Mathematics of bookmaking: Difference between revisions
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{{Short description|Theory of laying bets}} |
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{{one source|date=February 2012}} |
{{one source|date=February 2012}} |
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In [[gambling]] parlance, '''making a book''' is the practice of [[Betting exchange# |
In [[gambling]] parlance, '''making a book''' is the practice of [[Betting exchange#Backing and laying|laying bets]] on the various possible outcomes of a single event. The phrase 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'.<ref name=idSidney1976>{{Cite book |last=Sidney |first=Charles |title=The Art of Legging |publisher=Maxline International |year=1976}}</ref>{{rp|6}}<ref name="idSidney2003">{{Cite book |last=Sidney |first=Charles |title=The Art of Legging: The History, Theory, and Practice of Bookmaking on the English Turf |publisher=Rotex Publishing |year=2003 |edition=3rd}}</ref>{{rp|13,36}} |
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==Making a 'book' (and the notion of overround)== |
==Making a 'book' (and the notion of overround)== |
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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).<ref name=cortis> |
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.<ref>{{Cite journal |last=Itzkowitz |first=David C. |date=1988 |title=Victorian Bookmakers and Their Customers |url=https://www.jstor.org/stable/3828287 |journal=Victorian Studies |volume=32 |issue=1 |pages=7–30 |issn=0042-5222}}</ref> 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).<ref name=cortis> |
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{{cite |
{{cite journal |last1=Cortis |first1=Dominic |title=Expected Values and variance in bookmaker payouts: A Theoretical Approach towards setting limits on odds |url=http://ubplj.org/index.php/jpm/article/view/987/968 |year=2015 |volume=9 |series=1 |journal= Journal of Prediction Markets }}</ref> |
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The odds quoted for a particular event may be fixed |
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]]. |
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It is important to understand the relationship between fractional and decimal odds. Fractional odds are |
It is important to understand the relationship between fractional and decimal odds. Fractional odds are written ''a'' − ''b'' (''a''/''b'' or ''a'' to ''b''), meaning a winning bettor will receive their money back plus ''a'' units for every ''b'' units they bet. Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet. |
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| ⚫ | 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 × 2.5 = £100. We can convert fractional to decimal odds by the formula ''D'' = {{frac|(''b'' + ''a'')|''b''}}. Hence, fractional odds of ''a'' − 1 (ie. ''b'' = 1) can be obtained from decimal odds by ''a'' = ''D'' − 1. |
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Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet. |
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| ⚫ | For example, a bet of |
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It is also important to understand the relationship between odds and implied probabilities: |
It is also important to understand the relationship between odds and implied probabilities: |
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Fractional odds of |
Fractional odds of ''a'' − ''b'' (with corresponding decimal odds ''D'') represent an implied probability of {{frac|''b''|(''a'' + ''b'')}} = {{frac|1|''D''}}, e.g. 6-4 corresponds to {{frac|4|(6 + 4)}} = {{frac|4|10}} = 0.4 (40%). |
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An implied probability of |
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, {{frac|1|x}} − 1 to 1), and decimal odds of ''D'' = {{frac|1|''x''}}. |
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===Example=== |
===Example=== |
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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: |
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: |
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:5-1 corresponds to an implied probability of {{frac|1|6}} (16{{frac|2|3}}%) |
:5-1 corresponds to an implied probability of {{frac|1|6}} (16{{frac|2|3}}%) |
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By adding the percentages together a total 'book' of 100% is achieved (representing a ''fair'' book). The bookmaker |
By adding the percentages together a total 'book' of 100% is achieved (representing a ''fair'' book). The bookmaker will reduce these odds to ensure a profit. 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): |
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:Home: 4-6 |
:Home: 4-6 |
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By adding ''these'' percentages together a 'book' of 120% is achieved. |
By adding ''these'' percentages together a 'book' of 120% is achieved. |
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The amount by which the actual 'book' exceeds 100% is known as the 'overround', |
