• Bookmaking Mathematics Curriculum
• Bookmaking Mathematics Lesson
• Bookmaking Mathematics Definition
• Bookmaking Mathematics Books
• Mathematics Of Bookmaking

Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's The Doctrine of Chances (1718) treated the subject as a branch of mathematics. In his book Bernoulli introduced the idea of representing complete certainty as one and probability as a number between zero and one. Just another WordPress.com site. In gambling parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger (the 'book') and gives the English language the term bookmaker for the person laying the. Excerpt: In betting parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger (the 'book') and gives the English language the term bookmaker for the person laying the bets and thus 'making the book'. Definitive, and extensively revised and updated 3rd edition on the history, theory, practice and mathematics of bookmaking, plus the mathematics of off-course betting, bets and their computation and liability control. Further reading 'How to make a book', Phil Bull, London: Morrison &.

Most people who want to place bets on sports are fans to begin with. It isn’t unheard of for a gambler to place some sports bets, especially during big games like the Super Bowl or the NCAA basketball Final Four, but for the most part, sports bettors are sports fans looking to use their knowledge of a game or of a game’s players to earn a little extra cash. Being a fan of a particular sport, a team, a college or professional squad—these are all precursors to placing sports bet. Sports betting is also a way for a fan to get in on the action of the game, with something more than self-respect at stake.

All gambling is mathematics, even games of chance. If you understand the math behind the game, you understand the game and can give yourself an advantage. For many games, like penny slots or poorly placed roulette bets, are so bad that smart bettors earn their advantage by avoiding them altogether. In sports betting, the math is more complicated. Depending on your favorite sport, you may need to think about things like bye weeks, underdogs, quarterback ratings, and injuries with the same fervor other connoisseurs reserve for fancy winces.

So how difficult is sports betting math? The math behind placing a winning bet is fairly complicated, but the way to stay ahead of the bookmaker is rather straightforward. If you collect on 52.4% of your bets, you’ll break even. We’ll have more details on that number later, including why it takes more than 50% wins to break even, but first some general knowledge about sports gambling and the numbers behind it.

Sports Betting Basics

The easiest way to demonstrate the math behind a sports bet is to make up an example. Let’s say you and your buddy walk into a casino, each with $200 burning a hole in your pocket. There’s a big game on tonight, the Cowboys and the Redskins, so you wander into the sportsbook to check up on the latest news about the game. While you’re sitting there, you see the wagering board, with some funny numbers on it. It looks like this:

• 428 Cowboys +175
• 429 Redskins -4 -200 38

Some of this is easy enough to read. The Redskins -4 means the Redskins are favored to win and must do so by at least 5 points for a bet on the ‘Skins to pay out. The next number (-200) is the moneyline, in this case the Redskins are a 2/1 favorite. The last number (38) is the total, the over/under of the expected number of points scored in the game.

More on Placing Sports Bets

Look at that over/under number, in this case 38. If you or your buddy thinks this is going to be a particularly high or low scoring game, based on your knowledge of the team’s offenses and defenses, or information about a hurt player or bad playing conditions, you can place a wager on the total of points scored.

So how is a guy supposed to know how to literally lay down a sports bet? You need to know three things:

#1 – the type of bet you want to make
#2 – the number of the corresponding team you have chosen and
#3 – the amount you wish to wager

Knowing all that beforehand gives the ticket writer the details he needs to write the ticket without having to bend over backwards to process your bet.

Tipping and Sports Betting

We haven’t even gotten to the meat of the sports math yet, and we’re already talking about tipping the staff behind the window? Yep. Here’s why.

If you place two $100 bets, and you win, you’ll collect $440. You should consider leaving a tip around five percent of your winnings. Yes, that’s a $22 tip, but you just made a huge win, and surely you can spring for a twenty-spot for the guy who helped you win it. If you tip around the five percent mark regularly, when you win, you’re way more likely to get free drinks, which is about all you’re going to get comp-wise at the sportsbook.

So, back to the basic math of sports betting. You and your buddy, after much deliberation, decide to each place a $100 bet on your favorite team. What now?

To bet on the Redskins using the point spread, your bet is called “laying the points.” For your bet to pay off, the ‘Skins have to win by five or more to cover the spread. Remember, if the ‘Skins win by exactly four, the game is a push, and both sides recoup their bet. Another alternative is called “taking the points” with the Cowboys. That means the Cowboys have to lose by three or less for your bet to win, or if the Cowboys win outright. So you and your buddy go up to place your $100 bet, and you find out that the standard straight bet at any bookie pays 11/10. That means you have to bet $110 if you want to win $100. You and your buddy pay the bookie $110 and sit down with drinks to watch your bets come in.

These are deceptively simple bets. Deceptively because they make it look like the outcome of the football game is like the outcome of picking marbles out of a bag. Put one black marble and two white marbles in a bag, pull one out at random, and there’s your football game. After all, the odds are the same: 2/1 for white.

