Whether you’re betting on your favourite sports team or spinning the reels of the latest video slot, mathematics principles are always at work behind the scenes. Bookies use statistical probabilities to their advantage when calculating the odds for different bets, while advanced algorithms power the randomised outcomes of virtual casino games like blackjack and roulette. But why does this matter to you, the bettor?
The outcome of a wager is never guaranteed — that’s what makes gambling exciting, but means it also comes with risks. By understanding the relationship between mathematics and gambling, you’ll be able to place more informed bets, manage your risk, and get the most value out of your gambling.
There are many concepts to get your head around when it comes to gambling mathematics — odds and probability, RTP and house edge, and expected value are just a few. We’ll explain everything you need to know in this expert guide, and run you through some potential betting strategies based on mathematics too. Let’s dive in!
Mathematics in Sports Betting: Odds and Probability
Understanding odds is essential if you’re going to bet on sports. These numbers represent the likelihood of a particular outcome — as well as how much you stand to win if you back that outcome and it comes to fruition.
Bookies usually represent betting odds in one of three ways:
- Fractional Odds
This is the most common format at UK bookies. It looks like a fraction, such as 5/1 or 2/3. The top number (numerator) represents how much you could win with a bet equal to the bottom number (denominator). So, a £1 bet on Spain to win Euro 2024 at 4/1 would return £5 in total — £4 in profit plus your original £1 stake.
- Decimal Odds
Another popular format is decimal odds, such as 2.0 or 3.6, where the number represents the total payout. In this case, simply multiply your stake by the decimal odds. Take a £5 bet on Elena Rybakina to win the women’s Wimbledon singles at 3.5 — this bet would return 3.5 x £5 = £17.50 (£12.50 profit plus your original £5 stake).
- American Odds
Also known as moneyline odds, American odds are popular for betting on the NFL, NBA, NHL, and MLB and are represented with positive or negative numbers like +200 or -150. The positive odds are what you’d win from a £100 stake, while negative odds show what you’d need to stake to win £100. Moneylines are typically used in two-sided events — the negative indicates the favourite and the positive indicates the underdog.
Implied Probability
Implied probability is closely linked to the odds given by bookmakers. It’s a percentage that reflects how likely a particular outcome is according to the sportsbook odds. You can convert betting odds into implied probability with a few simple formulas.
For fractional odds, divide the denominator by the sum of the denominator and numerator and multiply by 100%, e.g. for 4/1 odds of Spain to win Euro 2024 outright:
- 1 / (4 + 1) x 100% = 20%
According to the betting odds, the implied probability Spain will lift the trophy is 20%.
For decimal odds, simply divide 1 by the odds and multiply by 100%, e.g. for 3.5 odds for Rybakina to win the Wimbledon 2024 ladies’ singles:
- 1 / 3.5 x 100% = 28.57%
This means the bookies rate Rybakina’s chance of winning at around 28–29%.
Keep in mind the implied probability is different to the true probability of an outcome. Take a West Ham vs Aston Villa game in the Premier League for example:
- West Ham to win at 8/5 = 38.5% implied probability.
- Draw at 14/5 = 26.3% implied probability.
- Aston Villa to win at 17/10 = 37% implied probability.
Add up these percentages and you get 101.8%. This is because bookmakers adjust their odds in order to take a commission or ‘vig’ on each bet in order to make money — more on that below — and that’s what causes the extra 1.8% in the example.
How Do Betting Sites Calculate Odds?
A pricing team behind the scenes calculates the hundreds of betting markets you see at online bookies, and many things go into how odds are calculated.
Bookmakers collect vast amounts of data, including team performance, player stats, injury reports, weather conditions, home advantage, and any other relevant factors. These are plugged into sophisticated models and algorithms that can identify patterns and trends.
Historical results are important too, with pricing teams examining previous matchups, win/loss records, plus current form and momentum as recent performance can be an indicator of future outcomes. Of course, it’s not just computers doing the work here — there’s a strong element of expert judgement involved too.
Bookmaker Vigorish
Finally, once the bookie has deduced what it believes to be a relatively accurate opening price, the house edge is added. Also known as vigorish (vig), juice, or overround, this is effectively the bookmaker’s profit margin that guarantees they make money regardless of the outcome. You can think of it as a commission or fee for accepting the bet. Then, as bettors wager on different teams, betting sites will adjust their odds to attract bets on all outcomes, minimising their risk by ‘balancing the books’.
