Better Bookmaker

The complete guide to how to become a bookmaker (bookie), where to start, handling adversity, and buying PPH / pay per head. Become an exceptional independent sports bookmaker. 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. Find the best 10 online sports betting sites in UK for 2021, Covers trusted since 1995 reviews feature ratings, their bonuses, free bets, and more. Better approach to bookmakers reviews If you are searching for the best online bookmaker that will satisfy your criteria, you are in the right place! On our website you will be able to find a detailed list of verified bookmakers, what each bookmaker has to offer, their pros and cons. Independently of your online gambling experience, this list. Ordered the bookmaker salad, chicken parm, garlic bread, and cannoli. 'World famous salad dressing' was OK, but it seemed to missing something (salt maybe?), and it had a chalky texture. For $2.50 extra, not sure the bookmaker salad was a wise upgrade but it did have quite a bit of shrimp, salami, and cheese. The chicken parm was kinda tough.

1. Better Bookkeepers In San Jose Ca
2. Better Bookmakers

The principle of value betting is not to bet when you think you will win, but when the probability of you winning is greater than that implied by the bookmaker's odds. To be a profitable bettor means that your judgement is better than your bookmaker's. In this article, Joseph Buchdahl explains how sportsbooks manage the odds and shows how beating the bookies works. Read on to find out.

In 2014 Pinnacle tested the Wisdom of the Crowd hypothesis by inviting people to guess the number of chocolate balls in a video. By the end of the test, the average guess was just 1.4% higher than the actual figure, despite only 1 person out of the 608 entries guessing correctly.

[Although] the results of sporting events cannot be known a priori, yet even under these conditions the crowd does generally provide an accurate assessment of the respective probabilities of the outcomes.

Of course, sporting outcomes are binary by nature: they either happen or they don't. However, it is the flow of money via which the opinions of bettors are expressed that is used by bookmakers to judge the respective outcome probabilities, by means of the odds. It is true that herds are prone to expressing systematic biased judgement when faced with uncertainty, leading to a collectively less wise opinion.

However, typically we find that the more liquid (or popular) the betting market, the better the collective wisdom. The greater the number of independently acting players expressing a range of diverse views about a sporting event there are, the more likely it is that a crowd will be wise and the betting odds accurate.

The wisdom of Pinnacle bettors

I have previously examined just how accurate Pinnacle's 1X2 soccer match betting odds are, and by extension how wise their market is, by means of comparing expected outcome probabilities (defined by the odds) to actual outcome percentages. This analysis demonstrated that Pinnacle's betting odds, on average, are highly efficient - that is to say, accurate.

The greater the number of independently acting players expressing a range of diverse views about a sporting event there are, the more likely it is that a crowd will be wise and the betting odds accurate.

For example, teams fairly priced at 2.00 (i.e. once Pinnacle's margin has been removed) typically win about 50% of the time. Teams priced at 4.00 win 25% of the time, and so on. Whilst such an observation is not conclusive proof of market efficiency, it is consistent with it.

Of course, not all bookmakers offer the same price for a team. For example, Pinnacle offered a price of 2.22 for Liverpool to beat Tottenham in their game played 11^th February 2017. This varied between 2.15 and 2.33 with other bookmakers. How do we know which price was more accurate?

Testing Pinnacle's wisdom against other bookmakers

One way we could test relative price efficiency of Pinnacle versus other bookmakers is to formulate the following hypothesis and accompanying test:

1)Assume Pinnacle's odds (with their margin removed) provide an exact measure of the true outcome probabilities.

2)Consequently, the ratio of another bookmaker's odds to Pinnacle's odds provides a measure of expected value or expected return.

3)Analyse actual returns across a range of expected values.

For example, with the margin removed, Pinnacle's estimated fair price for Liverpool to beat Tottenham was about 2.25, implying roughly a 44% outcome probability. Consequently, if our hypothesis is correct, the best market price of 2.33 would offer an expected return of about 1.035 or profit of +3.5% (2.33/2.25).

On the other hand, betting at the market low of 2.15 would entail an expected return of 0.956 or loss of -4.4% (2.15/2.25). If we then find that all bets with an expected profit of +3.5% (or loss of -4.4%) collectively return a profit of 3.5% (or loss of 4.4%), we would conclude that our hypothesis is correct, that is to say Pinnacle's odds, on average, are efficient, accurate or wise.