Bookmaking Mathematics Curriculum

But we, as sports fans, know that the mathematics of a sporting event is much more complex. Sports bettors deeply involved in their hobby will subscribe to weather bulletins from major cities that take part in their sport, making huge wagering decisions based on a few mph of wind in one direction or another. Then there’s the unknown—does a player get hurt in the first quarter? Does weather become a factor? Is a particular player “in the zone?”

How Do Bookies Make a Profit?

Just as we finish ruminating on the concept of the difficult math at play in the background of major sporting events, we’re going to turn right back towards the simpler side of sports betting. Bookies make a profit because of vigorish. What’s vigorish?

Look at the above example again. You and your buddy each paid $10 to the bookie to place your bet. That’s what the standard 11/10 odds in sports betting are all about. You bet the Cowboys and your buddy bet the Redskins, a total of $220 bet. The sportsbook has to pay back $210 to the winner, leaving a nice $10 profit no matter what happens on the football field. That $10 built-in profit is called the vigorish, and it’s the final monkey wrench in the gears of sports betting.

Obviously, sportsbooks are going to take more than two bets on any game, but this example is for simplicity’s sake. Looking at the total number of bets on different games over the course of a week and adjusting the moneyline and other numbers is another way the bookie makes a profit. Adjusting the odds a tiny percentage point in either direction will affect the balance of beats and make the book more likely to turn a profit no matter what.

Essentially, a bookie is a person who holds on to money from bettors then pays them if they win and keeps their money if they don’t. That’s what the job is boiled down to its essence.

When a bookie sets odds for games, he will build what bookies call an “over round” into his set of odds. Another slang term used for this formula is “the juice.” For the sake of simplicity, let’s look at a boxing match where both contenders are equally talented, of equal stature, etc. Since they both have an equal chance of winning, a casual bet may be even money. You put $20 on one guy; your friend puts $20 on the other. Whichever fighter wins awards the bettor with the total of $40.

Bookies don’t offer even money like friends in a casual betting situation. In the above example, with two evenly matched boxers, a smart bookie will offer 5/6 odds for each. That way, a $10 winning bet would only return $8.30 plus your stake. What does this do for the bookmaker? He can float an equal amount of money on both fighters, winning no matter which fighter actually wins. If they take $1,000 worth of bets on one boxer and $1,000 on the other, the bookie would take in $1,000 but only have to pay out $830, for a guaranteed $170 profit regardless of the outcome.

Bookies look at the weight of their books all the time and adjust odds and other factors to make sure their books balance. Though it isn’t possible to completely balance a book, bookies that go too far out on one side run the risk of losing money, and losing money in gambling is the fastest way to find yourself in another industry. All of these factors are why bookies generally root for the underdog—too many favorites winning in a sport with a short season (such as the NFL) can cause a bookmaker to lose money, while a bunch of upsets (like you generally see in college football) is a guaranteed profit for the bookmaker.

The short answer here is that bookies making money has nothing at all to do with your betting. It is almost unheard of for a single customer to be allowed to place enough bets to sink a single book all on his own. High rollers in sports betting get special privileges in terms of their maximum bet size, but these privileges often change with the bettor’s luck—maximums get raised after the bettor sees big losses and decreased (sharply) when the bettor starts to get lucky.

In short, a sportsbook’s profits aren’t necessarily impacted directly by the way an individual bet is called. Unlike casino games or slot machines, where it’s you against the house, sports bettors fuel the bookmaker’s business and only rarely is an individual bettor betting against the bookie.

Sports Betting Odds

Remember at the beginning when we talked about the magic number necessary to guarantee a break-even week in sports betting? If you read enough about sports betting, you’ll hear this number repeated often: 52.4%. If a bettor can win 52.4% of his bets, he’ll break even. Where does that number come from?

When betting the spread, you get odds of -110. Sometimes, sportsbooks will offer a -105 line as a promotion or to welcome new business. But for the most part, if you’re betting the spread, you’re getting -110.

We draw that 52.4% break even number right out of the odds. -110 is equivalent to 11/10. That means if you bet 21 games, you’d have to win eleven of them and lose ten of them to break completely even. Even at -105, you’d still have to win an astounding 51.2% of the time just to break even.

If you don’t trust the basic math behind this break-even principle, look at another real-world example. Let’s say you get really into sports betting after your Cowboys cream the Redskins and you go home with a nice fat wallet. You then bet on the next 10 Cowboys games, winning six times and losing four times.

That 60% betting record (with the odds of -110 that is traditional for against the spread bets in football) will leave you with a profit of $160. Think about it—your $600 profit from your 6 winning bets minus the $440 you lost on losing bets leaves $160. It took you $1,100 to win $160, meaning you have to bet $6.87 to win $1 on average. So you see the small differences between a 52.4% winning rate and a 60% winning rate—inside those 7.3 percentage points lies hundreds of dollars in profit.

