Betting is a number’s game.
Some people may bet successfully with their gut or be privy to insider information in certain sporting circles, but at the top of the betting game, it’s just math nerds!
The American Bill Benter who made $1 billion in the Hong Kong horse racing markets studied physics.
The soccer betting legend Tony Bloom studied Mathematics at Manchester University before working at Victor Chandler (Now BetVictor) and ultimately setting up his billion-dollar betting syndicate StarLizard.
Edward Thorp in 1961 pioneered the card-counting blackjack strategy. He was a mathematics professor by profession but soon made gambling his primary income.
It’s no surprise that in order to start profiting from betting and gambling you need to utilize several mathematical betting strategies.
However, not all of the popular ‘Mathematical’ strategies work, for example, the Martingale System is utter trash and anyone who sings its praise should not be taken seriously (I’ll explain why later!).
However, many mathematical betting strategies can very quickly turn you into a profitable sports bettor and we’re going to take a look through some of the best online sports betting strategies below.
Mathematical Betting Strategies that work!
Starting with the easier mathematical betting strategies which only require an understanding of the basics of betting and getting into the more complex mathematical betting strategies as we go along.
Finally, we’ll look at the strategies to avoid at the end of the article.
Arbitrage Betting
The easiest and quickest mathematical betting strategies to go from a losing bettor to a profitable bettor is through the arbitrage betting strategy.
This strategy uses some simple mathematics to compare sportsbook odds against each other to calculate whether betting on both sides of a sports game can guarantee a profit. No matter the outcome of a game or betting market.
To understand Arbitrage Betting we first need to take a look at how sportsbooks make money.
Sportsbooks make money by offering customers lower odds than what the true odds should be given the probability of an outcome winning. A simple example is a coin toss.
Assuming the coin is fair, the chance of the coin landing heads is 50% and the chance the coin lands tails is 50%. The true odds for both outcomes are +100
Bet on Heads: $100 @ +100 = $100 payout
Bet on Tails: £100 @ +100 = $100 payout
By betting on both heads and tails we’re guaranteed to break even. No matter which side the coin lands one bet will always win and cover the losing outcome.
However, sportsbooks will offer lower odds than the true odds of +100. Let’s say they offer us -110 for both heads and tails.
Bet on Heads: $100 @ -110 = $91 payout
Bet on Tails: £100 @ -110 = $91 payout
Now no matter which side the coin lands we’re guaranteed to lose $9 as our winning bets can only win $91. However, our losing bet will lose $100 each time. As a result, we make a guaranteed loss.
This is how sportsbooks earn a guaranteed profit. When two customers bet on either side of the sportsgame they make a profit from the set margin or house edge they have added. This is referred to as the sportsbooks ‘Vig’.
We can calculate the vig by converting the moneyline odds to implied probabilities using the formulas below:
Implied Probability = Negative Moneyline Odds / (Negative Moneyline odds + 100) * 100
Implied Probability = 100 / (Positive Moneyline Odds + 100) * 100
When we take both potential outcomes of a betting market and convert the odds on offer to an implied probability. They should add up to 100% as there is a 100% chance that one of these outcomes will win. The Vig is the percentage above 100% is the Vig. This is the percentage amount we will lose by betting on both sides.
With arbitrage betting, we use this same concept to guarantee a profit for ourselves. We can do this by betting on both sides of market at two different sportsbooks when the implied probabilities total below 100%. This is considered a Negative Vig
Sportsbook odds vary from sportsbook to sportsbook even on the same game. This can happen for several reasons such as the sportsbook having taken too much money on one outcome or the sportsbook calculating a different true probability than other sportsbooks.
For example, Colorado Avalanche are playing Seattle Kraken. Fanduel are offering odds of +105 on Colorado Avalanche
We can find these arbitrage bets by using software called ‘arbitrage bet finders’. Arbitrage bet finders scan the odds available at every sportsbook to find betting lines that have a negative vig (Meaning we can profit!). My favorite tool for finding arbitrage bets is Oddsjam’s Arbitrage Tool.
Oddsjam find hundreds of profitable betting opportunities per days.
It has a built-in arbitrage calculator which works out exactly how much money we should place on each team to make the same profit regardless of the outcome. So you just log in check the tool and then place your bets.
Matched Betting
Matched Betting is often referred to as bonus arbitrage or free bet conversion.
Similar to arbitrage betting, matched betting involves betting on all possible outcomes of a betting market. However, Instead of targeting overpriced odds and betting lines. With matched betting, we make a profit from bonus bets and sportsbook promotions.
For example, Let’s say Draftkings are offering a promotional bet $100 on Monday’s NFL games and get a $100 bonus bet for Tuesday’s NFL games.
We can use a low hold betting tool to find a match with a low vig between Draftkings and another sportsbook. We then bet $100 with DraftKings on an NFL game and bet at another sportsbook on the opposing team. Ensuring we break even from our initial $100 bet.
We didn’t make any money from those bets, but we’ve now qualified for a $100 bonus bet. We can then bet the bonus bets and place a matched bet to hedge out the profit.
Check out my Matched Betting Guide where I break down the process in a step-by-step video tutorial.
