In
gambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
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
A ledger is a book or collection of accounts in which account transactions are recorded. Each account has an opening or carry-forward balance, and would record each transaction as either a debit or credit in separate columns, and the ending or ...
(the 'book') and gives the English language the term
bookmaker
A bookmaker, bookie, or turf accountant is an organization or a person that accepts and pays off bets on sporting and other events at agreed-upon odds.
History
The first bookmaker, Ogden, stood at Newmarket in 1795.
Range of events
Bookma ...
for the person laying the bets and thus 'making the book'.
[ Sidney 2003, pp. 13, 36]
Making a 'book' (and the notion of overround)
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
In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which ...
and
coherence (philosophical gambling strategy)
In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent precisely if it does not expose the wagerer to certain loss regardless of the outcomes of events ...
. This is achieved primarily by adjusting what are determined to be the true
odds
Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.
Odds also have ...
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).
[
]
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
Parimutuel betting or pool betting is a betting system in which all bets of a particular type are placed together in a pool; taxes and the "house-take" or "vigorish" are deducted, and payoff odds are calculated by sharing the pool among all winnin ...
.
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 × 2.5 = £100. We can convert fractional to decimal odds by the formula ''D'' = . 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 = , e.g. 6-4 corresponds to = = 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 to 1), and decimal odds of ''D'' = .
Example
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
The Evens ( eve, эвэн; pl. , in Even and , in Russian; formerly called ''Lamuts'') are a people in Siberia and the Russian Far East. They live in regions of the Magadan Oblast and Kamchatka Krai and northern parts of Sakha east of th ...
: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 (50%)
:2-1 corresponds to an implied probability of (33%)
:5-1 corresponds to an implied probability of (16%)
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 (60%)
:6-4 corresponds to an implied probability of (40%)
:4-1 corresponds to an implied probability of (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',
[ Sidney 2003, pp. 126–130] 'bookmaker margin'
or the '
vigorish
Vigorish (also known as ''juice'', ''under-juice'', the ''cut'', the ''take'', the ''margin'', the ''house edge'' or simply the ''vig'') is the fee charged by a bookmaker (or ''bookie'') for accepting a gambler's wager. In American English, it can ...
' or 'vig':
it 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 what the actual outcome of the football match.
Examining how he potentially achieves this:
:A stake of £60.00 @ 4-6 returns £100.00 (exactly) for a home win.
:A stake of £40.00 @ 6-4 returns £100.00 (exactly) for a drawn match
:A stake of £20.00 @ 4-1 returns £100.00 (exactly) for an away win
Total stakes received — £120.00 and a maximum payout of £100.00 irrespective of the result. This £20.00 profit represents a 16 % 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
Bookmaker margin in English football leagues decreased in recent years. The study of six large bookmakers between 2005/06 season and 2017/2018 season showed that average margin in
Premier League
The Premier League (legal name: The Football Association Premier League Limited) is the highest level of the men's English football league system. Contested by 20 clubs, it operates on a system of promotion and relegation with the English Foo ...
decreased from 9% to 4%, in
English Football League Championship
The English Football League Championship (often referred to as the Championship for short or the Sky Bet Championship for sponsorship purposes) is the highest division of the English Football League (EFL) and second-highest overall in the En ...
,
English Football League One, and
English Football League Two from 11% to 6%, and in
National League
The National League of Professional Baseball Clubs, known simply as the National League (NL), is the older of two leagues constituting Major League Baseball (MLB) in the United States and Canada, and the world's oldest extant professional team s ...
from 11% to 8%.
Overround on multiple bets
When a punter (''bettor'') combines more than one selection in, for example, a
double
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* Th ...
,
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 × ( + ) 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 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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
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 . So the decimal odds of the double bet is × = 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 ''B''
1 and ''B''
2, expressed as decimals, by ''O''
D = ''B''
1 × ''B''
2 × 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
The term ''Yankee'' and its contracted form ''Yank'' have several interrelated meanings, all referring to people from the United States. Its various senses depend on the context, and may refer to New Englanders, residents of the Northern United St ...
, 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 ''B''
1, ''B''
2, ..., ''B''
''i'' given in terms of decimals, is calculated by ''B''
1 × ''B''
2 × ... × ''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
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
An each-way bet is a wager offered by bookmakers consisting of two separate bets: a win bet and a place bet. For the win part of the bet to give a return, the selection must win, or finish first, in the event. For the place part of the bet to gi ...
bets is calculated separately from the
win part; the method is identical but the odds are reduced by whatever the
place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own Municipality, municipal government
* "Place", a type of street or road ...
