Every branch of knowledge has its basic principles. In the gym you need to know how to select the load when training, when baking a cake you need to know the recipe. When gambling you should know some basic maths. In this article we shall be describing the concept of expected value and explain how it allows you to make profitable bets.
Supposing a friend were to offer you a coin flip. If it lands heads up you will win €1 but if lands tails up he will win €1. You could get lucky and win €1, but is this game worthwhile? You can determine that by calculating the expected value of your potential winnings.
Expected value is defined as the sum of all possible outcomes of an event weighted by the probability of their occurrence. In the case of a coin flip, both outcomes are equally likely, so you have a 50% chance of winning €1 and a 50% chance of losing €1. Therefore the expected value of this game is zero:
+€1 × 50% - €1 × 50% = 0
This agrees with the intuition that for every two games played on average you would win one and lose one, neither losing nor earning any money. Expected value equal to zero shows that the bet offered gives you no competitive advantage. Therefore, from a gambler’s point of view, it’s a waste of time.
This would not be the case if you were to win €2 for heads but pay only €1 for tails. The expected value of such a game is +€0.5:
+€2 × 50% - €1 × 50% = €0.5
The game would also be worthwhile if you know the coin is badly balanced and it lands heads up 51% of the time and tails up 49% of the time. Then, even making a symmetrical bet of €1, the game has a positive expected value:
+€1 × 51% - €1 × 49% = €0.02
Every gambler should seek games with a positive expected value. Positive expected value is the force that makes you a net winner. Even small advantages of the 51% to 49% magnitude in time pay enough to build cities like Las Vegas and Monte Carlo, or sponsor the biggest sports teams around the world. The concept of expected value applies to all gambling activities and almost always answers the question of whether a bet is worth making.
In the case of sports betting, you can usually bet on one of three outcomes – a win for team one (1), a draw (x), or a win for team two (2). Let’s take an example set of bookmaker odds:
| 1 | x | 2 |
|---|---|---|
| 1.44 | 5.19 | 6.38 |
It is clear that team one is the favourite as you are offered only €1.44 for every €1 you bet on their win, so this bet would be the most likely to pay. On the other hand a bet on a draw, or on team two winning, involve far more attractive odds. To assess which of the three betting opportunities is the most profitable, you need to consider what the chance of each of the three outcomes is.
If, in your opinion, the chance of team one winning is 60%, then for every €1 bet placed on their win you will win €1.44 with a 60% probability, and win nothing with a 40% probability. The expected value of this bet is therefore €0.86:
+€1.44 × 60% + €0 × 40% = €0.86
or net of the €1 cost of placing the bet it is -€0.14.
If the chances of a draw and team two winning are both 20%, the expected values of placing bets on these outcomes are respectively €1.04 and €1.28:
+€5.19 × 20% + €0 × 80% = €1.04
+€6.38 × 20% + €0 × 80% = €1.28
or €0.04 and €0.28 net.
As you can see, betting money on team one might have the highest probability of a payout, but it is a sucker’s bet – for each €1 placed on this team you would lose on average -€0.14. The most profitable choice is to bet money on team two winning which, if you were to trust the bookmaker’s odds, is the least likely outcome. There’s an 80% chance you will lose your money, but the odds are great enough to justify the risk.
Positive expected value can be thought of as a drift that over time brings the sum of your wins and losses into net profit territory. It is the key criteria for selecting any bet. The rationalbets.com tool is designed to carry out such an analysis and help you choose only the profitable bets within seconds. You will find it under the Bet tool tab in the top navigation bar.
It should be noted that the expected values calculated above are only as accurate as the assumption that chances of a team one win, a draw and a team two win are 60%, 20% and 20% respectively. However to make money betting it is not necessary that you judge the probabilities of event outcomes better than the bookmaker – it is enough that you judge them better than the average bettor, as we explain in the next article on how bookmakers set odds.
Bookmakers seem to be experts in everything. They quote every sport, every league, outcomes of political events and even who will win the Eurovision Song Contest. It is not only the extent of their knowledge that impresses, but also its precision. How do they know that in a match between Crystal Palace and Arsenal the correct odds for Crystal Palace are 4.13 and not 4.10 or 4.15? As we explain in this article, they know it from you and from other players.
Let’s imagine an alternative reality for 1986. A prime Mike Tyson is looking for his next opponent. A few months earlier he had beaten Marvis Frazier and now holds a flawless record of no losses and 25 wins – 23 of these by knock-out of which 15 are by knock-out in the first round. Marvis lasted less than 30 seconds before Mike unleashed a brutal combination, in which four of the punches came after Marvis started falling unconscious onto the canvas. There’s no doubt that the 100 kg Iron Mike is a brutal beast that will tear apart any opponent in the ring.