The amount by which the actual 'book' exceeds 100% is known as the 'overround',{{r|idSidney1976|p=96–104}}{{r|idSidney2003|p=126–130}} 'bookmaker margin'<ref name="cortis" /> or the '[[vigorish]]' or 'vig'<ref name="cortis" /> and represents the bookmaker's expected profit. 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 the actual outcome of the football match. Examining how he potentially achieves this: |
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Examining how he potentially achieves this: |
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:A stake of £60.00 @ 4-6 returns £100.00 (exactly) for a home win. |
:A stake of £60.00 @ 4-6 returns £100.00 (exactly) for a home win. |
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:A stake of £20.00 @ 4-1 returns £100.00 (exactly) for an away win |
:A stake of £20.00 @ 4-1 returns £100.00 (exactly) for an away win |
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Total stakes received |
Total stakes received are £120.00 with a maximum payout of £100.00 irrespective of the result. This £20.00 profit represents a 16{{frac|2|3}} % profit on turnover (20.00/120.00). |
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In reality, bookmakers use models of reducing that are more complicated than |
In reality, bookmakers use models of reducing that are more complicated than this model of the 'ideal' situation. |
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=== Bookmaker margin in English football leagues === |
=== Bookmaker margin in English football leagues === |
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Bookmaker margin in English football leagues decreased in recent years<ref>{{Cite journal|last=Marek|first=Patrice|date=September 2018|title= |
Bookmaker margin in English football leagues decreased in recent years.<ref>{{Cite journal|last=Marek|first=Patrice|date=September 2018|title=Bookmakers' Efficiency in English Football Leagues|url=https://www.researchgate.net/publication/327631361|journal=Mathematical Methods in Economics - Conference Proceedings|pages=330–335}}</ref> A study of six large bookmakers between the 2005/06 season and 2017/2018 season showed that average margin in [[Premier League]] decreased from 9% to 4%, in [[EFL Championship|English Football League Championship]], [[EFL League One|English Football League One]], and [[EFL League Two|English Football League Two]] from 11% to 6%, and in [[National League (English football)|National League]] from 11% to 8%. |
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==Overround on multiple bets== |
==Overround on multiple bets== |
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When a punter (''bettor'') combines more than one selection in, for example, a [[Glossary of bets offered by UK bookmakers#Double|double]], [[Glossary of bets offered by UK bookmakers#Treble|treble]] or [[Glossary of bets offered by UK bookmakers#Accumulator|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. |
When a punter (''bettor'') combines more than one selection in, for example, a [[Glossary of bets offered by UK bookmakers#Double|double]], [[Glossary of bets offered by UK bookmakers#Treble|treble]] or [[Glossary of bets offered by UK bookmakers#Accumulator|accumulator]] bet, then the effect of the overround in the book of each selection is compounded. This is 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. |
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For example, consider a double made by selecting the winners from two tennis matches: |
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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 |
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 × ({{frac|6|11}} + {{frac|6|11}}) i.e. 9.09% overround. |
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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 |
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 have been deliberately chosen to ensure that they are equally likely, the probability of each outcome occurring is {{frac|1|4}} or 0.25, and the fractional odds against each one occurring is 3-1. A bet of 100 units 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. |
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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 [[Independence (probability theory)|independent]] then the implied probability should be the [[Independence (probability theory)#Two events|product of the implied probabilities]] of the individual bets. In the above case with fractional odds of 5 |
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 [[Independence (probability theory)|independent]] then the implied probability should be the [[Independence (probability theory)#Two events|product of the implied probabilities]] of the individual bets. In the above case with fractional odds of 5 − 6, the decimal odds are {{frac|11|6}}. So the decimal odds of the double bet is {{frac|11|6}}×{{frac|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. |