So, how wise is Pinnacle's soccer match betting markets? I've analysed a sample of 35,570 league matches played throughout Europe since the start of the 2012/13 season, yielding 106,710 possible outcomes from the home/draw/away market.

For each, the expected value (or return) is calculated by the ratio of the odds from one of four leading bookmakers to that of Pinnacle's price with their margin removed, yielding 426,840 expected returns. Actual returns are then calculated for 0.01 intervals in expected return (for example 0.98, 0.99, 1.00, 1.01 etc) before a 5-point running average is used to smooth the variance in the data. Data is plotted in a scatter graph below, with very low and very high-expected returns removed for which there are understandably far fewer contributing data points.
The correlation between expected and observed returns is very strong and essentially 1:1. That is to say, when the expected return over a sample of matches is 90%, we actually return about 90% (or a loss of 10%). When the expected return is 105%, we actually return about 105% (or a profit of 5%)

Market folly or manipulation

Let's now reverse the process. This time, let's assume that our other bookmakers' odds (with their margins removed) provide an accurate measure of true outcome probabilities. How do actual returns, this time betting Pinnacle's odds, compare to those expected by the hypothesis? Take a look.

Now, correlation between expected and actual returns is completely absent. If a fair price is 2.00, whether Pinnacle offers 1.8 or 2.1 makes no difference: we lose about 2% regardless (which is roughly the size of Pinnacle's soccer match betting market margin). The implication is that the odds from the four other bookmakers used in this analysis do not, on average, provide any meaningful measure of the true outcome probabilities relative to Pinnacle's. It's Pinnacle's price, which provides the accurate measure.

What pricing model does your bookmaker use?

Presumably, there are two possible explanations for such a finding. Perhaps other bookmakers don't know how to set prices properly. Evidently, that is not a credible conclusion, given the longevity of success of these bookmakers. Alternatively, we could speculate that bookmakers are intentionally shifting prices away from market efficiency in favour of pursuing interests of their business models.

Pinnacle's pricing model utilises crowd wisdom and accepts sharp players to tighten them. Other bookmakers from Europe and the UK prefer to encourage a steady flow of squares via promotional offers, a wider variety of low-liquidity markets and the regular availability of best market prices (if not the lowest margins). With respect to the last of those, a casual perusal of any odds comparison will reveal numerous matches where bookmakers are significantly out of line with Pinnacle's market, and in the extreme offer loss-leading value to the player.

In my analysis sample, such positive expectation was available in 4.1% of the betting propositions. Naturally, this would account for why so many of these alternative brands, in contrast to Pinnacle, rely on account closures to mitigate the threat of players taking systematic advantage of these loss-leading prices either through value betting or arbitrage. It will also come as no surprise to learn that whilst Pinnacle openly accepts arbitrage players, other brands do not, and that typically the bookmaker on the negative expectation side of the arbitrage, from the perspective of the punter, is Pinnacle.

This is not to say Pinnacle's betting market is perfectly efficient; just that inefficiencies are harder to find and exploit. However, those sharp enough to find them can do so safe in the knowledge that their custom will always be welcome. It helps, indeed, to build the wise betting market that sets Pinnacle apart from its competitors.

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:

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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

Types of poker games. 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 How to gamble online without credit card.

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]

For example, teams fairly priced at 2.00 (i.e. once Pinnacle's margin has been removed) typically win about 50% of the time. Teams priced at 4.00 win 25% of the time, and so on. Whilst such an observation is not conclusive proof of market efficiency, it is consistent with it.

Of course, not all bookmakers offer the same price for a team. For example, Pinnacle offered a price of 2.22 for Liverpool to beat Tottenham in their game played 11^th February 2017. This varied between 2.15 and 2.33 with other bookmakers. How do we know which price was more accurate?

Testing Pinnacle's wisdom against other bookmakers

One way we could test relative price efficiency of Pinnacle versus other bookmakers is to formulate the following hypothesis and accompanying test:

1)Assume Pinnacle's odds (with their margin removed) provide an exact measure of the true outcome probabilities.