Let’s further imagine that the only person willing to fight him is an aspiring featherweight boxer – a 55 kg teenager who has trouble keeping his guard up longer than the first round. The pay is good and the contract is signed. Bookies start setting the odds.
In order to determine ‘fair’ odds for an event a bookmaker has to determine the probability of each potential outcome. For the sake of calculation, let’s assume that Mike’s chance of winning is 98%, while the chances of a draw and his opponent winning are each 1% – it cannot be ruled out that Mike will show his opponent mercy, or that he will fall to a miraculous right uppercut for ten long seconds.
The odds of each outcome are then determined by dividing 1 by the probability of the outcome. The ‘fair’ odds for Mike winning are therefore 1.02:
1 / 98% = 1.02
and the ‘fair’ odds for a draw or his opponent winning are 100:
1 / 1 % = 100
This way the expected net win for any bet placed is zero. For example if you bet €1 on Mike winning, you will win €1.02 with a 98% probability and win nothing with a 2% probability. The expected value of this bet is therefore €1:
+€1.02 × 98% + €0 × 1% = €1
or net of the €1 cost of placing the bet it is zero.
The bookmaker then charges a margin on these ‘fair’ odds and comes up with quotes like:
| Mike | Draw | Mike’s opponent |
|---|---|---|
| 1.01 | 30 | 30 |
In reality the bookmaker carries out such an analysis only when setting the initial odds. As his sole aim is to earn his margin regardless of the outcome of the fight, he dynamically corrects the odds based on how much money bettors place on the specific outcome of a particular event, not concerning himself with the topic of estimating real-world probabilities of these outcomes.
If the majority of people, carried away by a romantic vision of a fight between David and Goliath, start betting on Mike losing, the bookmaker will adjust his odds in order to have a hedged position. These new odds could quickly cease to reflect the initially calculated odds and therefore – the probabilities of event outcomes that justify them.
Let’s assume that the sum of bets placed on each of the fight outcomes is shown in the table below:
| Mike | Draw | Mike’s opponent |
|---|---|---|
| €10,000 | €20,000 | €70,000 |
The bookmaker has a budget of €100,000, of which he keeps €5,000 as his margin and the remaining €95,000 is prize money. If Mike wins, this €95,000 will go to the people who have placed a total of €10,000 in bets. So the bookmaker secures his margin by setting the odds of Mike winning at 9.5:
€95,000 / €10,000 = 9.5
Using this formula, the bookmaker sets a new set of odds that secure his margin:
| Mike | Draw | Mike’s opponent |
|---|---|---|
| 9.5 | 4.75 | 1.36 |
For example, in the event of a draw, for each of the €20,000 of bets placed on this outcome, the bookmaker will pay €4.75, giving a total payout of €95,000:
€20,000 × 4.75 = €95,000
and leaving him with the €5,000 fee.
Although real life odds are a little more complicated, as they change dynamically in line with the inflow of new bets, the idea stays the same – bets placed by the losing players are supposed to cover the bookmaker’s margin and a payout to the winning players. In extreme situations, where everyone is betting on only one outcome and a bookmaker is unable to secure his position, he sometimes reduces the odds for this outcome to the unsatisfying level of 1.01 or stops accepting bets on the given outcome altogether.
Even though bookmaker’s odds are quoted this way, they usually are not far off real-world probabilities. This is due to the fact that the bias of individual bettors averages out, giving a reasonable end result. Just like in the experiment in which people were asked to estimate the weight of an ox – individual estimates varied widely, but the average was only 0.1% away from the actual weight.
The key takeaway is that the bookmaker’s odds are not a product of his expectations but a product of the expectations of all the fans who have decided to bet money on the event outcome. Therefore a winning bettor does not have to be smarter than the bookmaker, just smarter than the average fan or more precisely, taking the bookmaker’s margin into account, at least 5% smarter than the average fan.
If this is the case for you, you will be able to identify bargain odds that don’t reflect the actual probability distribution of the event outcomes and gamble with a positive expected value. In our example for every €1 bet on Mike winning, you'd earn an expected €9.31:
€1 × 9.5 × 98% = €9.31
Helping you find these profitable odds is the main purpose of the rationalbets.com tool. With its use you can specify a probability distribution for any event and, using the automatically calculated expected values of your bets, compose a winning coupon.
With double chance bets you can bet on two of the three possible outcomes of an event. They are very popular among bettors as they increase the probability of a payout. Unfortunately they also decrease the odds.
To understand how they work, let’s look at an example set of odds where you can bet on a team one win (1), a draw (x) or a team two win (2):
| 1 | x | 2 |
|---|---|---|
| 1.84 | 3.73 | 3.87 |
With double chance bets you can simultaneously bet on a team one win or a draw (1x), a team two win or a draw (x2) and even a team one or team two win (12).