Value Betting (Positive Ev Betting)
Value Betting, also known as Positive EV Betting involves placing bets at higher odds than the true odds for a betting outcome.
Simply put, we bet when we believe a sportsbook has overpriced its odds and made a mistake.
As long as the odds we’re betting at are higher than the true odds we will make a profit in the long run due to the principle of positive expected value.
Take the example of a coin toss. If the coin is fair we have a 50% chance to win and a 50% chance to lose. As a result, the true odds of the coin flip should be +100.
Below I have simulated the results of 5000 coin tosses where we bet $100 at odds of +100
As you can see from the graph we have the potential to go on long winning or losing streaks but collective all these results average around to breakeven at $0 of profit.
Meaning betting at +100 when the true win probability is 50% results in a zero EV bet.
However, what if we get offered $120 if we win the coin toss for our $100 bet? How does this effect the simulation?
The graph now shows a clear positive profitable trend between all simulations and collectively the average profit is $50’000.
As we are getting $120 for each win but only losing $100 for each loss. The odds we’re being offered are +120. We expect to win 50% and Lose 50%.
We can calculate the Expected value of the bet using the following equation:
Expected Value = (Winning Profit x True Win Probability) – (Bet Stake x Loss Probability)
The expected value of the coin toss is then $10 for each $100 bet.
Expected Value = ($120 x 0.5) – ($100 x 0.5)
Expected Value = ($60) – ($50)
Expected Value = $10
Now in the real betting world we won’t be betting on coin flips. The real science of value betting is in using mathematical models to infer the true odds or ‘No Vig Odds’ of sports betting markets to make more profitable positive EV decisions.
We can use the idea of ‘Wisdom of Crowds’ to infer the true probability of a selection winning. The “Wisdom of Crowds” is a concept that suggests that the collective opinion or decision of a group of people is often more accurate and insightful than that of any individual within the group.
When applied to betting we can take the opinions (odds) offered by all sportsbooks (adjusting for each sportsbooks vig) and infer the true odds for a selection. There are several simple and advanced statistical models and methods to do this such as:
- The Multiplicative Method
- The Vig is spread proportionately between outcomes according to their implied probability.
- The Additive Method
- The Vig is distributed evenly between all outcomes regardless of their implied probability.
- The Power Method
- This is a combination of both Multiplicative and Additive methods where we raise the probabilities to a constant power.
- The Shin Method
- The Shin method is an iterative method that corrects for long-shot bias and assumes an unknown level of informed/insider traders (z). There is a python implementation of shins method available along with an R implementation
Explanations for all of these methods in the following paper. If you would like to incorporate them into your own mathematical betting strategies
Apart from simply removing the vig from each sportsbook we also need to consider which sportsbook is setting the odds. Not all sportsbook odds are equal. We can divide sportsbooks into two categories:
- Sharp Books: These are sportsbooks that allow consistently winning players
- Soft Books: These are sportsbooks that will ban and limit consistently winning players.
As a result we should apply more weight to odds originating from ‘sharp books’ as these odds tend to update quicker and account for more information than ‘soft books’. This isdue to taking bets from more informed bettors and factoring these into updated odds.
To find Positive Ev Bets at scale and speed you can either build your own custom software to aggregate sportsbook odds and devig them allowing you to determine value bets. Alternatively, you can subscribe to Positive Ev Betting software tools such as Oddsjam or Rebel Betting who will do this for you 24/7.
Market Making
Market making is a betting strategy that we can employ on betting exchanges, such as Betfair, Sporttrade or Prophet Exchange.
A betting exchange is like a stock market for odds where all users can add or take money from the market. Unlike a sportsbook where we only have the option of taking the odds offered by the sportsbook. On a betting exchange, we can enter our own odds into the market for other users of the betting exchange to bet against.
This allows us to essentially become the sportsbook. We can offer money into the market on both sides of a betting market and add our own ‘vig’ this means that when a user bets on one outcome to win and another user bets on the opposing outcome we make a guaranteed profit.
For example, let’s say we have an NFL game of Baltimore Ravens vs New York Giants. Both team are evenly matched. We can place a $100 betting offer into the betting exchange on Baltimore Ravens for +105 and also place a $100 betting offer on the New York Giants for +105. If both of these bets get matched to other bettors of the exchange we make a $5 regardless of who wins (similar to an arbitrage bet).
One of the great aspects of the market-making strategy is that we don’t have to try and ‘beat the market’. We’re simply providing money into the betting exchange markets to other users and in return we get a small profit for providing this service.
As a betting exchange is a peer-to-peer trading platform it allows winning players. So we can do this over and over again! Day in day out!
Some people even automate this strategy and bet via API. Below is a video of my in-play soccer market maker running on Betfair.
Bayesian Inference
I’m going to give you a little sports betting hack!… are you ready?
If you want to beat the best betting syndicates you have to learn from the best. The likes of Starlizard, SmartOdds, Longshot Systems, Mustard Systems. Go on webarchive.org and review their previous quantitative analyst/developer job listing.
One concept appears again and again and again… Bayesian Inference
So what exactly is Bayesian Inference?
