Chapter 4 · Worked solutions
Chapter 4 workings: What Is Value, Actually?
Five scenarios, three numbers each: the break-even probability, the expected value per pound and the edge. Then a ruling, and the reason for it.
Two formulas do all the work on this page. For a fractional price a/b, the break-even probability is p* = b ÷ (a + b), and the expected value of a £1 bet at your probability p is p × (a/b) − (1 − p). When the EV is positive, the price is generous at your number; when it is negative, it is not, however exciting the payout looks.
Here is the whole exercise on one table before the working.
| Scenario | Your p | Price | p* | EV per £1 | Ruling |
|---|---|---|---|---|---|
| 1 | 50% | evens | 50.0% | 0p | No bet |
| 2 | 30% | 5/2 | 28.6% | +5p | Bet |
| 3 | 55% | 4/5 | 55.6% | −1p | No bet |
| 4 | 8% | 9/1 | 10.0% | −20p | No bet |
| 5 | 50% | 11/10 | 47.6% | +5p | Bet |
The exercises
Scenario 1
A home win you rate at 50 per cent, priced at evens.
Break-even probability
Divide the right-hand number by the sum of both: 1 ÷ (1 + 1) gives 1/2, or 50.00 per cent. That is what evens claims, and the line your estimate has to clear.
Expected value per £1
EV = p × (a/b) − (1 − p) = 0.50 × 1 − 0.50 = 0.00, so 0p per pound.
Edge and cushion
The edge per £1 staked is the EV: 0 per cent. Your estimate sits 0.00 points level with the claim.
Ruling: no bet
Zero on paper is not zero in practice. Your 50 is an estimate with error bars, and the bet only has value if the estimate is exactly right. A fair-looking price is usually a polite way of saying a slightly bad one.
The answer
p* 50.0 per cent, EV 0p, no bet.
Scenario 2
An away win you rate at 30 per cent, priced at 5/2.
Break-even probability
Divide the right-hand number by the sum of both: 2 ÷ (5 + 2) gives 2/7, or 28.57 per cent. That is what 5/2 claims, and the line your estimate has to clear.
Expected value per £1
EV = p × (a/b) − (1 − p) = 0.30 × 2.5 − 0.70 = +0.05, so +5p per pound.
Edge and cushion
The edge per £1 staked is the EV: +5 per cent. Your estimate sits 1.43 points above the claim.
Ruling: bet
A bet, provided you trust the 30. The cushion is thin: your number only has to be 1.43 points too high for the value to vanish. Edges this size are real, and they are fragile.
The answer
p* 28.6 per cent, EV +5p, bet.
Scenario 3
Both teams to score, which you rate at 55 per cent, priced at 4/5.
Break-even probability
Divide the right-hand number by the sum of both: 5 ÷ (4 + 5) gives 5/9, or 55.56 per cent. That is what 4/5 claims, and the line your estimate has to clear.
Expected value per £1
EV = p × (a/b) − (1 − p) = 0.55 × 0.8 − 0.45 = −0.01, so −1p per pound.
Edge and cushion
The edge per £1 staked is the EV: −1 per cent. Your estimate sits 0.56 points below the claim.
Ruling: no bet
No bet, and this is the instructive one. 4/5 sits below your 55 per cent by eye and still loses, because the price claims 55.56. Gut feel would have taken it. Arithmetic would not.
The answer
p* 55.6 per cent, EV −1p, no bet.
Scenario 4
A struggling side away at the leaders, which you rate at 8 per cent, priced at 9/1.
Break-even probability
Divide the right-hand number by the sum of both: 1 ÷ (9 + 1) gives 1/10, or 10.00 per cent. That is what 9/1 claims, and the line your estimate has to clear.
Expected value per £1
EV = p × (a/b) − (1 − p) = 0.08 × 9 − 0.92 = −0.20, so −20p per pound.
Edge and cushion
The edge per £1 staked is the EV: −20 per cent. Your estimate sits 2.00 points below the claim.
Ruling: no bet
No bet, and by a distance. The payout is large and the value is poor, which is the longshot problem chapter 5 is built around: you are 2 points short on a 10-point claim, so a fifth of your stake evaporates on average.
The answer
p* 10.0 per cent, EV −20p, no bet.
Scenario 5
A home win you rate at 50 per cent, priced at 11/10.
Break-even probability
Divide the right-hand number by the sum of both: 10 ÷ (11 + 10) gives 10/21, or 47.62 per cent. That is what 11/10 claims, and the line your estimate has to clear.
Expected value per £1
EV = p × (a/b) − (1 − p) = 0.50 × 1.1 − 0.50 = +0.05, so +5p per pound.
Edge and cushion
The edge per £1 staked is the EV: +5 per cent. Your estimate sits 2.38 points above the claim.
Ruling: bet
A bet. Same team, same opinion as scenario 1, a different verdict, because only the price moved. You were never betting on the team; you were betting on the gap.
The answer
p* 47.6 per cent, EV +5p, bet.
These are the workings. The argument for them, and the exercises themselves, are in chapter 4 of the book, What Is Value, Actually?.
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