on a single number is (1/38 x $35) – $1 = -$0.05.

This shows the bet has a negative expectation – you’ll lose 5 cents on average per $1 wagered. Knowing the math behind odds is key to making smart bets – and it’s also pretty interesting to learn more about once you’ve got the basics covered!

Why the House Always Wins

The house always wins – this age-old gambling phrase rings true because casinos don’t leave winning to chance. Through precise mathematical calculations and careful game design, the odds are always stacked in favor of the house.

The mathematics of games and gambling ensures casinos have a built-in mathematical edge that ensures they’ll turn a profit over time – usually between 2% to 15%. The mathematics of gambling all comes down to the house edge, actually – and there are several factors at play that guarantee the house edge stays consistent.

Firstly, payouts on winning bets are usually less than the true odds. This difference between the true and paid odds is where the house gets its edge. For example, the true odds of rolling a 6 on a die are 1 in 6. However, in a fictional casino game where you just bet on the outcome of a dice – in this example, a casino may only pay out 5:1 on this rather than 6:1.

Games themselves also have mechanics that can benefit the casino more than players. Take blackjack, for example. In blackjack, if you go “bust” – draw cards totaling more than 21 – you lose immediately, even if the dealer also ends up busting his or her hand. In another casino table game, craps, a 12 is a tie, while several other numbers favor the house only.

Of course, individual players can come out ahead short-term – that’s why it’s possible for you to have winning sessions – but in the long run, the casino will always win. Even on games where the house edge is small, casinos don’t care; they rely on volume. With enough bets, the law of large numbers kicks in and the expected probabilities become more likely – and this is really where the casino starts to make their money.

Best and Worst Casino Games

Regardless of whether you play at land-based casinos in Vegas or the top US online casinos you should be aware that when it comes to casino games, some offer significantly better odds and lower house edges than others. Knowing which games to play – and which to avoid – is vital if you want to minimize the house edge and boost your chances of winning.

One of the best casino games to play is blackjack – and, when played perfectly, the house edge can be reduced to as low as 0.5%! Craps is another notable contender, with bets like Pass/Don’t Pass boasting a low house edge of 1.4%. Baccarat also offers pretty favorable odds on certain tables.

However, while there are some good casino games – there are also some awful ones! Keno is undoubtedly the worst, and, on average, it has a staggering house edge of 25%! European roulette wheels have a house edge of 2.7% – not bad – but play an American roulette wheel and this jumps to 5.26%!

Ultimately, blackjack and craps give savvy players the best odds overall. Table game variants with side bets may offer higher win potential – but with this win potential comes a greater house edge, too! Below, we’ll show you a quick table with a few of the best and worst casino games to play from a house edge point of view:

Best House Edge Worst House Edge
Blackjack (0.5%) Keno (25%)
Craps (1.4%) American roulette (5.26%)
Baccarat (1.06%) Slots (up to 15%)

Frequently Asked Questions

Still have questions about the mathematical and statistical side of gambling? This FAQ section aims to tie up any loose ends by addressing additional topics around gambling mathematics, odds calculations, probability percentages, payout setting, impacts of betting systems, and ways players can tilt the odds to improve their chances. Math plays a crucial role in predicting gambling outcomes, so read on if you want to build your numerical knowledge to better understand the games.

What are the core mathematical concepts behind gambling?

The general mathematical concepts in gambling are odds, probability, house edge, and expected value. Odds determine the likelihood of particular outcomes occurring. Probability calculates the percentage chances of outcomes. The house edge represents the mathematical advantage built into games through payouts below true odds. Expected value helps you determine your average expected return on a bet based on the odds and payouts.

How does math give casinos an advantage?

Casinos gain an important mathematical advantage through the house edge that's built into casino games. This can be achieved in several ways, although most commonly, your winning bets aren't paid at the true odds. Some casino games - baccarat is a good example - take a commission on certain bets, too. This is a very small amount, but it adds up over time - and that's what casinos rely on, don't forget; volume and long-term numbers!

How do odds relate to the payouts on bets?

Odds directly correlate to the likelihood of a particular outcome occurring. Lower odds equate to less chance of that outcome happening, while higher odds mean there's a greater chance of something happening. Generally speaking, when you see lower odds, the potential payouts are bigger - although this isn't always the case.

Why do casinos always win overall long-term?

The phrase "the house always wins" stems from the fact that casinos - and the games they offer - always have a mathematical edge over you, the player. This means that while they may lose in the short-term, in the long run, when probability takes its due course, they can be almost guaranteed a profit.

What casino games offer the best odds for players?

Blackjack, craps, baccarat, and video poker offer some of the lowest house edges, giving savvy players the best odds. On the flip side, games like keno, American roulette, and slots can have very high house advantages. Choosing games strategically based on house edge percentages, and using optimal strategy, is key to maximizing your chances of winning as a player - although, again, you shouldn't think about it too much in the short term.
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