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.
[ Sidney 2003, pp. 163–177]
Singles
Win single
E.g. £100
single
Single may refer to:
Arts, entertainment, and media
* Single (music), a song release
Songs
* "Single" (Natasha Bedingfield song), 2004
* "Single" (New Kids on the Block and Ne-Yo song), 2008
* "Single" (William Wei song), 2016
* "Single", by ...
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 ( 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
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.
[ Sidney 2003, pp. 170–171] The first example below shows the two different approaches to settling these types of bets.
Double
[ Sidney 1976, pp. 153–168][ Sidney 2003, pp. 169–177]
E.g. £100 each-way
double
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* Th ...
with winners at 2-1 ( odds a place) and 5-4 ( 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
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
E.g. £100 each-way fivefold
accumulator with winners at Evens ( odds a place), 11-8 ( odds), 5-4 ( odds), 1-2 (all up to win) and 3-1 ( 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
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
Canadians (french: Canadiens) are people identified with the country of Canada. This connection may be residential, legal, historical or cultural. For most Canadians, many (or all) of these connections exist and are collectively the source of ...
,
Heinz,
Super Heinz and
Goliath
Goliath ( ) ''Goləyāṯ''; ar, جُليات ''Ǧulyāt'' (Christian term) or (Quranic term). is a character in the Book of Samuel, described as a Philistine giant
In folklore, giants (from Ancient Greek: ''gigas'', cognate giga-) a ...
form a family of bets known as
full cover bet A full cover bet is any bet which consists of all available multiple bets over a given number of selections.
Examples of full cover bets:
* '' Trixie'' - three selections
* ''Yankee'' - four selections
* '' Canadian or Super Yankee'' - five selecti ...
s 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
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
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).
[ Sidney 1976, p. 166][ Sidney 2003, pp. 169, 176]
Examples
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.
[ Sidney 1976, p. 161][ Sidney 2003, p. 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
-1, 5-2, 7-2 (all odds a place) and 6-4 ( odds a place)and a further placed horse
-2 ( 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
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
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).
Examples
*OM (2 winners) = (2''a'' − 1) + (2''b'' − 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)br />= (''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
*
Statistical association football predictions
*
Glossary of bets offered by UK bookmakers
This is a non-exhaustive list of traditional and popular bets offered by bookmakers in the United Kingdom. The 'multiple-selection' bets in particular are most often associated with horse racing selections but since the advent of fixed-odds bett ...
Notes
References
*
*
*. ''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
* "''Finding an Edge''",
Ron Loftus Ron is a shortening of the name Ronald.
Ron or RON may also refer to:
Arts and media
* Big Ron (''EastEnders''), a TV character
* Ron (''King of Fighters''), a video game character
*Ron Douglas, the protagonist in ''Lucky Stiff'' played by Joe ...
, US-SC-North Charleston: Create Space., 2011, 144pp.
* "''How to make a book''",
Phil Bull
Phil Bull (9 April 1910 – 11 June 1989), born West Yorkshire, England, was a professional gambler, racehorse owner and publisher, who founded the Timeform private handicapping system for British horseracing. Since 1948, Timeform have produce ...
, London: Morrison & Gibb Ltd., 1948, 160pp.
* "''The book on bookmaking''", Ferde Rombola, California: Romford Press, 1984, 147pp. .
* ''The Art of Bookmaking'', Malcolm Boyle, High Stakes Publishing 2006.
* ''Secrets of Successful Betting'', Michael Adams, Raceform, 2002.
* ''
The Mathematics of Games and Gambling
''The Mathematics of Games and Gambling'' is a book on probability theory and its application to games of chance. It was written by Edward Packel, and published in 1981 by the Mathematical Association of America as volume 28 of their New Mathemat ...
'', Edward W. Packel,
Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
, 2006.
* ''The Mathematics of Gambling'', Edward O. Thorp, L. Stuart, 1984.
* "Maximin Hedges", Jean-Claude Derderian, ''
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
'', volume 51, number 3. (May, 1978), pages 188–192.
* "Carnap and de Finetti on Bets and the Probability of Singular Events: The
Dutch Book
In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which ...
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
Management science (or managerial science) is a wide and interdisciplinary study of solving complex problems and making strategic decisions as it pertains to institutions, corporations, governments and other types of organizational entities. It is ...
'', 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
''The Economic Journal'' is a peer-reviewed academic journal of economics published on behalf of the Royal Economic Society by Oxford University Press. The journal was established in 1891 and publishes papers from all areas of economics.The edito ...
'', 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. .
{{Gambling
Gambling mathematics
Wagering