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Mathematics of Gambling
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Math and gambling go hand in hand. Games of chance are built around probability; and all casino games come with a mathematical house advantage built directly into the rules.
Our guide to mathematics and gambling looks at key concepts, strategies, and misconceptions. It also explains the math of popular casino games, so keep reading to learn how it all adds up.
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Key Mathematics of Gambling Concepts
It’s common to hear terms like house edge, RTP, odds, and volatility when talking about gambling, but what do these concepts mean? Keep reading to learn the math behind some common gambling terms.
House Edge & RTP
House edge is a built-in advantage casinos (also known as the house) place on games to ensure they always turn a profit in the long run. The advantage is displayed as a percentage. For example, if a casino game has a house edge of 5%, the house expects to keep $5 for every $100 wagered on that particular game.
Return to player (RTP) describes the same concept from the player’s perspective. The RTP represents how much money wagered on a game is expected to be returned to players. So, a game with a 95% RTP is expected to return $95 of every $100 wagered back to all players.
Probability
Gambling is built on the player’s probability of winning or losing in various games of chance. Every bet you make has a likelihood to win, and some casino games have rules that tip that likelihood in the house’s favor.
For example: the ability to split Aces in blackjack can positively affect the probability of not only winning the hand, but increasing your payout as well. That’s why many casinos don’t allow players to double down on Aces.
Another example is the number of decks used in blackjack. The more decks that are used, the harder it is for players to keep track of which cards have been played; which is why most blackjack games use 6–8 decks.
Betting Odds vs. Payout Odds
Odds are obviously an important aspect of gambling mathematics. However, it’s important to understand the fundamental difference between betting odds and payout odds. Betting odds are the chances of an event happening, while payout odds are how much money the casino pays you for betting correctly.
Take flipping a coin for example. The betting odds of landing heads is 50%. So, if the casino offered a 1:1 payout, you’d win the same amount of money that you wagered; a $10 bet would win you $10 — for a total of $20. But if the casino lowered the payout odds to 2:3, you’d only win $6.66 — for a total of $16.66.
This critical difference helps ensure that the casino always comes out ahead by offering payout odds that favor themselves in the long run.
Volatility
Volatility in gambling relates to the ups and downs players experience while playing casino games. Most slot machines, for example, state the volatility rating as high, low, or medium in the game’s “about” section.
High volatility games often see players losing many rounds in a row before hitting a large win. A good example of high volatility games are progressive slots with enormous jackpots. Low volatility games, meanwhile, win more often — but with lower payouts.
Low volatility games produce smaller, more predictable wins, while high volatility games are less predictable but come with bigger payouts.
Gambling Math of Popular Casino Games
Mathematics are built into every casino game, including these popular games that are frequently found at both land-based and online casinos.
Slots
Slots use random number generators (RNGs) to determine their outcomes. They use pre-made probabilities to set how often winning symbols and symbol combinations appear. Slot software providers simulate a near-infinite number of games to set the RTP percentage too, as well as each game’s volatility rating.
Blackjack
Blackjack relies on probability, since the player’s odds of winning or losing heavily depends on which cards they’re dealt, combined with the dealer’s upcard. Players can study blackjack strategy charts to reduce the house edge by knowing when to hit, stand, double down, etc.
Casinos, however, can cancel out this slight player advantage by increasing the number of decks used in blackjack games, adjusting payout odds in their favor, and instituting rule variations. Example rule variations include forbidding players from splitting Aces, restricting when players can double down, and allowing dealers to hit on soft 17s.
Roulette
Roulette relies on probability the same way blackjack does, with casinos inserting 1–2 zero pockets on wheels to tilt the odds in their favor. Betting on even money bets like red/black, odd/even, and high/low seem to have 50/50 odds on the surface.
However, casinos insert one zero on European roulette wheels and two zeroes on American roulette wheels to reduce the 50% odds of these bets to 47.37% and 48.65%, respectively. So what seems like 50/50 bets, are more like 48/50 bets; with the discrepancy favoring the house.
This is because zeroes are neither odd nor even, nor do they qualify for high/low bets. And since these pockets are green, you can’t win if you place a red/black bet and it lands on the green zero.
Craps
Craps mathematics is built on the probable outcomes of two rolled dice, and two of the most common wagers are Pass Line and Don’t Pass bets.
Pass Line bets win if the first roll is a 7 or 11, and lose if the roll is a 2, 3, or 12. Any other number becomes the “Point” number, and players must roll the Point number before a 7 to win. Don’t Pass bets win on a 2 or 3, lose on 7 or 11, and tie on 12. Any other number becomes the Point number, and the bet wins if a 7 is rolled before the Point.
Pass Line Bets have a 1.41% house edge, and Don’t Pass bets have a slightly lower 1.36% house advantage. They’re the most popular bets at Craps tables since they have the lowest house edge, while Craps 2 and Craps 12 carry the highest house edge at 13.89%
By learning how math works in craps you can understand the volatility of which bets pay out smaller and more consistently vs. which ones pay out larger but less often.