Identifying where a bookmaker is offering a price that’s better than the true probability of an outcome is one way to find value if you want to make money betting over the longer term. Say a bookmaker is offering even money (50% implied probability) on a particular team winning, but their chances are actually more like 60%. When the true odds are greater than the implied odds, the wager has a ‘positive expected value’, indicating it’s profitable over the long run.
The Mathematics of Online Casino Games
Just like sports betting, table games in online casinos rely heavily on mathematics to determine odds, probabilities, and payouts. Although there are similarities to sports betting, the underlying principles do differ somewhat due to the nature of the games and role of the house.
Roulette Mathematics
Roulette is a great example, offering various betting options with different probabilities and payouts. We’ll use European roulette with a single zero pocket as an example. Here are a few of the available bets:
- Single number – 1/37 chance of winning, pays out 35/1.
- Red/black, odd/even, high/low – 18/37 chance of winning, pays out 1/1.
- 1st, 2nd, or 3rd dozen – 12/37 chance of winning, pays out 2/1.
What you may notice is the implied probability according to the payout doesn’t reflect the true probability. In roulette, payouts are based on a wheel with 36 outcomes — but once you include the green zero pocket, there are actually 37 possible numbers for the ball to land on. Just like in sports betting, there’s a built-in house edge that ensures the casino makes money. The key difference to sports betting is odds and payouts in roulette are fixed, so it’s ultimately impossible to make a profit over the long term. In other words, the expected value is always negative.
In the case of European roulette, the house edge is 2.7%. What does this mean in practice? Say you bet £1,000 on roulette over a session. Statistically speaking, you can expect to lose £27 on average. Of course, this number won’t bear out over shorter sessions.
Blackjack Mathematics
Blackjack introduces an interesting dynamic to gambling mathematics as there’s an element of skill involved, unlike roulette which is purely chance-based.
Savvy blackjack players employ what’s called basic strategy, a set of rules that dictate the most statistically favourable action to take based on your hand. The specific rules of the variant you’re playing also affect probabilities in blackjack. Typically, the house edge sits at around 0.5% to 1% when played with perfect strategy, so the game offers better value than many others in the long run.
A major mathematical mechanic underlying blackjack is the probability of drawing specific cards. In single-deck blackjack for example, you have a 7.69% (4/52) chance of drawing an Ace, assuming none have been drawn already. If one has been drawn, the chance drops to 5.88% (3/53).
Some players use a strategy called card counting to keep track of what’s been drawn, gaining an advantage over the house. However, many blackjack games use more than one deck.
How does that affect the maths in an eight-deck game? You have the same 7.69% (32/416) chance of drawing an Ace to begin with, but it only drops to 7.45% (31/415) after one has been drawn. As drawing any one specific card has less impact on what’s going to come up next, this mitigates the advantage of card counting.
How Do Online Casinos Decide Game Outcomes?
Under the hood of your favourite casino games, there’s something called a Random Number Generator (RNG) at work.
RNGs use complex algorithms to produce sequences of numbers that lack any discernible patterns, ensuring that every spin of the roulette wheel or deal of the blackjack cards is fair — just as it would be in a physical casino.
Slot Mechanics and Mathematics
Online slots introduce yet more mathematical principles into the mix. Due to the varying numbers of symbols, pay lines, and reels across titles, things can get incredibly complex. That said, there are a few simpler concepts that are useful to understand.
Slot RTP and House Edge
RTP stands for Return to Player. It’s effectively the inverse of house edge, and represents how much a slot pays out as winnings in the long run compared to what’s staked on it.
For example, a slot with 96% RTP pays out £96 on average for every £100 bet on it. The house edge can be calculated as 100% minus RTP. In this case, that’s a house edge of 4%.
Keep in mind these numbers are usually theoretical and calculated over millions of spins, so they won’t be noticeable during your session. That said, it’s wise to play slots with higher RTPs as these offer better value longer term due to the lower house edge.
Slot Volatility Explained
Volatility, also known as variance, describes the probabilities and sizes of payouts on a given slot:
- High variance slots pay out less frequently but in larger amounts.
- Low variance slots pay out more frequently but in smaller amounts.