Now imagine instead that you lost one of those six winning bets, leaving you with a 50% betting record. You spent a total of $1,100, won $500, and lost $550. That means overall your 50% record drained your wallet by $50. That’s where the vigorish will get you. Not even winning half the time is good enough to break even in sports betting.

Professional Sports Bettors

Believe it or not, some people really do bet on sports for a living. Maybe they work part time at a sportsbook or in some other marginal job in the casino industry, but there is a group of gamblers who bet on sports for their life’s work. With all the math swirling around in our heads after the last bit of the article, it’s hard to imagine anyone wanting to do this for a living.

If you know that a 52.4% record will mean you break even, the simplest way to turn sports betting into a career is to bet enough so that a 53% winning record will bring in the kind of money you want to make.

Another example. After your successful Cowboys experiment, you decide to invest $10,000 in sports gambling over the first four months of the following football season. That $10,000 is set aside to win or lose in sportsbooks.

You plan on betting on 160 games during your investment period. You dream of a 55% winning record because your win-loss with a 55% winning record would give you an 88-72 record. That’s an expected profit of +8.8 units. How did we get to that number? To calculate your units, subtract the total of your losses (multiplied by 1.1 to include the vig) from your wins and you’ll get your unit profit.

Placing $460 bets on each of these games, a number pulled from some quick and dirty math about how much you could afford to bet in a single week’s NFL play without blowing your bankroll, would result in a $4,048 profit if you maintain that 55% winning record. Turning $10,000 into $14,048 in just four months is an investment return of 40.48%. I dare you to ask your bank for that kind of return on your savings account.

But that’s all assuming you can pick the winner 55% of the time. Do your research, look into the records of professional sports gamblers. 55%, while not impossible, would place you among the elite sports bettors in the country, if not the world.

Professional sports bettors have to worry about variance more than any other type of gambler. Working against the forces of variance means managing your bankroll over the course of the season to avoid the negative possibilities that could totally empty your wagering account. Professional sports bettors have the time and resources necessary to calculate these variances, and there are even a few pieces of software out there that can help you figure out your ideal bet in the face of negative variance. But the bottom line is that professional sports bettors would dream of having a 55% winning record, simply because it guarantees you’re beating the house.

Pro bettors make their money on bets that sportsbooks offer that give them even the slightest betting advantage. The key to becoming a profitable sports bettor is being able to find advantages, opportunities where the line a book is offering is vulnerable.

This is why many long-term sports bettors are math freaks. Good sports bettors understand statistics, particularly what are called inferential statistics, though any higher math will help when it comes time to place a bet.

Here is what a professional baseball bettor might do in his head. After looking over statistics from MLB (kept religiously by all sorts of bloggers, data archives, and magazines) between the years 2000-2010, he notices a particular statistic pop out. For example: when the home team starts a left-handed pitcher the day after a loss, that team wins 59% of the time. Good sports bettors can do this sort of math in their head or very quickly on paper. From that bit of information comes a new betting theory—look for game situations that mirror the above example and bet on them. That means he’ll only bet games where the home team starts a left-handed pitcher the day after a loss. Does he just jump in and start betting based on this back of the napkin math? No way. More statistical analysis is required—he may find that this was a fluke for that particular decade and isn’t a trustworthy statistics, or he may find an even more advantageous bet based on his original theory.

Pro sports bettors also keep near-obsessive records of their bets. Obviously, no edge in sports betting lasts longer than a single game. Taking proper records will also help you test theories, like the above one about left-handed pitchers and losses. Without taking good records, no sports bettor’s bankroll will last very long.

What Is a Good Record for Sports Bettors

So, at the end of the day, what could you call a “good” record for a sports bettor? Most casual gamblers looking into sports betting see a pro advertising his 1100-900 record and shake their head a little. How could such an abysmal record be something to be proud of? That’s a 55% winning percentage, and it indicates to those in the know that this bettor is actually turning a profit placing bets on sports.

A good record for a sports bettor is any record equal to or larger than 52.4%, because that number or anything higher means you’re not losing money. A 53% winning record, while not impressive on paper, means you’re actually beating the sportsbook and putting money back in your pocket. Ask your friends that play the slots or play online poker how often they end up putting money back in their pocket.

A -110 wager, standard for spread bets in the NFL, gives the house a built-in advantage of 10%. It means that even if you do win, and you line up to collect your $100, some sucker behind you just spent $10 to hand the casino $100.

A good record for sports bettors is any record that ensures they at least break-even. If you bet 16 games this NFL season and you won 9 and lost 7, you probably made money. And taking money away from a casino is always something to be proud of.