This reduces your risk dramatically. For example by betting on a team one win or a draw, if the first team is a strong favourite, you have a very big chance of a payout. This will of course be reflected in low odds. These odds are often quoted by bookmakers directly, but they can also be calculated synthetically as shown below.
Let’s assume that you want to bet €10 on a team one win or a draw based on the odds presented in the table above. Since you want to win money in the event of both outcomes, you have to place a bet on each of them. In other words, you have to bet €x on a team one win and the remaining €10 - €x on a draw. If team one wins, you will win
1.84 × €x
and in the event of a draw you will win
3.73 × (€10 - €x)
The aim is to find such an amount €x, so the payout is the same in both cases. This leads to the equation:
1.84 × €x = 3.73 × (€10 - €x)
which with the help of a little algebra gives:
€x = €6.70
Indeed, if you bet this amount on a team one win your potential payout is €12.32:
1.84 × €6.70 = €12.32
and if we bet the remaining amount on a draw your potential payout is (nearly) the same:
3.73 × (€10 - €6.70) = €12.31
Your bet was €10, so the double chance odds work out to be 1.23:
€12.32 / €10 = 1.23
These odds are considerably lower than both the odds for a team one win and the odds for a draw, which agrees with the general intuition that the lower the risk, the lower the potential profits.
In the general case, the formula for calculating double chance odds is:
double chance odds = (first odds × second odds) / (first odds + second odds)
The rationalbets.com tool allows you to analyse the profitability of placing double bets by calculating them synthetically, or by adding a custom event you can analyse double chance odds quoted by your bookmaker directly.
With a multiple bet, also called an accumulator or a parlay, you can bet on the outcome of several events simultaneously. The odds of such a bet are the product of odds for the individual bets, but all bets must be correct. This can lead to very high odds but also decreases the probability of a payout.
Let’s look at two basketball games which the bookmaker quotes as follows:
Match one
| Warriors | Lakers |
|---|---|
| 1.39 | 3.01 |
Match two
| Celtics | Bucks |
|---|---|
| 1.55 | 2.45 |
Placing a multiple bet on a Lakers win and a Bucks win you get multiple odds of 7.37:
3.01 × 2.45 = 7.37
significantly higher than the odds available for any single bet. However, the probability of a payout is considerably lower – to get paid, you have to bet correctly on the results of both matches.
You could of course place a multiple bet for more than just two events, which might seem tempting, as the odds increase exponentially. For example, betting €1 on two events in a multiple bet, each at odds of 2, will lead to a potential €4 payout. The potential payout for a multiple bet of five such events is already €32, and for 10 – €1024.
As each odds are subject to a bookmaker’s margin of usually 5% to 10%, when the odds multiply, this margin also multiplies. And just like the odds, it increases exponentially. The more events you bet in a multiple bet, the greater the margin you will be playing on. For this reason most bettors believe multiple bets are unprofitable. However, they do not take into account that the bookmaker’s odds are a product of bets placed by other bettors and they might not represent the real-world probabilities of the events’ outcomes.
If the real-world probability of Lakers winning the first game is 40% and the real-world probability of Bucks winning the second game is 50%, these probabilities would imply fair margin-free odds as presented below:
Match one – hypothetical fair odds
| Warriors | Lakers |
|---|---|
| 1.67 | 2.05 |
Match two – hypothetical fair odds
| Celtics | Bucks |
|---|---|
| 2 | 2 |
In such an arrangement, despite the bookmaker charging a margin, the available odds are greater than the fair margin-free odds. So with respect to the factual probability of potential outcomes, the bookmaker is charging a negative margin due to the bias of his customers. This negative margin will compound in your favour.
Specifically, if you are correct about the actual probability distribution of the event outcomes, the expected value of winning when betting €10 on a multiple bet is €14.74:
€10 × 7.37 × 40% × 50% = €14.74
In comparison, the expected value of a €10 bet on Lakers is €12.04:
€10 × 3.01 × 40% = €12.04
and a €10 bet on Bucks is €12.25:
€10 × 2.45 × 50% = €12.25
This observation should, however, be treated with caution. Even if you are able to estimate the real-world probabilities correctly 80% of the time, placing five separate bets will yield a positive expected payout for four of them and a negative for only one. If you place a multiple bet on the same events, this one misjudged event can reverse the profitability of your entire strategy. Also these are high odds low probability bets that might draw you away from achieving your expected wins, as we explain in the article on variance.
The rationalbets.com tool calculates the expected value of a multiple bet composed of all the events you are planning to bet on. The result is presented beneath your coupon. You can exclude individual events from this bet without dropping them out of the basket by pressing the monkey icon on the event fiche.
As explained in the first article, expected value is the main profitability criterion when placing a bet. As we mentioned therein, it is the drift that brings the sum of our wins and losses into net profit territory. Variance answers the question – How bumpy will the road be?
