Chapter 5 · Worked solutions
Chapter 5 workings: The Longshot Tax
The working for chapter five's four exercises: what 5000/1 claims, whether Leicester was value, how badly 100/1 shots are priced, and when a cash-out offer is worth refusing.
Every exercise here comes down to the same move: turn the price into the probability it claims, then compare it with a probability you formed separately. Longshots make the comparison harder to feel, which is exactly why it needs doing on paper.
The exercises
What 5000/1 claims
Turn 5000/1 into a probability, then into “one season in how many”.
Use b ÷ (a + b)
1 ÷ (5000 + 1) = 1/5001 = 0.00019996.
Make it a percentage
Multiply by 100: 0.020 per cent, which rounds to 0.02 per cent.
Make it a season count
One title in every 5,001 seasons. The Premier League would need to run for roughly five thousand years to see it once.
The answer
1/5001, about 0.02 per cent: once in 5,001 seasons.
Was Leicester value?
Decide whether Leicester at 5000/1 was a value bet.
Notice what is missing
Value compares the price's claim with your own probability, formed before the result. The question as asked has only the price and the result in it, so it cannot be answered as it stands.
Ask the answerable version
At what probability would 5000/1 have been value? Anything above 1/5001. The EV of £1 at a true chance p is p × 5001 − 1, and the table shows how quickly that swings.
Keep the verdict and the outcome apart
Suppose your honest August 2015 estimate was better than one in 5,001. Then the bet was good by your own numbers, and it would have been just as good in every imagined season where they finished mid-table.
| True chance | As a percentage | EV per £1 |
|---|---|---|
| 1 in 10,000 | 0.010 per cent | −0.50 |
| 1 in 5,001 | 0.020 per cent | 0.00 |
| 1 in 2,000 | 0.050 per cent | +1.50 |
| 1 in 1,000 | 0.100 per cent | +4.00 |
The answer
Only if your own pre-season chance beat 0.02 per cent. The trophy cannot decide it.
Pricing the 100/1 shot
In the Snowberg and Wolfers data, backing everything at 100/1 or bigger lost around 61 per cent of the money staked. Work out the true winning chance that implies for a 100/1 shot, and set it beside the price's own claim.
Turn the return into pence
Minus 61 per cent means each £1 comes back as 39p on average.
Write the average return
A winner at 100/1 returns £101 for £1. Average return = p × 101, so p × 101 = 0.39.
Solve
p = 0.39 ÷ 101 = 0.00386, which is 0.39 per cent.
Compare with the claim
100/1 claims 1/101, which is 0.99 per cent. 0.0099 ÷ 0.0039 = 2.56, so the price credits the horse with about two and a half times its real chance.
The answer
True chance about 0.39 per cent; the price claims about 0.99 per cent.
The cash-out offer
£50 at 5000/1 would return £250,050 in full. The firm offers £72,335 to settle now. Above what title chance is letting it run the higher-EV choice?
Write both options as expected cash
Cash out: £72,335 for certain. Let it run: p × £250,050, plus nothing in the other branch.
Set them equal
p × 250,050 = 72,335, so p = 72,335 ÷ 250,050 = 0.2893.
Read it
Letting it run has the higher EV whenever the true chance is above 28.9 per cent. The offer is, in effect, the firm's price for buying your ticket back.
Then ask whether EV should decide it
A sum this size can change a life, and a certain £72,335 is worth more to most people than a coin flip at twice the money. That is a fair reason to take less EV, as long as you know the number you are giving up. Chapter 11 goes further.
| True chance | EV of letting it run | Cash now | Higher EV |
|---|---|---|---|
| 20.0 per cent | £50,010 | £72,335 | Cash out |
| 28.93 per cent | £72,335 | £72,335 | Level |
| 40.0 per cent | £100,020 | £72,335 | Ride |
The answer
About 28.9 per cent. Above that, letting it run has the higher EV.
These are the workings. The argument for them, and the exercises themselves, are in chapter 5 of the book, The Longshot Tax.
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