Other Advanced Sports Betting Strategy Articles:
» Future Betting Strategy
» NFL Bye Week Betting Strategy
» Parlay Betting Strategy

Sports Betting Break Even Video:

In the video above I go over the break even % for sports betting, and we take a look at the difference between hitting 52% and 53%. I also quickly show the amounts of profits you can expect if you can hit 55% consistently.

In gambling parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger (the 'book') and gives the English language the term bookmaker for the person laying the bets and thus 'making the book'.^[1][2]

Making a 'book' (and the notion of overround)[edit]

A bookmaker strives to accept bets on the outcome of an event in the right proportions in order to make a profit regardless of which outcome prevails. See Dutch book and coherence (philosophical gambling strategy). This is achieved primarily by adjusting what are determined to be the true odds of the various outcomes of an event in a downward fashion (i.e. the bookmaker will pay out using his actual odds, an amount which is less than the true odds would have paid, thus ensuring a profit).^[3]

The odds quoted for a particular event may be fixed but are more likely to fluctuate in order to take account of the size of wagers placed by the bettors in the run-up to the actual event (e.g. a horse race). This article explains the mathematics of making a book in the (simpler) case of the former event. For the second method, see Parimutuel betting.

It is important to understand the relationship between fractional and decimal odds. Fractional odds are those written a-b (a/b or a to b) mean a winning bettor will receive their money back plus a units for every b units they bet. Multiplying both a and b by the same number gives odds equivalent to a-b. Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet. For example, a bet of £40 at 6-4 (fractional odds) will pay out £40 + £60 = £100. The equivalent decimal odds are 2.5; £40 x 2.5 = £100. We can convert fractional to decimal odds by the formula D=​^b+a⁄_b. Hence, fractional odds of a-1 (ie. b=1) can be obtained from decimal odds by a=D-1.

It is also important to understand the relationship between odds and implied probabilities:Fractional odds of a-b (with corresponding decimal odds D) represent an implied probability of ​^b⁄_a+b=​¹⁄_D, e.g. 6-4 corresponds to ​⁴⁄₆₊₄ = ​⁴⁄₁₀ = 0.4 (40%).An implied probability of x is represented by fractional odds of (1-x)/x, e.g. 0.2 is (1-0.2)/0.2 = 0.8/0.2 = 4/1 (4-1, 4 to 1) (equivalently, ​¹⁄_x - 1 to 1), and decimal odds of D=​¹⁄_x.

Example[edit]

In considering a football match (the event) that can be either a 'home win', 'draw' or 'away win' (the outcomes) then the following odds might be encountered to represent the true chance of each of the three outcomes:

Home: Evens

Draw: 2-1

Away: 5-1

These odds can be represented as implied probabilities (or percentages by multiplying by 100) as follows:

Evens (or 1-1) corresponds to an implied probability of ​¹⁄₂ (50%)

2-1 corresponds to an implied probability of ​¹⁄₃ (33​¹⁄₃%)

5-1 corresponds to an implied probability of ​¹⁄₆ (16​²⁄₃%)

By adding the percentages together a total 'book' of 100% is achieved (representing a fair book). The bookmaker, in his wish to avail himself of a profit, will invariably reduce these odds. Consider the simplest model of reducing, which uses a proportional decreasing of odds. For the above example, the following odds are in the same proportion with regard to their implied probabilities (3:2:1):

Home: 4-6

Draw: 6-4

Away: 4-1

4-6 corresponds to an implied probability of ​³⁄₅ (60%)

6-4 corresponds to an implied probability of ​²⁄₅ (40%)

4-1 corresponds to an implied probability of ​¹⁄₅ (20%)

By adding these percentages together a 'book' of 120% is achieved.

The amount by which the actual 'book' exceeds 100% is known as the 'overround',^[4][5] 'bookmaker margin' ^[3] or the 'vigorish' or 'vig':^[3] it represents the bookmaker's expected profit.^[3] Thus, in an 'ideal' situation, if the bookmaker accepts £120 in bets at his own quoted odds in the correct proportion, he will pay out only £100 (including returned stakes) no matter what the actual outcome of the football match.Examining how he potentially achieves this:

A stake of £60.00 @ 4-6 returns £100.00 (exactly) for a home win.

A stake of £40.00 @ 6-4 returns £100.00 (exactly) for a drawn match

A stake of £20.00 @ 4-1 returns £100.00 (exactly) for an away win

Total stakes received — £120.00 and a maximum payout of £100.00 irrespective of the result. This £20.00 profit represents a 16​²⁄₃ % profit on turnover (20.00/120.00).

In reality, bookmakers use models of reducing that are more complicated than the model of the 'ideal' situation.

Bookmaker margin in English football leagues[edit]

Bookmaker margin in English football leagues decreased in recent years.^[6] The study of six large bookmakers between 2005/06 season and 2017/2018 season showed that average margin in Premier League decreased from 9% to 4%, in English Football League Championship, English Football League One, and English Football League Two from 11% to 6%, and in National League from 11% to 8%.