Bayes Theorem is a mathematical model for updating prior assumptions as we receive new raw data.
P(A|B) = P(A) P(B|A)P(B)
For Example, We’re flipping a coin under the assumption that it’s a fair 50-50 coin? How should we adjust our assumptions about the coin if we keep flipping heads too often. At what point do we have to throw out the assumption that the coin is fair? Well, Bayes Theorem provides us a way of quantifying the probability that the coin isn’t fair after each flip.
How can we think about this in terms of sports and betting markets?
Let’s say we have a soccer match / football match between Manchester United and Manchester City. It’s the first match of the English Premier League and both teams have made big changes to their squads.
We take the industry odds at kickoff and convert these into an expected goals value for both teams (You can do this with the follow R package https://opisthokonta.net/?p=1760). Expected Goals is a representation of the average goals each team would score after n-games against each other
We now have an expected goals value for each team. This will be our prior assumption for each team. As both teams hit the pitch we can use Bayes theorem to adjust the expected goals per game for each team as they attack and take shots on goal. Each time a team take a shot on goal the shot is assigned an Expected Goal (xG) value (Either by ourselve or we can use a third party figure such as StatsBomb(https://statsbomb.com/)) whether they score or not (xG represents the average number of goals that shot would score in n-attempts).
We can then adjust our total expected goals for each team as the match elapses. Using Poisson distribution we can then convert these expected goals back into probabilities of each team winning.
This is just a brief overview of the basic concept of Bayesian inference. You can apply the same concept to any sports whether it’s CS:GO, Darts, Ice-Hockey. We can also go in-depth and assign an xG to every action a player takes on a pitch.
The Bayesian Inference is an advanced betting strategy. The key take away is that quantifying new information in live games provides an opportunity to find profitable bets.
Sports Betting Systems to Avoid
There are many betting systems and mathematical betting strategies in sports betting that claim to help you win. However, I’ve noticed a pattern that these systems are usually just staking systems or risk management systems.
Which don’t generate any positive expected value but rather adjust the variance so that you win small amounts often and lose bets less frequently but when you do lose you lose your whole bankroll.
I call this picking up pennies in front of a steamroller
Progressive Betting Systems
Martingale Betting System
The Martingale strategy revolves around the concept of doubling your stakes after each loss, with the intention of recovering previous losses and making a profit equal to the original stake. The underlying assumption is that, eventually, a win will occur, and the accumulated losses will be recouped.
On the surface, the Martingale system seems like a logical approach, especially to novice gamblers. The prospect of recovering losses with a single winning bet appears enticing, creating an illusion of a foolproof strategy.
The main drawback of the Martingale betting strategy is it’s reliant on an infinite bankroll. In reality, no gambler possesses an unlimited source of funds, and hitting a losing streak. As you need to double your money after each loss this creates an exponentially increasing bet size that compounds both your losses but also the cash you need to recover to break even incredibly quickly.
Fibonacci Betting System
The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. In the context of gambling, the Fibonacci Betting System utilizes this sequence to determine the size of bets. After each loss, the player is supposed to move one step forward in the sequence and bet the corresponding amount. Following a win, the player regresses two steps in the sequence.
The fibonacci system has the same flaw as the Martingale sytem:
- Neither system creates an edge or advantage over the market
- both systems rely on an infinite bankroll.
- These systems compound your losses exponentially.
People get intrigued by the system as it sounds mathematical and logical. However these systems pluck out a mathematical sequence from a high school maths textbook. In reality, it’s just a structured way of chasing your losses.
Paroli system
The Paroli system is another progressive betting strategy which has a few betting rules to follow:
- Select a bet stake or percentage of your betting bankroll
- Double the bet stake after a win
- Return to the initial stake after a loss
- Stop doubling stakes after 3 wins and return to the original stake.
The Paroli strategy aims to maximize win streaks by increasing stakes after consecutive wins. Whilst betting flat stakes during consecutive losses and losing streaks to minimize losses. However, the system does alter the underlying advantage the casino or sportsbook has against the bettor.
Again you may end more betting sessions with a win but overall across thousands of betting sessions the system still results in a loss. This is because the system itself doesn’t allow us to bet when we have an advantage over the sportsbook or casino. No matter what staking or bet size system we use if the underlying bets have a long term negative value we will lose.
However, It is worth mentioning that we should consider the amount we are betting as part of good bankroll management. A stake management system will never create an edge for us, but it can maximizer the edges and profitable bets that we do find. The hands down best system for this is the kelly criterion
Kelly Criterion (Stake Management that works!)m
The Kelly Criterion is a bet stake management system used to grow your bankroll at the optimal rate. Kelly Criterion calculates the optimal amount we should bet on a Positive Ev Bet given the size of the betting odds, the true win probability and our bankroll size.
The Kelly Criterion Formula is:
Optimal Bet Amount = (((Decimal odds -1) x Win Probabilty) – Loss Probability) / (Decimal odds -1)
Kelly criterion or kelly staking is not just used in betting decisions during the mid-200’s it became the chosen staking strategy for many investment firms looking to maximize their stock market returns.
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