2)Consequently, the ratio of another bookmaker's odds to Pinnacle's odds provides a measure of expected value or expected return.

3)Analyse actual returns across a range of expected values.

For example, with the margin removed, Pinnacle's estimated fair price for Liverpool to beat Tottenham was about 2.25, implying roughly a 44% outcome probability. Consequently, if our hypothesis is correct, the best market price of 2.33 would offer an expected return of about 1.035 or profit of +3.5% (2.33/2.25).

On the other hand, betting at the market low of 2.15 would entail an expected return of 0.956 or loss of -4.4% (2.15/2.25). If we then find that all bets with an expected profit of +3.5% (or loss of -4.4%) collectively return a profit of 3.5% (or loss of 4.4%), we would conclude that our hypothesis is correct, that is to say Pinnacle's odds, on average, are efficient, accurate or wise.

So, how wise is Pinnacle's soccer match betting markets? I've analysed a sample of 35,570 league matches played throughout Europe since the start of the 2012/13 season, yielding 106,710 possible outcomes from the home/draw/away market.

For each, the expected value (or return) is calculated by the ratio of the odds from one of four leading bookmakers to that of Pinnacle's price with their margin removed, yielding 426,840 expected returns. Actual returns are then calculated for 0.01 intervals in expected return (for example 0.98, 0.99, 1.00, 1.01 etc) before a 5-point running average is used to smooth the variance in the data. Data is plotted in a scatter graph below, with very low and very high-expected returns removed for which there are understandably far fewer contributing data points.
The correlation between expected and observed returns is very strong and essentially 1:1. That is to say, when the expected return over a sample of matches is 90%, we actually return about 90% (or a loss of 10%). When the expected return is 105%, we actually return about 105% (or a profit of 5%)

Market folly or manipulation

Let's now reverse the process. This time, let's assume that our other bookmakers' odds (with their margins removed) provide an accurate measure of true outcome probabilities. How do actual returns, this time betting Pinnacle's odds, compare to those expected by the hypothesis? Take a look.

Now, correlation between expected and actual returns is completely absent. If a fair price is 2.00, whether Pinnacle offers 1.8 or 2.1 makes no difference: we lose about 2% regardless (which is roughly the size of Pinnacle's soccer match betting market margin). The implication is that the odds from the four other bookmakers used in this analysis do not, on average, provide any meaningful measure of the true outcome probabilities relative to Pinnacle's. It's Pinnacle's price, which provides the accurate measure.

What pricing model does your bookmaker use?

Presumably, there are two possible explanations for such a finding. Perhaps other bookmakers don't know how to set prices properly. Evidently, that is not a credible conclusion, given the longevity of success of these bookmakers. Alternatively, we could speculate that bookmakers are intentionally shifting prices away from market efficiency in favour of pursuing interests of their business models.

Pinnacle's pricing model utilises crowd wisdom and accepts sharp players to tighten them. Other bookmakers from Europe and the UK prefer to encourage a steady flow of squares via promotional offers, a wider variety of low-liquidity markets and the regular availability of best market prices (if not the lowest margins). With respect to the last of those, a casual perusal of any odds comparison will reveal numerous matches where bookmakers are significantly out of line with Pinnacle's market, and in the extreme offer loss-leading value to the player.

In my analysis sample, such positive expectation was available in 4.1% of the betting propositions. Naturally, this would account for why so many of these alternative brands, in contrast to Pinnacle, rely on account closures to mitigate the threat of players taking systematic advantage of these loss-leading prices either through value betting or arbitrage. It will also come as no surprise to learn that whilst Pinnacle openly accepts arbitrage players, other brands do not, and that typically the bookmaker on the negative expectation side of the arbitrage, from the perspective of the punter, is Pinnacle.

This is not to say Pinnacle's betting market is perfectly efficient; just that inefficiencies are harder to find and exploit. However, those sharp enough to find them can do so safe in the knowledge that their custom will always be welcome. It helps, indeed, to build the wise betting market that sets Pinnacle apart from its competitors.

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:

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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

Types of poker games. 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 How to gamble online without credit card.

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

Better Bookmakers

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]

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]

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]

• 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.

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]