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In general, the combined overround on a double (O<sub>D</sub>), expressed as a percentage, is calculated from the individual books ''B''<sub>1</sub> and ''B''<sub>2</sub>, expressed as decimals, by O<sub>D</sub> = |
In general, the combined overround on a double (''O''<sub>D</sub>), expressed as a percentage, is calculated from the individual books ''B''<sub>1</sub> and ''B''<sub>2</sub>, expressed as decimals, by ''O''<sub>D</sub> = ''B''<sub>1</sub> × ''B''<sub>2</sub> × 100 - 100. |
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In the example we have O<sub>D</sub> = |
In the example we have ''O''<sub>D</sub> = 1.0909 × 1.0909 × 100 - 100 = 19.01%. |
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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 |
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 [[Glossary of bets offered by UK bookmakers#Yankee|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 ''[[Glossary of bets offered by UK bookmakers#Lucky 15|Lucky 15]]'', ''[[Glossary of bets offered by UK bookmakers#Lucky 31|Lucky 31]]'' and ''[[Glossary of bets offered by UK bookmakers#Lucky 63|Lucky 63]],'' offering double the odds for one winner and increasing percentage bonuses for two, three and more winners. |
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In general, for any accumulator bet from two to ''i'' selections, the combined percentage overround of books of ''B''<sub>1</sub>, ''B''<sub>2</sub>, ..., ''B''<sub>''i''</sub> given in terms of decimals, is calculated by ''B''<sub>1</sub> |
In general, for any accumulator bet from two to ''i'' selections, the combined percentage overround of books of ''B''<sub>1</sub>, ''B''<sub>2</sub>, ..., ''B''<sub>''i''</sub> given in terms of decimals, is calculated by ''B''<sub>1</sub> × ''B''<sub>2</sub> × ... × ''B''<sub>''i''</sub> × 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%. |
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==Settling winning bets== |
==Settling winning bets== |
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In settling winning bets either decimal odds are used or one is added to the fractional odds |
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 [[Glossary of bets offered by UK bookmakers#Each-Way|each-way]] bets is calculated separately from the [[Glossary of bets offered by UK bookmakers#Win|win]] part; the method is identical but the odds are reduced by whatever the [[Glossary of bets offered by UK bookmakers#Place|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 are 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.{{r|idSidney1976|p=138–147}}{{r|idSidney2003|p=163–177}} |
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===Singles=== |
===Singles=== |
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'''Win single''' |
'''Win single''' |
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E.g. £100 [[Glossary of bets offered by UK bookmakers#Single|single]] at 9 |
E.g. £100 [[Glossary of bets offered by UK bookmakers#Single|single]] at 9 − 2; total staked = £100 |
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:Returns = £100 |
:Returns = £100 × (9/2 + 1) = £100 × 5.5 = £550 |
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'''Each-way single''' |
'''Each-way single''' |
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E.g. £100 each-way single at 11 |
E.g. £100 each-way single at 11 − 4 ( {{frac|1|5}} odds a place); total staked = £200 |
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:Returns (win) = £100 |
:Returns (win) = £100 × (11/4 + 1) = £100 × 3.75 = £375 |
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:Returns (place) = £100 |
:Returns (place) = £100 × (11/20 + 1) = £100 × 1.55 = £155 |
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:Total returns if selection wins = £530; if only placed = £155 |
:Total returns if selection wins = £530; if only placed = £155 |
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===Multiple bets=== |
===Multiple bets=== |
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Each- |
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.{{r|idSidney1976|p=155–156}}{{r|idSidney2003|p=170–171}} The first example below shows the two different approaches to settling these types of bets. |
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'''Double''' |
'''Double'''{{r|idSidney1976|p=153–168}}{{r|idSidney2003|p=169–177}} |
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E.g. £100 each-way [[Glossary of bets offered by UK bookmakers#Double|double]] with winners at 2-1 ( {{frac|1|5}} odds a place) and 5-4 ( {{frac|1|4}} odds a place); total staked = £200 |
E.g. £100 each-way [[Glossary of bets offered by UK bookmakers#Double|double]] with winners at 2-1 ( {{frac|1|5}} odds a place) and 5-4 ( {{frac|1|4}} odds a place); total staked = £200 |
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{{hidden begin|title="''Win to Win, Place to Place''"}} |
{{hidden begin|title="''Win to Win, Place to Place''"}} |
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:Returns (win double) = £100 |
:Returns (win double) = £100 × (2/1 + 1) × (5/4 + 1) = £675 |
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:Returns (place double) = £100 |