Overround on multiple bets[edit]

When a punter (bettor) combines more than one selection in, for example, a double, treble or accumulator then the effect of the overround in the book of each selection is compounded to the detriment of the punter in terms of the financial return compared to the true odds of all of the selections winning and thus resulting in a successful bet.

To explain the concept in the most basic of situations an example consisting of a double made up of selecting the winner from each of two tennis matches will be looked at:

In Match 1 between players A and B both players are assessed to have an equal chance of winning. The situation is the same in Match 2 between players C and D. In a fair book in each of their matches, i.e. each has a book of 100%, all players would be offered at odds of Evens (1-1). However, a bookmaker would probably offer odds of 5-6 (for example) on each of the two possible outcomes in each event (each tennis match). This results in a book for each of the tennis matches of 109.09...%, calculated by 100 × (​⁶⁄₁₁ + ​⁶⁄₁₁) i.e. 9.09% overround.

There are four possible outcomes from combining the results from both matches: the winning pair of players could be AC, AD, BC or BD. As each of the outcomes for this example has been deliberately chosen to ensure that they are equally likely it can be deduced that the probability of each outcome occurring is ​¹⁄₄ or 0.25 and that the fractional odds against each one occurring is 3-1. A bet of 100 units (for simplicity) on any of the four combinations would produce a return of 100 × (3/1 + 1) = 400 units if successful, reflecting decimal odds of 4.0.

The decimal odds of a multiple bet is often calculated by multiplying the decimal odds of the individual bets, the idea being that if the events are independent then the implied probability should be the product of the implied probabilities of the individual bets. In the above case with fractional odds of 5-6, the decimal odds are ​¹¹⁄₆. So the decimal odds of the double bet is ​¹¹⁄₆×​¹¹⁄₆=1.833...×1.833...=3.3611..., or fractional odds of 2.3611-1. This represents an implied probability of 29.752% (1/3.3611) and multiplying by 4 (for each of the four equally likely combinations of outcomes) gives a total book of 119.01%. Thus the overround has slightly more than doubled by combining two single bets into a double.

In general, the combined overround on a double (O_D), expressed as a percentage, is calculated from the individual books B₁ and B₂, expressed as decimals, by O_D = B₁ × B₂ × 100 − 100.In the example we have O_D = 1.0909 × 1.0909 × 100 − 100 = 19.01%.

This massive increase in potential profit for the bookmaker (19% instead of 9% on an event; in this case the double) is the main reason why bookmakers pay bonuses for the successful selection of winners in multiple bets: compare offering a 25% bonus on the correct choice of four winners from four selections in a Yankee, for example, when the potential overround on a simple fourfold of races with individual books of 120% is over 107% (a book of 207%). This is why bookmakers offer bets such as Lucky 15, Lucky 31 and Lucky 63; offering double the odds for one winner and increasing percentage bonuses for two, three and more winners.

In general, for any accumulator bet from two to i selections, the combined percentage overround of books of B₁, B₂, ..., B_i given in terms of decimals, is calculated by B₁ × B₂ × ... × B_i × 100 − 100. E.g. the previously mentioned fourfold consisting of individual books of 120% (1.20) gives an overround of 1.20 × 1.20 × 1.20 × 1.20 × 100 − 100 = 107.36%.

Settling winning bets[edit]

In settling winning bets either decimal odds are used or one is added to the fractional odds: this is to include the stake in the return. The place part of each-way bets is calculated separately from the win part; the method is identical but the odds are reduced by whatever the place factor is for the particular event (see Accumulator below for detailed example). All bets are taken as 'win' bets unless 'each-way' is specifically stated. All show use of fractional odds: replace (fractional odds + 1) by decimal odds if decimal odds known. Non-runners are treated as winners with fractional odds of zero (decimal odds of 1). Fractions of pence in total winnings are invariably rounded down by bookmakers to the nearest penny below. Calculations below for multiple-bet wagers result in totals being shown for the separate categories (e.g. doubles, trebles etc.), and therefore overall returns may not be exactly the same as the amount received from using the computer software available to bookmakers to calculate total winnings.^[7][8]

Singles[edit]

Win single

E.g. £100 single at 9-2; total staked = £100

Returns = £100 × (9/2 + 1) = £100 × 5.5 = £550

Each-way single

E.g. £100 each-way single at 11-4 ( ​¹⁄₅ odds a place); total staked = £200

Returns (win) = £100 × (11/4 + 1) = £100 × 3.75 = £375

Returns (place) = £100 × (11/20 + 1) = £100 × 1.55 = £155

Total returns if selection wins = £530; if only placed = £155

Multiple bets[edit]

Each-Way multiple bets are usually settled using a default 'Win to Win, Place to Place' method, meaning that the bet consists of a win accumulator and a separate place accumulator (Note: a double or treble is an accumulator with 2 or 3 selections respectively). However, a more uncommon way of settling these type of bets is 'Each-Way all Each-Way' (known as 'Equally Divided', which must normally be requested as such on the betting slip) in which the returns from one selection in the accumulator are split to form an equal-stake each-way bet on the next selection and so on until all selections have been used.^[9][10] The first example below shows the two different approaches to settling these types of bets.