Baccarat
Baccarat seems like a simple game on the surface, and in some ways it is. It revolves around probability. In standard Baccarat games the banker has the best chance of winning: 45.86%. The player isn’t far behind at 44.62%.
Since bankers only have an approximate 1% advantage over players, most Baccarat games charge a 5% commission on banker wins. So if you successfully bet $10 on the banker, you’ll only win $9.50. This commission on banker wins ensures more money ends up in the casino’s hands.
Betting Systems & Strategies
No betting strategy is ever 100% guaranteed, as the house always wins in the end. Many gamblers deploy math-based systems to try and gain an edge over the house, however, including using the following strategies. You can also learn more in-depth information about betting systems by visiting our roulette betting strategies page.
Card Counting
Card Counting in blackjack helps players track which cards have been used and which ones are still in the deck. When there are more high cards left the player is at an advantage since they give players a better chance to play strong hands; and when there’s more low cards left the casino has an advantage since there’s a better chance players will go bust.
In its simplest form, card counting involves assigning values to each card and keeping a running count in your head as hands are played. Deploying card counting strategies in blackjack doesn’t guarantee you’ll win, but it can help you make smarter betting decisions.
Casinos combat card counting by using 6–8 decks in most blackjack games. Single deck blackjack games involve keeping track of 52 cards, while multi-deck blackjack requires players to keep track of 312–416 cards — a much harder task.
It’s also worth mentioning that card counting is strongly discouraged at both land-based and online casinos. While you won’t be prosecuted for counting cards, being caught doing so is almost guaranteed to get you banned from the casino for life.
The Martingale Strategy
The Martingale strategy is a straightforward betting tactic that involves doubling your bet every time you lose. The idea is that by doubling your bet after each loss, you’ll cover all of your losses and then some when you finally win. After winning you return to the base bet amount and start again.
One main problem with the Martingale strategy is that it requires a large bankroll to continually cover losses. Tables at casinos also have betting limits too, so there’s a decent chance you’ll hit the betting limit before you’re able to recover your losses.
Doubling your bet every time you lose will quickly drain your bankroll with no guarantee that you’ll win it back. And even if you do win it back, the profit from doing so isn’t likely to be worth your time.
Other Betting Systems
The d’Alembert system is like the Martingale strategy, however you add the base bet amount to the next wager after each loss, rather than doubling it.
The Fibonacci betting strategy revolves around wagering using the Fibonacci sequence of numbers, which is when each number is the sum of its two preceding numbers. The basic strategy is to move up a number in the sequence when you lose, and down two numbers when you win.
While mathematically both systems have upsides, they also run into the same budget and table limit problems that the Martingale strategy does.
Misconceptions About Gambling Mathematics
Even though most players know casinos build advantages for themselves directly into games, a lot of people don’t understand the mathematics behind some of these advantages. Here are a few commonly misunderstood concepts about math and gambling.
RTP Means Guaranteed Money
While RTP does stand for Return-to-Player, these percentages don’t apply to each and every individual round. They’re calculated by simulating millions and millions of games to make sure the house always comes out ahead. In the short term your wins and losses can vary wildly.
A game with 96% RTP doesn’t mean you’ll always win at least 96% of your money back, it means that over the long term the casino should keep at least 4% of the money that all players wager.
This is why RTP doesn’t guarantee that you’ll win money, because the statistic describes how much the casino expects to earn — not how much you’re expected to win every time you gamble.
Past Outcomes Dictate Future Results
It’s common for slot players to think that certain machines are “due for a win” when they haven’t hit in a while. You might also hear roulette players say things along the lines of “the last five spins were black, so red is coming up soon.
However, the truth of the matter is that past results have absolutely no effect on future results. As such, using previous hands to dictate future betting strategies isn’t ideal and should be avoided.
Why The House Always Wins
All casino games and sports bets are created to make the casino money. From RTP in slots and blackjack rules to the zeroes in roulette, every single game of chance has a mathematical advantage built into it.
That’s not to say that winning is impossible in gambling, because players frequently score both small and large wins. Just keep in mind that the house always wins, and cashing out while you’re ahead is the only true way to leave the casino with more money than you started with.
Bankroll Management
Managing your gambling budget is extremely important no matter what kind of gambling you do. It’s critical to set a budget before you start gambling and stick to it. Once you’re gambling budget is gone, simply walk away. Many players like to set win limits as well, to ensure they walk away with potential winnings rather than losing them by gambling more.
Once your budget has been set, only use a small percentage of it on each bet — no more than 1%–5% of the total budget.
The closer you stick to the gambling budget you set for yourself, the better off you’ll be. As a matter fact, many people consider themselves to “spend” money gambling rather than “wagering” it. This can help you consider gambling sessions as spending money on entertainment, rather than an investment meant to return a profit.
Mathematics and Gambling Go Hand in Hand
Gambling is built on mathematical concepts like odds, probability, and variance. It’s important to understand terms like RTP, house edge, and volatility, so you can have a better idea of what to expect at the casino. No betting strategy is ever 100% fool proof, but understanding how gambling math works can potentially reduce the house’s advantage so you can win a bit of money.