:Returns (place double) = £100 × (2/5 + 1) × (5/16 + 1) = £183.75 |
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:Total returns = £858.75 |
:Total returns = £858.75 |
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{{hidden begin|title="''Each-Way all Each-Way''"}} |
{{hidden begin|title="''Each-Way all Each-Way''"}} |
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:Returns (first selection) = £100 |
:Returns (first selection) = £100 × (2/1 + 1) + £100 × (2/5 + 1) = £440 which is split equally to give a £220 each-way bet on the second selection) |
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:Returns (second selection) = £220 |
:Returns (second selection) = £220 × (5/4 + 1) + £220 × (5/16 + 1) = £783.75 |
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:Total returns = £783.85 |
:Total returns = £783.85 |
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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. |
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. |
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'''Treble''' |
'''Treble'''{{r|idSidney1976|p=153-168}}{{r|idSidney2003|p=169-177}} |
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E.g. £100 [[Glossary of bets offered by UK bookmakers#Treble|treble]] with winners at 3-1, 4-6 and 11-4; total staked = £100 |
E.g. £100 [[Glossary of bets offered by UK bookmakers#Treble|treble]] with winners at 3-1, 4-6 and 11-4; total staked = £100 |
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:Returns = £100 |
:Returns = £100 × (3/1 + 1) × (4/6 + 1) × (11/4 + 1) = £2500 |
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'''Accumulator''' |
'''Accumulator'''{{r|idSidney1976|p=153-168}}{{r|idSidney2003|p=169-177}} |
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E.g. £100 each-way fivefold [[Glossary of bets offered by UK bookmakers#Accumulator|accumulator]] with winners at Evens ( {{frac|1|4}} odds a place), 11-8 ( {{frac|1|5}} odds), 5-4 ( {{frac|1|4}} odds), 1-2 (all up to win) and 3-1 ( {{frac|1|5}} odds); total staked = £200 |
E.g. £100 each-way fivefold [[Glossary of bets offered by UK bookmakers#Accumulator|accumulator]] with winners at Evens ( {{frac|1|4}} odds a place), 11-8 ( {{frac|1|5}} odds), 5-4 ( {{frac|1|4}} odds), 1-2 (all up to win) and 3-1 ( {{frac|1|5}} odds); total staked = £200 |
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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. |
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. |
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:Returns (win fivefold) = £100 |
:Returns (win fivefold) = £100 × (1/1 + 1) × (11/8 + 1) × (5/4 + 1) × (1/2 + 1) × (3/1 + 1) = £6412.50 |
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:Returns (place fivefold) = £100 |
:Returns (place fivefold) = £100 × (1/4 + 1) × (11/40 + 1) × (5/16 + 1) × (1/2 + 1) × (3/5 + 1) = £502.03 |
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:Total returns = £6914.53 |
:Total returns = £6914.53 |
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{{hidden begin|title=E.g. £10 [[Glossary of bets offered by UK bookmakers#Trixie|Trixie]] with winners at 4-7, 2-1 and 11-10; total staked = £40}} |
{{hidden begin|title=E.g. £10 [[Glossary of bets offered by UK bookmakers#Trixie|Trixie]] with winners at 4-7, 2-1 and 11-10; total staked = £40}} |
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:Returns (3 doubles) = £10 |
:Returns (3 doubles) = £10 × [(4/7 + 1) × (2/1 + 1) + (4/7 + 1) × (11/10 + 1) + (2/1 + 1) × (11/10 + 1)] = £143.14 |
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:Returns (1 treble) = £10 |
:Returns (1 treble) = £10 × (4/7 + 1) × (2/1 + 1) × (11/10 + 1) = £99.00 |
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:Total returns = £242.14 |
:Total returns = £242.14 |
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{{hidden begin|title=E.g. £10 [[Glossary of bets offered by UK bookmakers#Yankee|Yankee]] with winners at 1-3, 5-2, 6-4 and Evens; total staked = £110}} |
{{hidden begin|title=E.g. £10 [[Glossary of bets offered by UK bookmakers#Yankee|Yankee]] with winners at 1-3, 5-2, 6-4 and Evens; total staked = £110}} |
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:Returns (6 doubles) = £10 |
:Returns (6 doubles) = £10 × [(1/3 + 1) × (5/2 + 1) + (1/3 + 1) × (6/4 + 1) + (1/3 + 1) × (1/1 + 1) + (5/2 + 1) × (6/4 + 1) + (5/2 + 1) × (1/1 + 1) + (6/4 + 1) × (1/1 + 1)] = £314.16 |
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:Returns (4 trebles) = £10 |
:Returns (4 trebles) = £10 × [(1/3 + 1) × (5/2 + 1) × (6/4 + 1) + (1/3 + 1) × (5/2 + 1) × (1/1 + 1) + (1/3 + 1) × (6/4 + 1) × (1/1 + 1) + (5/2 + 1) × (6/4 + 1) × (1/1 + 1)] = £451.66 |
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:Returns (1 fourfold) = £10 |
:Returns (1 fourfold) = £10 × (1/3 + 1) × (5/2 + 1) × (6/4 + 1) × (1/1 + 1) = £233.33 |
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:Total returns = £999.15 |
:Total returns = £999.15 |
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{{hidden begin|title=E.g. £2 [[Glossary of bets offered by UK bookmakers#Patent|Patent]] with winners at 4-6, 2-1 and 11-4; total staked = £14}} |
