Mathematics of bookmaking

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

Making a ‘book’ (and the notion of over round)

A bookmaker strives to accept bets on the outcome of an event in the right proportions so that he makes 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 hopefully ensuring a profit).

The odds quoted for a particular event may be fixed (as in the case of a football match for example) or may fluctuate 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 odds and relative probabilities: Thus, odds of a–b (a/b or a to b) represent a relative probability of b/(a + b), e.g. 6-4 (6 to 4) is 4/(6 + 4) = 4/10 = 0.4 (or 40%). A relative probability of x represents 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).

Example

In considering a soccer 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 relative probabilities (or percentages by multiplying by 100) as follows:

Evens (or 1-1) corresponds to a relative probability of ¹⁄₂ (50%)

2-1 corresponds to a relative probability of ¹⁄₃ (33¹⁄₃%)

5-1 corresponds to a relative probability of ¹⁄₆ (16²⁄₃%)

By adding the percentages together a total ‘book’ of 100% is achieved (representing a fair book). The bookmaker, in his wish to avail himself of a profit, will invariably reduce these odds – possibly to the following:

Home: 4-5

Draw: 9-5

Away: 4-1

4-5 corresponds to a relative probability of ⁵⁄₉ (55⁵⁄₉%)

9-5 corresponds to a relative probability of ⁵⁄₁₄ (35⁵⁄₇%)

4-1 corresponds to a relative probability of ¹⁄₅ (20%)

By adding these percentages together a ‘book’ of 111¹⁷⁄₆₃%, or more commonly 111.27%, is achieved.

The amount by which the actual ‘book’ exceeds 100% is known as the ‘overround’:^[3][4] it represents the bookmaker’s potential profit if he is fortunate enough to accept bets in the exact proportions required. Thus, in an ideal situation, if the bookmaker accepts £111.27 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 £55.56 @ 4-5 returns £100.00 (rounded down to nearest penny) for a home win.

A stake of £35.71 @ 9-5 returns £ 99.98 (rounded down to nearest penny) for a drawn match

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

Total stakes received — £111.27 and a maximum payout of £100 irrespective of the result. This £11.27 profit represents a 10.1% profit on turnover (11.27 × 100/111.27).

Overround on multiple bets

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

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

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

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

As detailed below, the actual return on any of these winning doubles is obtained by multiplying stake × (‘odds plus one’ from each single bet) together. Thus for a stake of 100 units we get a return of 100 × (5/6 + 1) × (5/6 + 1) = 336.11… units, representing odds of 2.3611-1 which is far less than the true 3-1. Odds of 2.3611-1 represent a percentage of 29.752% (100/3.3611) and multiplying by 4 for the total number of equally likely outcomes gives a total book of 119.01%. Thus the overround has slightly more than doubled by combining two single bets into a double.

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

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

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

Settling winning bets

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

Singles

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

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" (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.^[7][8] The first example below shows the two different approaches to settling these types of bets.DoubleE.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