Variance can be defined as the expected square of the deviation from the expected value. In other words it is a measure of how far on average the returns achieved are from the expected ones. Without going into details you can assume that bets with lower variance are the ones with lower odds and higher probability of a payout, while bets with higher variance are the ones with higher odds and lower probability of a payout.
Let’s analyse both a low and high variance bet with parameters as presented in the tables below.
Low variance bet
| Bet | Odds | Probability of a payout |
|---|---|---|
| €1 | 2 | 55% |
High variance bet
| Bet | Odds | Probability of a payout |
|---|---|---|
| €1 | 11 | 10% |
As you can easily assess, the expected values of both bets are equal at €0.1:
€1 × 2 × 55% - €1 = €0.1
€1 × 11 × 10% - €1 = €0.1
therefore both these bets give you the same expected profits. But these expected profits are just the upward drift from which you will more or less deviate.
If you look at the sample earnings of a portfolio composed of placing a thousand consecutive low variance bets and a thousand consecutive high variance bets, as presented in the graphs below, you will notice that although in both cases the earnings drift towards the expected value indicated by the black dashed line, bets with a small variance oscillate around it fairly tightly, while bets with a large variance are much more scattered, with large profits but also large losses along the way.
Chance shows no mercy – you can go broke placing bets with a positive expected value just like you can make money placing the most stupid bets. The good news is you can reduce the risk of this deviation from the expected value of your bets by choosing bets with a lower variance, i.e. bets with lower odds but a higher probability of a payout. Unfortunately, you then also reduce the ‘risk’ of large deviations from the expected value in your favour, but these often turn out to be wishful thinking.
Arbitrage is the achievement of a certain profit by taking advantage of differences in odds quoted by different bookmakers. Arbitrage opportunities occur from time to time. They are eliminated fairly quickly and most bookmakers ban bettors who play this way. Nevertheless the presence of such opportunities confirms that the odds quoted by bookmakers do not have to be unattractive.
Let’s examine a Nadal vs. Federer tennis match. As it is difficult to say who is the favourite, the odds may vary from one bookmaker to another. Let’s assume that bookmaker X set odds as follows:
| Nadal | Federer |
|---|---|
| 1.9 | 1.8 |
while bookmaker Y is quoting:
| Nadal | Federer |
|---|---|
| 1.6 | 2.2 |
Each bookmaker has a positive margin. For example to earn a certain profit of €10 at bookmaker X, using the formula
bet amount = expected winnings / odds
you can calculate that you should bet €5.26 on a Nadal win and €5.56 on a Federer win. With these bets you are certain to win €10, no matter what the outcome is:
€5.26 × 1.9 = €10
€5.56 × 1.8 = €10
The problem is that winning a certain €10 costs you €10.82:
€5.26 + €5.56 = €10.82
This €0.82, or 8.2% of the payout, is the bookmaker’s margin, thanks to which, unlike his customers, he enjoys an almost certain profit. By analogy, you can calculate that bookmaker Y operates on a very similar margin of 8%.
It turns out that even you can’t make a certain profit with one bookmaker, placing bets with different bookmakers you sometimes can. In this example you can take advantage of the highest available odds by placing a bet on Nadal’s win with bookmaker X and a bet on Federer’s win with bookmaker Y. In this setup, to enjoy a sure win of €10 you will have to bet €5.26 on Nadal’s victory and €4.55 on Federer’s victory. The total cost of the strategy is €9.81:
€5.26 + €4.55 = €9.81
and therefore gives you a certain €0.19 profit.
If you read the article about how bookmakers set odds, you may ask yourself why arbitrage opportunities are so rare when every bookmaker has his own customers for whom he sets the odds? There are a few good reasons.
First of all the law of large numbers is at work – the average expectations of a large number of bettors at each bookmaker tend to be similar, as individual biases net out. Any deviation from this rule, as in our example, will be quickly exploited by arbitrageurs who, by betting on an arbitrage opportunity, will quickly move the odds of both bookmakers into unison, if this was not done in advance by the bookmaker himself, who observes his competitors and tries to predict such situations. Also most bookmakers don’t like arbitrageurs – people betting unusual amounts like €5.56 are likely to be banned.
Although some bettors claim to earn money this way, it seems looking for arbitrage opportunities is quite a hassle. The occasions are rare and the profits are limited. You will probably be better off placing bets with positive expected value, which you can read about in the article on expected value and easily identify using the rationalbets.com tool.
The vision of profits can be deceiving. Anyone who has placed €10 on a multiple bet with odds of 500 and wondered whether it will be better to spend the winnings on a new car or a trip around the Bahamas knows this. That is why it is worth remembering a few basic rules that all gamblers should follow. These can save you from making unwise decisions, especially when emotions take over. Fate is blind and not always fair.