{{hidden begin|title=E.g. £2 [[Glossary of bets offered by UK bookmakers#Patent|Patent]] with winners at 4-6, 2-1 and 11-4; total staked = £14}} |
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:Returns (3 singles) = £2 |
:Returns (3 singles) = £2 × [(4/6 + 1) + (2/1 + 1) + (11/4 + 1)] = £16.83 |
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:Returns (3 doubles) = £2 |
:Returns (3 doubles) = £2 × [(4/6 + 1) × (2/1 + 1) + (4/6 + 1) × (11/4 + 1) + (2/1 + 1) × (11/4 + 1)] = £45.00 |
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:Returns (1 treble) = £2 |
:Returns (1 treble) = £2 × (4/6 + 1) × (2/1 + 1) × (11/4 + 1) = £37.50 |
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:Total returns = £99.33 |
:Total returns = £99.33 |
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==Algebraic interpretation== |
==Algebraic interpretation== |
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Returns on any bet may be considered to be calculated as 'stake unit' |
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. |
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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). |
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).{{r|idSidney1976|p=166}}{{r|idSidney2003|p=169, 176}} |
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===Examples=== |
===Examples=== |
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3 selections with decimal odds ''a'', ''b'' and ''c''. |
3 selections with decimal odds ''a'', ''b'' and ''c''. |
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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''. |
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''. |
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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. |
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.{{r|idSidney1976|p=161}}{{r|idSidney2003|p=176}} |
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E.g. The winning Patent described earlier can be more quickly and simply evaluated by the following: |
E.g. The winning Patent described earlier can be more quickly and simply evaluated by the following: |
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:Total returns = £2 |
:Total returns = £2 × [(4/6 + 2) × (2/1 + 2) × (11/4 + 2) − 1] = £99.33 |
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Ignoring any bonuses, a 50 pence each-way Lucky 63 (total stake £63) with 4 winners [2-1, 5-2, 7-2 (all {{frac|1|5}} odds a place) and 6-4 ({{frac|1|4}} odds a place)] and a further placed horse [9-2 ({{frac|1|5}} odds a place)] can be relatively easily calculated as follows: |
Ignoring any bonuses, a 50 pence each-way Lucky 63 (total stake £63) with 4 winners [2-1, 5-2, 7-2 (all {{frac|1|5}} odds a place) and 6-4 ({{frac|1|4}} odds a place)] and a further placed horse [9-2 ({{frac|1|5}} odds a place)] can be relatively easily calculated as follows: |
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:Returns (win part) = 0.50 |
:Returns (win part) = 0.50 × [(2/1 + 2) × (5/2 + 2) × (7/2 + 2) × (6/4 + 2) − 1] = £172.75 |
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:<small>or more simply as 0.50 |
:<small>or more simply as 0.50 × (4 × 4.5 × 5.5 × 3.5 − 1)</small> |
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:Returns (place part) = 0.50 |
:Returns (place part) = 0.50 × [(2/5 + 2) × (5/10 + 2) × (7/10 + 2) × (6/16 + 2) × (9/10 + 2) − 1] = £11.79 |
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:<small>or more simply as 0.50 |
:<small>or more simply as 0.50 × (2.4 × 2.5 × 2.7 × 2.375 × 2.9 − 1)</small> |
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:Total returns = £184.54 |
:Total returns = £184.54 |
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| Line 218: | Line 216: | ||
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: |
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: |
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:£10 |
:£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 |
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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 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. |
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| Line 234: | Line 232: | ||
{{hidden begin|title=E.g. £20 Up and Down with winners at 7-2 and 15-8; total staked = £40}} |
{{hidden begin|title=E.g. £20 Up and Down with winners at 7-2 and 15-8; total staked = £40}} |
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:Returns (£20 single at 7-2 ATC £20 single at 15-8) = £20 |
:Returns (£20 single at 7-2 ATC £20 single at 15-8) = £20 × 7/2 + £20 × (15/8 + 1) = £127.50 |
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:Returns (£20 single at 15-8 ATC £20 single at 7-2) = £20 |
:Returns (£20 single at 15-8 ATC £20 single at 7-2) = £20 × 15/8 + £20 × (7/2 + 1) = £127.50 |
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:Total returns = £255.00 |
:Total returns = £255.00 |
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| Line 245: | Line 243: | ||
{{hidden begin|title=E.g. £10 Up and Down with a winner at 5-1 and a loser; total staked = £20}} |