Double^[11][12]

E.g. £100 each-way double with winners at 2-1 ( ​¹⁄₅ odds a place) and 5-4 ( ​¹⁄₄ odds a place); total staked = £200

Note: 'Win to Win, Place to Place' will always provide a greater return if all selections win, whereas 'Each-Way all Each-Way' provides greater compensation if one selection is a loser as each of the other winners provide a greater amount of place money for subsequent selections.

Treble^[11][12]

E.g. £100 treble with winners at 3-1, 4-6 and 11-4; total staked = £100

Returns = £100 × (3/1 + 1) × (4/6 + 1) × (11/4 + 1) = £2500

Accumulator^[11][12]

E.g. £100 each-way fivefold accumulator with winners at Evens ( ​¹⁄₄ odds a place), 11-8 ( ​¹⁄₅ odds), 5-4 ( ​¹⁄₄ odds), 1-2 (all up to win) and 3-1 ( ​¹⁄₅ odds); total staked = £200

Note: 'All up to win' means there are insufficient participants in the event for place odds to be given (e.g. 4 or fewer runners in a horse race). The only 'place' therefore is first place, for which the win odds are given.

Returns (win fivefold) = £100 × (1/1 + 1) × (11/8 + 1) × (5/4 + 1) × (1/2 + 1) × (3/1 + 1) = £6412.50

Returns (place fivefold) = £100 × (1/4 + 1) × (11/40 + 1) × (5/16 + 1) × (1/2 + 1) × (3/5 + 1) = £502.03

Total returns = £6914.53

Full-cover bets[edit]

Trixie

Yankee

Trixie, Yankee, Canadian, Heinz, Super Heinz and Goliath form a family of bets known as full cover bets which have all possible multiples present. Examples of winning Trixie and Yankee bets have been shown above. The other named bets are calculated in a similar way by looking at all the possible combinations of selections in their multiples. Note: A Double may be thought of as a full cover bet with only two selections.

Should a selection in one of these bets not win, then the remaining winners are treated as being a wholly successful bet on the next 'family member' down. For example, only two winners out of three in a Trixie means the bet is settled as a double; only four winners out of five in a Canadian means it is settled as a Yankee; only five winners out of eight in a Goliath means it is settled as a Canadian. The place part of each-way bets is calculated separately using reduced place odds. Thus, an each-way Super Heinz on seven horses with three winners and a further two placed horses is settled as a win Trixie and a place Canadian. Virtually all bookmakers use computer software for ease, speed and accuracy of calculation for the settling of multiples bets.

Full cover bets with singles[edit]

Patent

Patent, Lucky 15, Lucky 31, Lucky 63 and higher Lucky bets form a family of bets known as full cover bets with singles which have all possible multiples present together with single bets on all selections. An examples of a winning Patent bet has been shown above. The other named bets are calculated in a similar way by looking at all the possible combinations of selections in their multiples and singles.

Should a selection in one of these bets not win, then the remaining winners are treated as being a wholly successful bet on the next 'family member' down. For example, only two winners out of three in a Patent means the bet is settled as a double and two singles; only three winners out of four in a Lucky 15 means it is settled as a Patent; only four winners out of six in a Lucky 63 means it is settled as a Lucky 15. The place part of each-way bets is calculated separately using reduced place odds. Thus, an each-way Lucky 63 on six horses with three winners and a further two placed horses is settled as a win Patent and a place Lucky 31.

Algebraic interpretation[edit]

Returns on any bet may be considered to be calculated as 'stake unit' × 'odds multiplier'. The overall 'odds multiplier' is a combined decimal odds value and is the result of all the individual bets that make up a full cover bet, including singles if needed. E.g. if a successful £10 Yankee returned £461.35 then the overall 'odds multiplier' (OM) is 46.135.

If a, b, c, d... represent the decimal odds, i.e. (fractional odds + 1), then an OM can be calculated algebraically by multiplying the expressions (a + 1), (b + 1), (c + 1)... etc. together in the required manner and subtracting 1. If required, (decimal odds + 1) may be replaced by (fractional odds + 2).^[13][14]

Bookmaking Mathematics Lesson

Examples[edit]

3 selections with decimal odds a, b and c.Expanding (a + 1)(b + 1)(c + 1) algebraically gives abc + ab + ac + bc + a + b + c + 1. This is equivalent to the OM for a Patent (treble: abc; doubles: ab, ac and bc; singles: a, b and c) plus 1.Therefore to calculate the returns for a winning Patent it is just a case of multiplying (a + 1), (b + 1) and (c + 1) together and subtracting 1 to get the OM for the winning bet, i.e. OM = (a + 1)(b + 1)(c + 1) − 1. Now multiply by the unit stake to get the total return on the bet.^[15][16]

