Chapter 7 · Worked solutions
Chapter 7 workings: The House's Head Start
Booksum, overround and margin worked out for chapter seven's four markets, plus the audit worksheet to fill in with a Saturday coupon of your own.
Three numbers describe how much a market charges. The booksum is the sum of every outcome's implied probability; the overround is how far it sits above 100; the margin is the share of a balanced stake the firm keeps, 1 − 1/booksum. The overround is the bigger-looking number, which is perhaps why it is the one nobody advertises.
The exercises
Match odds
The prices: 21/20 for home; 12/5 for draw; 11/4 for away.
Convert to decimals
Home 21/20 = 21 ÷ 20 + 1 = 2.05; Draw 12/5 = 12 ÷ 5 + 1 = 3.40; Away 11/4 = 11 ÷ 4 + 1 = 3.75.
Booksum
1/2.05 + 1/3.40 + 1/3.75 = 104.86 per cent.
Overround and margin
Overround is the excess over 100: 4.86 points. Margin is 1 − 1/booksum = 4.63 per cent.
Return per pound
1 ÷ 1.0486 = 0.954, so 95.4p back per £1. A pound spread across all three outcomes in proportion to their prices comes back as this much on average, whatever happens. The other few pence are the rent.
The answer
Booksum 104.86 per cent, margin 4.63 per cent, 95.4p back per pound.
Both teams to score
The prices: 8/11 for yes; 21/20 for no.
Convert to decimals
Yes 8/11 = 8 ÷ 11 + 1 = 1.7273; No 21/20 = 21 ÷ 20 + 1 = 2.05.
Booksum
1/1.7273 + 1/2.05 = 106.68 per cent.
Overround and margin
Overround is the excess over 100: 6.68 points. Margin is 1 − 1/booksum = 6.26 per cent.
Return per pound
1 ÷ 1.0668 = 0.937, so 93.7p back per £1. Two outcomes, and a higher margin than the three-way market. Fewer outcomes usually means a smaller margin, but that is a tendency, not a law: markets that fewer people compare get priced more lazily.
The answer
Booksum 106.68 per cent, margin 6.26 per cent, 93.7p back per pound.
Total goals 2.5
The prices: 5/6 for over; 20/21 for under.
Convert to decimals
Over 5/6 = 5 ÷ 6 + 1 = 1.8333; Under 20/21 = 20 ÷ 21 + 1 = 1.9524.
Booksum
1/1.8333 + 1/1.9524 = 105.76 per cent.
Overround and margin
Overround is the excess over 100: 5.76 points. Margin is 1 − 1/booksum = 5.45 per cent.
Return per pound
1 ÷ 1.0576 = 0.945, so 94.5p back per £1. Three markets on the same afternoon, three different rents. That spread is the reason the audit is worth doing before the bet rather than after it.
The answer
Booksum 105.76 per cent, margin 5.45 per cent, 94.5p back per pound.
The double
Combine the both-teams-to-score and total-goals markets into a two-leg double. What booksum and margin is the combined bet priced to?
List every combination
Two outcomes times two outcomes gives four doubles. Each double's decimal price is the product of its legs, so its claim is the product of the legs' claims.
Add up the four claims
The four claims sum to the product of the two booksums: 1.0668 × 1.0576 = 1.1283, a booksum of 112.83 per cent.
Margin
The combined margin is 11.37 per cent (1 − 1/1.1283): two rents of five or six per cent became one of eleven the moment the legs were stapled together.
One caution
The multiplication treats the legs as independent, as two separate matches would be. If both legs come from the same match they are related (goals help both), and firms price same-match combinations as bet builders with their own, usually larger, margin.
| Double | Decimal price | Claims |
|---|---|---|
| BTTS yes and over | 3.1667 | 31.58% |
| BTTS yes and under | 3.3723 | 29.65% |
| BTTS no and over | 3.7583 | 26.61% |
| BTTS no and under | 4.0024 | 24.99% |
| Total | 112.83% |
The answer
Booksum 112.83 per cent, margin 11.37 per cent.
The audit worksheet
Print this page or copy the table. Take one Saturday's coupon, choose six markets you would actually bet on, and fill in a row for each. The margin column is the one to look at first; the margin remover will check your sums.
| Market | Prices | Decimals | Booksum | Overround (points) | Margin | Back per £1 |
|---|---|---|---|---|---|---|
| Match odds | 21/20 / 12/5 / 11/4 | 2.05, 3.40, 3.75 | 104.86% | 4.86 | 4.63% | 95.4p |
| Both teams to score | 8/11 / 21/20 | 1.7273, 2.05 | 106.68% | 6.68 | 6.26% | 93.7p |
| Total goals 2.5 | 5/6 / 20/21 | 1.8333, 1.9524 | 105.76% | 5.76 | 5.45% | 94.5p |
These are the workings. The argument for them, and the exercises themselves, are in chapter 7 of the book, The House's Head Start.
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