{{hidden begin|title=E.g. £10 Up and Down with a winner at 5-1 and a loser; total staked = £20}} |
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:Returns (£10 single at 5-1 ATC £10 single on 'loser') = £10 |
:Returns (£10 single at 5-1 ATC £10 single on 'loser') = £10 × 5/1 = £50 |
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:Note: This calculation of a bet where the stake is not returned is called "receiving the odds to the stake" on the winner; in this case receiving the odds to £10 (on the 5-1 winner). |
:Note: This calculation of a bet where the stake is not returned is called "receiving the odds to the stake" on the winner; in this case receiving the odds to £10 (on the 5-1 winner). |
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| Line 264: | Line 262: | ||
==Algebraic interpretation== |
==Algebraic interpretation== |
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If ''a'', ''b'', ''c'', ''d''... represent the ''decimal odds'', i.e. (fractional odds |
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).{{r|idSidney1976|p=166}}{{r|idSidney2003|p=169-176}} |
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===Examples=== |
===Examples=== |
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| Line 276: | Line 274: | ||
{{hidden begin|title=3 selections with decimal odds ''a'', ''b'' and ''c'' in a Round Robin.}} |
{{hidden begin|title=3 selections with decimal odds ''a'', ''b'' and ''c'' in a Round Robin.}} |
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*'''OM''' (3 winners) = (''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''' |
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*'''OM''' (2 winners) = (''a'' + 1) |
*'''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'''<br /><small>or more simply as '''OM''' = '''''ab'' + 3(''a'' + ''b'') − 4'''</small> |
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*'''OM''' (1 winner) = 2 |
*'''OM''' (1 winner) = 2 × (''a'' − 1) = '''2(''a'' − 1)''' |
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{{hidden end}} |
{{hidden end}} |
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{{hidden begin|title=4 selections with decimal odds ''a'', ''b'', ''c'' and ''d'' in a Flag.}} |
{{hidden begin|title=4 selections with decimal odds ''a'', ''b'', ''c'' and ''d'' in a Flag.}} |
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*'''OM''' (4 winners) = (''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)]<br />= '''(''a'' + 1)(''b'' + 1)(''c'' + 1)(''d'' + 1) + 5(''a'' + ''b'' + ''c'' + ''d'') − 13''' |
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*'''OM''' (3 winners) = (''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) = '''(''a'' + 1)(''b'' + 1)(''c'' + 1) + 4(''a'' + ''b'' + ''c'') − 10''' |
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*'''OM''' (2 winners) = (''a'' + 1) |
*'''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'''<br /><small>or more simply as '''OM''' = '''''ab'' + 4(''a'' + ''b'') − 6'''</small> |
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*'''OM''' (1 winner) = 3 |
*'''OM''' (1 winner) = 3 × (''a'' − 1) = '''3(''a'' − 1)''' |
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{{hidden end}} |
{{hidden end}} |
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| Line 293: | Line 291: | ||
*[[Glossary of bets offered by UK bookmakers]] |
*[[Glossary of bets offered by UK bookmakers]] |
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== |
==References== |
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{{reflist}} |
{{reflist}} |
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==References== |
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*{{wikicite|id=idCortis2015|reference=Cortis, D. (2015). "Expected Values and variance in bookmaker payouts: A Theoretical Approach towards setting limits on odds". Journal of Prediction Markets. 1. 9.}} |
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*{{wikicite|id=idSidney1976|reference=Sidney, C (1976). ''The Art of Legging'', Maxline International.}} |
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*{{wikicite|id=idSidney2003|reference=Sidney, C (2003). ''The Art of Legging: The History, Theory, and Practice of Bookmaking on the English Turf'', 3rd edition, Rotex Publishing 2003, 224pp. {{ISBN|978-1-872254-06-7}}}}. ''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''. |
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==Further reading== |
==Further reading== |
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| Line 307: | Line 300: | ||
* ''The Art of Bookmaking'', Malcolm Boyle, High Stakes Publishing 2006. |
* ''The Art of Bookmaking'', Malcolm Boyle, High Stakes Publishing 2006. |
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* ''Secrets of Successful Betting'', Michael Adams, Raceform, 2002. |
* ''Secrets of Successful Betting'', Michael Adams, Raceform, 2002. |
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* ''The Mathematics of Games and Gambling'', Edward W. Packel, [[Mathematical Association of America]], 2006. |
* ''[[The Mathematics of Games and Gambling]]'', [[Edward W. Packel]], [[Mathematical Association of America]], 2006. |