E.g. The winning Patent described earlier can be more quickly and simply evaluated by the following:

Total returns = £2 × [(4/6 + 2) × (2/1 + 2) × (11/4 + 2) − 1] = £99.33

Ignoring any bonuses, a 50 pence each-way Lucky 63 (total stake £63) with 4 winners [2-1, 5-2, 7-2 (all ​¹⁄₅ odds a place) and 6-4 (​¹⁄₄ odds a place)] and a further placed horse [9-2 (​¹⁄₅ odds a place)] can be relatively easily calculated as follows:

Returns (win part) = 0.50 × [(2/1 + 2) × (5/2 + 2) × (7/2 + 2) × (6/4 + 2) − 1] = £172.75

or more simply as 0.50 × (4 × 4.5 × 5.5 × 3.5 − 1)

Returns (place part) = 0.50 × [(2/5 + 2) × (5/10 + 2) × (7/10 + 2) × (6/16 + 2) × (9/10 + 2) − 1] = £11.79

or more simply as 0.50 × (2.4 × 2.5 × 2.7 × 2.375 × 2.9 − 1)

Total returns = £184.54

For the family of full cover bets that do not include singles an adjustment to the calculation is made to leave just the doubles, trebles and accumulators. Thus, a previously described winning £10 Yankee with winners at 1-3, 5-2, 6-4 and Evens has returns calculated by:

£10 × [(1/3 + 2) × (5/2 + 2) × (6/4 + 2) × (1/1 + 2) − 1 − [(1/3 + 1) + (5/2 + 1) + (6/4 + 1) + (1/1 + 1)]] = £999.16

In effect, the bet has been calculated as a Lucky 15 minus the singles. Note that the total returns value of £999.16 is a penny higher than the previously calculated value as this quicker method only involves rounding the final answer, and not rounding at each individual step.

In algebraic terms the OM for the Yankee bet is given by:

OM = (a + 1)(b + 1)(c + 1)(d + 1) − 1 − (a + b + c + d)

In the days before software became available for use by bookmakers and those settling bets in Licensed Betting Offices (LBOs) this method was virtually de rigueur for saving time and avoiding the multiple repetitious calculations necessary in settling bets of the full cover type.

Settling other types of winning bets[edit]

Up and down

Round Robin

A Round Robin with 3 winners is calculated as a Trixie plus three Up and Down bets with 2 winners in each.

A Round Robin with 2 winners is calculated as a double plus one Up and Down bet with 2 winners plus two Up and Down bets with 1 winner in each.

A Round Robin with 1 winner is calculated as two Up and Down bets with one winner in each.

Flag and Super Flag bets may be calculated in a similar manner as above using the appropriate full cover bet (if sufficient winners) together with the required number of 2 winner- and 1 winner Up and Down bets.

Note: Expert bet settlers before the introduction of bet-settling software would have invariably used an algebraic-type method together with a simple calculator to determine the return on a bet (see below).

Algebraic interpretation[edit]

Bookmaking Mathematics Definition

If a, b, c, d... represent the decimal odds, i.e. (fractional odds + 1), then an 'odds multiplier' OM can be calculated algebraically by multiplying the expressions (a + 1), (b + 1), (c + 1)... etc. together in the required manner and adding or subtracting additional components. If required, (decimal odds + 1) may be replaced by (fractional odds + 2).^[13][14]

Examples[edit]

• OM (2 winners) = (2a − 1) + (2b − 1) = 2(a + b − 1)
• OM (1 winner) = a − 1

• OM (3 winners) = (a + 1) × (b + 1) × (c + 1) − 1 − (a + b + c) + 2 × [(a + b − 1) + (a + c − 1) + (b + c − 1)] = (a + 1)(b + 1)(c + 1) + 3(a + b + c) − 7
• OM (2 winners) = (a + 1) × (b + 1) − 1 − (a + b) + 2 × (a + b − 1) + (a − 1) + (b − 1) = (a + 1)(b + 1) + 2(a + b) − 5
  or more simply as OM = ab + 3(a + b) − 4
• OM (1 winner) = 2 × (a − 1) = 2(a − 1)

