Chapter 16 · Worked solutions
Chapter 16 workings: Poisson and Chips
Chapter sixteen worked: the away side's +0.5 and +1.5 handicaps, settling a £40 quarter-line slip, and repricing the match when the home striker drops out.
All three exercises use the specimen match from the chapter, with λ = 2.2425 for the home side and μ = 1.188 for the away side, both kept unrounded. Every probability below is a sum of cells from the same correct-score grid, built to 25 goals a side so nothing measurable is left out.
The Poisson predictor builds the same grid for any pair of numbers; the chapter 16 example is its first preset.
| Goal margin | Probability |
|---|---|
| home by 2+ | 38.86% |
| home by 1 | 22.31% |
| draw | 19.62% |
| away by 1 | 11.82% |
| away by 2+ | 7.38% |
The exercises
The away side at +0.5 and +1.5
Price the visitors' +0.5 Asian handicap line, then their +1.5 line.
Turn each line into a result
Give the away side half a goal and the bet wins on any result except a home win; there is no push, because no whole-number score lands on a half. Give them a goal and a half and the bet survives a one-goal home win as well.
Read the +0.5 off the margin table
Draw plus every away win = 19.62 per cent + 19.21 per cent = 38.83 per cent.
Add one more row for +1.5
Now a home win by exactly one also counts: 38.83 per cent + 22.31 per cent = 61.14 per cent.
Price both
Fair decimal odds are one divided by the probability: +0.5 at 2.58, about 13/8 (the book says close to 8/5), +1.5 at 1.64, about 8/13. The second is almost exactly the home win price turned round, because the two events are near mirror images on this grid.
The answer
+0.5 wins 38.8 per cent of the time, fair about 8/5; +1.5 wins 61.1 per cent, fair about 8/13.
Settling £40 on home −0.75 at 5/6
Settle a £40 bet on the home side at −0.75, struck at 5/6, first for a 2-0 finish and then for 1-0.
Split the stake
A quarter line is two bets: £20 on −0.5 and £20 on −1, both at 5/6.
2-0: a two-goal win
−0.5 wins (they won). −1 also wins, because 2-0 minus one goal is still a win. Profit = 2 × £20 × 5/6 = £33.33; return = £73.33.
1-0: a one-goal win
−0.5 wins: £20 × 5/6 = £16.67 profit. −1 leaves an adjusted 0-0, so that £20 comes back untouched. Return = £20 + £16.67 + £20 = £56.67.
Anything else
A draw or an away win loses both halves, and the whole £40.
The answer
2-0 returns £73.33 (£33.33 profit); 1-0 returns £56.67 (£16.67 profit).
The striker is out
Cut the home side's expected goals by 15 per cent for a late injury, leave μ alone, and reprice home, draw and away.
Adjust the input, not the output
New λ = 2.2425 × 0.85 = 1.9061. μ stays at 1.188.
Rebuild the grid
Every home-goals probability changes, so the whole grid is rebuilt and summed again, below the diagonal, on it and above it.
Compare
The table shows the move. About seven points leave the home column and are shared between the draw and the away win.
| Result | Before | After | Change | Fair price after |
|---|---|---|---|---|
| Home | 61.17% | 54.13% | −7.0 pts | 1.85, about 5/6 |
| Draw | 19.62% | 22.29% | +2.7 pts | 4.49, about 7/2 |
| Away | 19.21% | 23.58% | +4.4 pts | 4.24, about 10/3 |
The answer
Home 54.1 per cent, draw 22.3 per cent, away 23.6 per cent: roughly 6/7, 7/2 and 13/4.
These are the workings. The argument for them, and the exercises themselves, are in chapter 16 of the book, Poisson and Chips.
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