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* ''The Mathematics of Gambling'', Edward O. Thorp, L. Stuart, 1984. |
* ''The Mathematics of Gambling'', [[Edward O. Thorp]], L. Stuart, 1984. |
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* "Maximin Hedges", Jean-Claude Derderian, ''[[Mathematics Magazine]]'', volume 51, number 3. (May, 1978), pages 188–192. |
* "Maximin Hedges", Jean-Claude Derderian, ''[[Mathematics Magazine]]'', volume 51, number 3. (May, 1978), pages 188–192. |
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* "Carnap and de Finetti on Bets and the Probability of Singular Events: The [[Dutch book|Dutch Book]] Argument Reconsidered" Klaus Heilig, ''The British Journal for the Philosophy of Science'', volume 29, number 4. (December, 1978), pages 325–346. |
* "Carnap and de Finetti on Bets and the Probability of Singular Events: The [[Dutch book|Dutch Book]] Argument Reconsidered" Klaus Heilig, ''The British Journal for the Philosophy of Science'', volume 29, number 4. (December, 1978), pages 325–346. |
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| Line 318: | Line 311: | ||
* "A Mathematical Perspective on Gambling", Molly Maxwell, ''[[MIT Undergraduate Journal of Mathematics]]'', volume 1, (1999), pages 123–132. |
* "A Mathematical Perspective on Gambling", Molly Maxwell, ''[[MIT Undergraduate Journal of Mathematics]]'', volume 1, (1999), pages 123–132. |
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* "''Probability Guide to Gambling: The Mathematics of dice, slots, roulette, baccarat, blackjack, poker, lottery and sport bets''", Catalin Barboianu, Infarom, 2006, 316pp. {{ISBN|973-87520-3-5}}. |
* "''Probability Guide to Gambling: The Mathematics of dice, slots, roulette, baccarat, blackjack, poker, lottery and sport bets''", Catalin Barboianu, Infarom, 2006, 316pp. {{ISBN|973-87520-3-5}}. |
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==External links== |
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{{Gambling}} |
{{Gambling}} |
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{{Mathematics of}} |
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[[Category:Gambling mathematics]] |
[[Category:Gambling mathematics]] |
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[[Category:Wagering]] |
[[Category:Wagering]] |
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Latest revision as of 22:57, 12 July 2024
 | This article relies largely or entirely on a single source. (February 2012) |
In gambling parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The phrase 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]: 6 [2]: 13, 36
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.[3] 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).[4]
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 written a − b (a/b or a to b), meaning a winning bettor will receive their money back plus a units for every b units they bet. 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 × 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 (331⁄3%)
- 5-1 corresponds to an implied probability of 1⁄6 (162⁄3%)
By adding the percentages together a total 'book' of 100% is achieved (representing a fair book). The bookmaker will reduce these odds to ensure a profit. 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',[1]: 96–104 [2]: 126–130 'bookmaker margin'[4] or the 'vigorish' or 'vig'[4] and represents the bookmaker's expected profit. 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 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 are £120.00 with a maximum payout of £100.00 irrespective of the result. This £20.00 profit represents a 162⁄3 % profit on turnover (20.00/120.00).
In reality, bookmakers use models of reducing that are more complicated than this model of the 'ideal' situation.
Bookmaker margin in English football leagues
[edit]Bookmaker margin in English football leagues decreased in recent years.[5] A study of six large bookmakers between the 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 bet, then the effect of the overround in the book of each selection is compounded. This is 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.
For example, consider a double made by selecting the winners from two tennis matches:
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 have been deliberately chosen to ensure that they are equally likely, the probability of each outcome occurring is 1⁄4 or 0.25, and the fractional odds against each one occurring is 3-1. A bet of 100 units 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 (OD), expressed as a percentage, is calculated from the individual books B1 and B2, expressed as decimals, by OD = B1 × B2 × 100 - 100. In the example we have OD = 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, ..., Bi given in terms of decimals, is calculated by B1 × B2 × ... × Bi × 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 are 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.[1]: 138–147 [2]: 163–177
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.[1]: 155–156 [2]: 170–171 The first example below shows the two different approaches to settling these types of bets.