Here’s how the outcomes of seven spins might look at high and low volatilities.
| Variance | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| High | lose £1 | lose £1 | lose £1 | lose £1 | lose £1 | lose £1 | win £6 |
| Low | lose £1 | win £1 | lose £1 | lose £1 | win £2 | win £1 | lose £1 |
As you can see, the high volatility slot had a six-spin loss streak followed by one big £6 win. The low volatility slot had four losses and three smaller wins. In both examples, the player broke even.
Ultimately, it’s up to you to decide what you prefer in terms of risk vs reward. Smaller budgets are better suited to low volatilities as you’re more likely to hit some smaller wins before you use up your bankroll. However, some players enjoy the added excitement of a bigger potential payout, albeit with a greater risk of dry spells.
How Do Paylines Affect Slot Odds?
Paylines are an integral slot mechanic, and are the lines on which you need a combination of symbols to land in order to win:
- More paylines increases the chances of winning per spin, but usually requires a higher stake.
- Fewer paylines decreases the probability of a win, but you won’t have to bet as much.
Again, there’s a balance to be struck here according to your preferred playstyle. One final consideration is in progressive slots — you’ll sometimes have to bet the maximum paylines to be in with a chance of winning the jackpot.
Mathematical Strategy in Online Casino Gambling
Wondering how you can actively use the power of mathematics in your own gambling? There are a few ways that are primarily applicable to casino games.
For starters, you could learn to count cards. This is a strategy used by blackjack players to determine whether they hold an advantage over the house by keeping track of how many high- and low-value cards have been dealt. It’s not much use these days, as many blackjack games use multiple decks to mitigate the strategy. Furthermore, online blackjack games typically shuffle after every hand, rendering count counting ineffective.
You can still bring maths in, however. A number of betting strategies have been devised over the years, most of which can be applied to even money bets such as red/black, odd/even, or high/low in roulette. These strategies use mathematical systems to adjust stake sizes based on the outcome of each round.
It’s important to clarify one thing – you’re never guaranteed a win in gambling, no matter what. No strategy will ensure you come out on top, nor are the probabilities of winning impacted by previous events. Believing otherwise is a concept known as gambler’s fallacy.
That said, the following betting strategies can provide a structured basis for stake amounts, helping you manage your bankroll or simply increasing your enjoyment of gambling. Let’s take a deeper look.
Martingale Betting
One of the most popular mathematical betting strategies is the Martingale system. The premise is simple — start with a baseline stake. If you win, keep betting this amount. If you lose, you double your bet.
Here’s an example with a base stake of £1 on an even-money bet in roulette:
- £1 bet – loses, down £1.
- £2 bet – loses, down £3.
- £4 bet – loses, down £7.
- £8 bet – wins, up £1.
The idea with Martingale betting is that you’ll always recoup your total losses on your next win by doubling your stake after each losing bet. This might sound foolproof, but unfortunately it’s not — a loss streak can quickly burn through your bankroll or hit table betting limits as your stake grows exponentially. If you want to try Martingale betting for yourself, be sure to start out small.
D’Alembert Betting
Another strategy you can use on even-money bets is the D’Alembert system. It’s similar to the Martingale, but a little more conservative when increasing the stake. You start with a base betting unit. If you lose, you add a single unit to the stake. If you win, you subtract a unit.
Say you’re playing blackjack and you start with a £1 base stake:
- £1 bet – loses, down £1. Add one unit.
- £2 bet – loses, down £3. Add one unit.
- £3 bet – wins, back to £0 total. Subtract one unit.
- £2 bet – loses, down £2. Add one unit.
- £3 bet – wins, up £1. Subtract one unit.
As the stakes don’t increase as rapidly with D’Alembert compared to Martingale, it’s suitable for smaller budgets or lower table limits. Of course, it’s still not foolproof and longer loss streaks can bankrupt the player.
Fibonacci Betting
If you really want to get into the numbers, you can use a betting strategy based on the Fibonacci sequence in which each number is the sum of the previous two:
- 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
Pick a base stake unit — say £1 again — and use the sequence as how many units to bet. If your bet loses, you move along one number in the sequence. If it wins, you move two numbers back. Here’s an example on an even-money bet:
- £1 bet – loses, down £1.
- £1 bet – loses, down £2.
- £2 bet – loses, down £4.
- £3 bet – loses, down £7.
- £5 bet – loses, down £12.
- £8 bet – wins, down £4. Move back two numbers in the sequence.
- £3 bet – wins, down £1. Move back two numbers.
- …and so on.