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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".
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 = . 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 = .
These odds can be represented as implied probabilities (or percentages by multiplying by 100) as follows:
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):
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", "bookmaker margin" or the "vigorish" or "vig" 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:
Total stakes received are £120.00 with 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 this model of the "ideal" situation.
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 × ( + ) 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 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 . 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 ( 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, ..., 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%.
E.g. £100 single at 9 − 2; total staked = £100
Each-way single
E.g. £100 each-way single at 11 − 4 ( odds a place); total staked = £200
Double
E.g. £100 each-way double with winners at 2-1 ( odds a place) and 5-4 ( odds a place); total staked = £200
Note: " Win to Win, Place to Place" will always provide a greater return if all selections win, whereas " Each-Way all Each-Way" provides greater compensation if one selection is a loser as each of the other winners provide a greater amount of place money for subsequent selections.
Treble
E.g. £100 treble with winners at 3-1, 4-6 and 11-4; total staked = £100
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.
Yankee
Trixie, Yankee, Canadian, Heinz, Super Heinz and Goliath form a family of bets known as full cover bets which have all possible multiples present. Examples of winning Trixie and Yankee bets have been shown above. The other named bets are calculated in a similar way by looking at all the possible combinations of selections in their multiples. Note: A Double may be thought of as a full cover bet with only two selections.
Should a selection in one of these bets not win, then the remaining winners are treated as being a wholly successful bet on the next "family member" down. For example, only two winners out of three in a Trixie means the bet is settled as a double; only four winners out of five in a Canadian means it is settled as a Yankee; only five winners out of eight in a Goliath means it is settled as a Canadian. The place part of each-way bets is calculated separately using reduced place odds. Thus, an each-way Super Heinz on seven horses with three winners and a further two placed horses is settled as a win Trixie and a place Canadian. Virtually all bookmakers use computer software for ease, speed and accuracy of calculation for the settling of multiples bets.
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.
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).
E.g. The winning Patent described earlier can be more quickly and simply evaluated by the following:
Ignoring any bonuses, a 50 pence each-way Lucky 63 (total stake £63) with 4 winners 2-1, and a further placed horse 9-2 can be relatively easily calculated as follows:
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:
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:
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.
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).
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