Double[1]: 153–168 [2]: 169–177
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
- Returns (win double) = £100 × (2/1 + 1) × (5/4 + 1) = £675
- Returns (place double) = £100 × (2/5 + 1) × (5/16 + 1) = £183.75
- Total returns = £858.75
- Returns (first selection) = £100 × (2/1 + 1) + £100 × (2/5 + 1) = £440 which is split equally to give a £220 each-way bet on the second selection)
- Returns (second selection) = £220 × (5/4 + 1) + £220 × (5/16 + 1) = £783.75
- Total returns = £783.85
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[1]: 153-168 [2]: 169-177
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[1]: 153-168 [2]: 169-177
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
- Returns (3 doubles) = £10 × [(4/7 + 1) × (2/1 + 1) + (4/7 + 1) × (11/10 + 1) + (2/1 + 1) × (11/10 + 1)] = £143.14
- Returns (1 treble) = £10 × (4/7 + 1) × (2/1 + 1) × (11/10 + 1) = £99.00
- Total returns = £242.14
Yankee
- Returns (6 doubles) = £10 × [(1/3 + 1) × (5/2 + 1) + (1/3 + 1) × (6/4 + 1) + (1/3 + 1) × (1/1 + 1) + (5/2 + 1) × (6/4 + 1) + (5/2 + 1) × (1/1 + 1) + (6/4 + 1) × (1/1 + 1)] = £314.16
- Returns (4 trebles) = £10 × [(1/3 + 1) × (5/2 + 1) × (6/4 + 1) + (1/3 + 1) × (5/2 + 1) × (1/1 + 1) + (1/3 + 1) × (6/4 + 1) × (1/1 + 1) + (5/2 + 1) × (6/4 + 1) × (1/1 + 1)] = £451.66
- Returns (1 fourfold) = £10 × (1/3 + 1) × (5/2 + 1) × (6/4 + 1) × (1/1 + 1) = £233.33
- Total returns = £999.15
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
- Returns (3 singles) = £2 × [(4/6 + 1) + (2/1 + 1) + (11/4 + 1)] = £16.83
- Returns (3 doubles) = £2 × [(4/6 + 1) × (2/1 + 1) + (4/6 + 1) × (11/4 + 1) + (2/1 + 1) × (11/4 + 1)] = £45.00
- Returns (1 treble) = £2 × (4/6 + 1) × (2/1 + 1) × (11/4 + 1) = £37.50
- Total returns = £99.33
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).[1]: 166 [2]: 169, 176
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.[1]: 161 [2]: 176
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
- Returns (£20 single at 7-2 ATC £20 single at 15-8) = £20 × 7/2 + £20 × (15/8 + 1) = £127.50
- Returns (£20 single at 15-8 ATC £20 single at 7-2) = £20 × 15/8 + £20 × (7/2 + 1) = £127.50
- Total returns = £255.00
- Note: This is the same as two £20 single bets at twice the odds; i.e. £20 singles at 7-1 and 15-4 and is the preferred manual way of calculating the bet.
- Returns (£10 single at 5-1 ATC £10 single on 'loser') = £10 × 5/1 = £50
- Note: This calculation of a bet where the stake is not returned is called "receiving the odds to the stake" on the winner; in this case receiving the odds to £10 (on the 5-1 winner).
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).[1]: 166 [2]: 169-176
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]References
[edit]- ^ a b c d e f g h i j Sidney, Charles (1976). The Art of Legging. Maxline International.
- ^ a b c d e f g h i j Sidney, Charles (2003). The Art of Legging: The History, Theory, and Practice of Bookmaking on the English Turf (3rd ed.). Rotex Publishing.
- ^ Itzkowitz, David C. (1988). "Victorian Bookmakers and Their Customers". Victorian Studies. 32 (1): 7–30. ISSN 0042-5222.
- ^ a b c Cortis, Dominic (2015). "Expected Values and variance in bookmaker payouts: A Theoretical Approach towards setting limits on odds". Journal of Prediction Markets. 1. 9.
- ^ Marek, Patrice (September 2018). "Bookmakers' Efficiency in English Football Leagues". Mathematical Methods in Economics - Conference Proceedings: 330–335.
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. ISBN 978-0-935536-37-9.
- 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. ISBN 973-87520-3-5.
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