The Mathematics of Gambling: How Odds Are Calculated

Understanding how the odds work, and their relation to the real probability of winning, is crucial for any player. Specifically, players who want to optimise their bankroll and play games or bets with the lowest house edge. It may just be a matter of a few additional cents on top of your wins. But that little change can make all the difference between turning a profit or going into the red.

Casino games are all designed to give the house a small edge, just enough to ensure a profit in the long run. We know that, and accept it. Our main goal is to find out exactly how the odds work to benefit the casino, and how we can reduce that edge to get better value on our gaming bankroll. With basic probability, preplanning, and insights into casino games, you will learn all you need to play smarter.

Studying Odds in Casino Games

For the most part, the odds on classic casino games are pretty much fixed. Such as when you place any bet on a roulette wheel, pick the winning hand in baccarat, or play blackjack. Sure, there may be variants that have slightly different rules, with varying payouts. In blackjack, for example, hitting a blackjack may pay out at 6:5, or in a better scenario, 3:2. Or in baccarat, where you have banker bets that pay 1:1 minus a commission that ranges from 4% up to 6%. However, some baccarat has zero commission. Casinos take out the commission in those games, but introduce other rules to ensure that they still have the edge.

Then, there are games such as video poker, where you can find the same game at two online casinos, but with two different paytables. The slight discrepancies in the paytables can make a tremendous difference. We are talking about increasing a 0.5% house edge up to 5%. Slots are the most diverse when it comes to paytables, and due to the nature of the games, it is practically impossible to calculate the actual probabilities of winning. There is a great difference between slots and the other casino games. It all revolves around the real probability and the probability implied by the odds.

Probability and Odds

Probability is the chance of something happening. Mathematicians define this number as a fraction, but for practical purposes, we can also specify it as a percentage. Such as, you have a 50% chance (1/2) of flipping heads in heads or tails. Or guessing the correct suit in a drawn card has a 25% chance (1/4) of succeeding.

To calculate probability, divide the number of winning outcomes by the total number of possible outcomes. In most cases, there is just 1 winning outcome, but there are exceptions, which we will look into below. The formula, in most cases, is the following:

[ 1 / Total Outcomes ] x 100 = Probability [%]

For example, a dice roll can bring 6 possible outcomes.

1 / 6 x 100 = 16.6%

However, if you place a bet on the dice to land on exactly 1 or 2, then 2 outcomes can win.

2 / 6 x 100 = 33.3%

The chances of landing a straight-up number in European or French roulette are 1 in 37. When drawing cards in blackjack, the approximate odds of landing a 10 valued card is 4/13. These are quite easy to calculate. When we introduce more rules and ways of winning, the probabilities become harder to calculate. You can check out any of our guides to find out more about your real chances of winning at each of the following games.

• Blackjack
• Baccarat
• Video Poker
• Roulette
• Craps

There is a clear distinction to be made between these probabilities, and the probability defined by the odds. The odds are not going to represent your actual chances of winning, and therein lies the house edge.

Odds and Implied Chances of Winning

The odds in any casino game will not pay you out exactly the amount that the probability would imply. It is easiest to explain in a simple example. You are drawing cards and guessing whether they will be red or black. The probability is 1/2 so the odds should be 2 – as you have a 50-50 chance of winning. However, the casino will not provide odds of 2 on either red or black. Instead, let’s say they give you odds of 1.9.

This means, if you play 20 rounds and win 10 – you will be at a loss. Wagering $10 per round, you will spend $200 in total and only receive $190 back. Therefore, you will need to win more often than the real probability suggests to make a profit. We can see this clearly in games such as roulette. Betting on red or black pays out at 1:1 (2x), but it is not a 50-50 bet. There are 18 red, 18 black, and 1 green segment – the 0 on French or European roulette. Therefore, your real odds of winning are 18/37, or 48.64%. The remaining percent belongs to the house, and that is how it gains an edge.

But this does not just apply to roulette. The game rules and payout odds of each and every casino game are designed to favour the house. It does not mean you can’t make a profit through your gaming. It just means you need a little luck, and know when to quit while you are ahead. But in some casino games, we don’t have the real life information to calculate the odds. We can calculate the probabilities in card-based games, the physical limitations of dice, and roulette wheels, but crash games and slots fall into a category of their own.

RTP vs Probability

In slots, we cannot break open the machines and calculate the various segments. Or, make formulas to define the centrifugal force, friction, gravity and other effects on the spinning reels. Instead, we rely on a value called RTP. Return to Player is a percentage of how much the slot can theoretically payout. This is a number that all licensed online casinos must provide for their games. However, they are not the ones who calculate RTP. The value is defined by independent game auditors, who make countless simulations to figure out how likely the slot machine is to pay out. It is not fault-proof, and there are no guarantees that you will get the same returns as the RTP would suggest.

In these cases, we cannot really count the probability. Slots are extremely complex games, and there is great variance between what you can win. Forming a partial payout may reward you with a value under your original stake. But you can form complete, or nearly complete paylines and go into a profit. Then, there are all kinds of aspects such as bonus rounds, free spins, cascading reels, and expanding symbols that add more variance. Therefore, we don’t have the real information that says how likely you are to form a specific payline.

Instead, we rely on the odds, and by factoring in the RTP, we can calculate a rough approximation. Slots use powerful algorithms to ensure that each outcome is fully randomised. So even the testing and RTP are not 100% accurate.

What You Can Do to Overcome the Odds

There are no ways to completely avoid house edge, though in games where you have an element of control, you can reduce it greatly. A basic blackjack strategy can take the house edge of a game from around 4% down to just 0.5%. If you use a mathematically optimised video poker strategy, you can similarly achieve a house edge of under 1%. Bringing the house edge down to a minimum is the first step.

But ultimately, the best way to prepare is through preplanning and incentivising your gameplay. Make a bankroll for your gaming, and allocate enough money to sustain longer gaming sessions. If you just have $20 to play with, don’t start with a stake of $1. If you are playing slots, where you can be playing 20 rounds per minute (without autoplay). You may want to pick a game with a minimum limit of a few cents to stretch out your bankroll. Playing with 5 cents per game, that $20 should last you at least 20 minutes (400 rounds).

Casino Game Mastery

We aren’t saying you won’t win at all, but remain open to any eventuality. The mastery comes in when you decide to quit. As the odds are stacked against you, the chances are that the longer you play, the more likely you will lose your money. All kinds of anomalies can occur during your gameplay. Such as winning multiple rounds in a row, or landing a rare combination that doubles up your stake.

If you have a large enough bankroll, the idea is that you keep playing and hope that some good luck comes your way. If it ever does, you need to quit while you are up. Should you keep losing and never get into the green, you should have a loss limit too, to avoid pumping in all your money.

You won’t always make a profit on your casino gaming, so you should also set loss limits to avoid going completely bust. Sometimes, it is better to take a break and come back later when you feel ready to test your luck again. Playing too long can tire you out. The games can play tricks on you, and lead into all kinds of cognitive biases, such as you are close to the next win, or you can predict the next outcome based on what happened in previous rounds. Avoid these gambler’s fallacies, and keep your gaming sessions under control. That way, you will give yourself the best possible chance to cash in on a string of good luck, and make a profit on your gaming.