• OM (4 winners) = (a + 1) × (b + 1) × (c + 1) × (d + 1) − 1 − (a + b + c + d) + 2 × [(a + b − 1) + (a + c − 1) + (a + d − 1) + (b + c − 1) + (b + d − 1) + (c + d − 1)]
  = (a + 1)(b + 1)(c + 1)(d + 1) + 5(a + b + c + d) − 13
• OM (3 winners) = (a + 1) × (b + 1) × (c + 1) − 1 − (a + b + c) + 2 × [(a + b − 1) + (a + c − 1) + (b + c − 1)] + (a − 1) + (b − 1) + (c − 1) = (a + 1)(b + 1)(c + 1) + 4(a + b + c) − 10
• OM (2 winners) = (a + 1) × (b + 1) − 1 − (a + b) + 2 × (a + b − 1) + 2 × [(a − 1) + (b − 1)] = (a + 1)(b + 1) + 3(a + b) − 7
  or more simply as OM = ab + 4(a + b) − 6
• OM (1 winner) = 3 × (a − 1) = 3(a − 1)

See also[edit]

Notes[edit]

1. ^Sidney 1976, p.6
2. ^Sidney 2003, p.13,36
3. ^ ^abcdCortis, Dominic (2015). Expected Values and variance in bookmaker payouts: A Theoretical Approach towards setting limits on odds. Journal of Prediction Markets. 1. 9.
4. ^Sidney 1976, p.96-104
5. ^Sidney 2003, p.126-130
6. ^Marek, Patrice (September 2018). 'Bookmakers' Efficiency in English Football Leagues'. Mathematical Methods in Economis - Conference Proceedings: 330–335.
7. ^Sidney 1976, p.138-147
8. ^Sidney 2003, p.163-177
9. ^Sidney 1976, p.155-156
10. ^Sidney 2003, p.170-171
11. ^ ^abcSidney 1976, p.153-168
12. ^ ^abcSidney 2003, p.169-177
13. ^ ^abSidney 1976, p.166
14. ^ ^abSidney 2003, p.169,176
15. ^Sidney 1976, p.161
16. ^Sidney 2003, p.176

References[edit]

Bookmaking Mathematics Books

• Cortis, D. (2015). 'Expected Values and variance in bookmaker payouts: A Theoretical Approach towards setting limits on odds'. Journal of Prediction Markets. 1. 9.
• Sidney, C (1976). The Art of Legging, Maxline International.
• Sidney, C (2003). The Art of Legging: The History, Theory, and Practice of Bookmaking on the English Turf, 3rd edition, Rotex Publishing 2003, 224pp. ISBN978-1-872254-06-7. Definitive, and extensively revised and updated 3rd edition on the history, theory, practice and mathematics of bookmaking, plus the mathematics of off-course betting, bets and their computation and liability control.

Mathematics Of Bookmaking

Further reading[edit]

• 'Finding an Edge', Ron Loftus, US-SC-North Charleston: Create Space., 2011, 144pp.
• 'How to make a book', Phil Bull, London: Morrison & Gibb Ltd., 1948, 160pp.
• 'The book on bookmaking', Ferde Rombola, California: Romford Press, 1984, 147pp. ISBN978-0-935536-37-9.
• The Art of Bookmaking, Malcolm Boyle, High Stakes Publishing 2006.
• Secrets of Successful Betting, Michael Adams, Raceform, 2002.
• The Mathematics of Games and Gambling, Edward W. Packel, Mathematical Association of America, 2006.
• The Mathematics of Gambling, Edward O. Thorp, L. Stuart, 1984.
• 'Maximin Hedges', Jean-Claude Derderian, Mathematics Magazine, volume 51, number 3. (May, 1978), pages 188–192.
• 'Carnap and de Finetti on Bets and the Probability of Singular Events: The Dutch Book Argument Reconsidered' Klaus Heilig, The British Journal for the Philosophy of Science, volume 29, number 4. (December, 1978), pages 325–346.
• 'Tests of the Efficiency of Racetrack Betting Using Bookmaker Odds', Ron Bird, Michael McCrae, Management Science, volume 33, number 12 (December, 1987), pages 152–156.
• 'Why is There a Favourite-Longshot Bias in British Racetrack Betting Markets', Leighton Vaughan Williams, David Paton. The Economic Journal, volume 107, number 440 (January, 1997), pages 150–158.
• Optimal Determination of Bookmakers' Betting Odds: Theory and Tests, by John Fingleton and Patrick Waldron, Trinity Economic Paper Series, Technical Paper No. 96/9, Trinity College, University of Dublin, 1999.
• 'Odds That Don't Add Up!', Mike Fletcher, Teaching Mathematics and its Applications, 1994, volume 13, number 4, pages 145–147.
• 'Information, Prices and Efficiency in a Fixed-Odds Betting Market', Peter F. Pope, David A. Peel, Economica, New Series, volume 56, number 223, (August, 1989), pages 323–341.
• 'A Mathematical Perspective on Gambling', Molly Maxwell, MIT Undergraduate Journal of Mathematics, volume 1, (1999), pages 123–132.
• 'Probability Guide to Gambling: The Mathematics of dice, slots, roulette, baccarat, blackjack, poker, lottery and sport bets', Catalin Barboianu, Infarom, 2006, 316pp. ISBN973-87520-3-5.

External links[edit]