Fair odds and implied probability

Odds formats in two lines

Fractional odds show profit against stake: 5/1 returns £5 profit for every £1 staked, plus the stake back. Decimal odds show the total return per £1, stake included: 5/1 is 6.0.

To convert fractional to decimal, divide and add one: 5/1 → 5 + 1 = 6.0; 4/6 → 0.667 + 1 = 1.67; evens (1/1) → 2.0. Decimal odds make everything below easier, which is why exchanges use them.

Implied probability

Every price carries a probability: the chance of winning at which that price would be exactly fair — no profit or loss for either side in the long run.

This is the single most useful conversion in racing. It turns "5/1" from a payout into a statement — "this price assumes about a one-in-six chance" — that you can agree or disagree with.

The bookmaker's margin (overround)

If you add up the implied probabilities of every runner in a race, the total should be 100% — exactly one horse wins. In a bookmaker's market it is always more. The excess is the overround: the bookmaker's built-in margin.

Margins vary. They tend to be bigger in large fields than small ones, and bigger with bookmakers than on a busy exchange — where the back and lay prices sit close together and the cost is mostly the gap between them, plus commission on winnings.

From market prices to fair odds

To estimate the market's margin-free view, remove the overround. The simplest method is to divide each implied probability by the book total:

This spreads the margin evenly across the field. In practice it is not spread evenly: longshots tend to be priced proportionally shorter than favourites (the favourite–longshot bias), so a proportional method probably overstates the outsiders' fair chances a little and understates the favourite's. More elaborate methods exist; all of them are approximations.

Your own fair odds

The same arithmetic works in reverse. If you estimate a horse's chance, fair odds = 1 ÷ your probability. Think a horse has a 25% chance? Fair odds are 1 ÷ 0.25 = 4.0 (3/1). Any price longer than that is, on your estimate, better than fair.

That comparison — your probability against the price — is the basis of value. The weak link is always the probability. A model's or a person's "fair price" is an estimate with its own error, not a discovered truth, and the market price it is compared with was made by a lot of people with a lot of information.

Model probability and market probability

A model that outputs probabilities can be checked in two separate ways, and it is worth keeping them apart:

• Calibration. Of all the horses it gives a 20% chance, do about 20% win? A calibrated model's numbers mean what they say.
• Beating the market. Are its probabilities more accurate than the market's, by enough to overcome the margin? Most are not. A model can be perfectly calibrated and still no better than the prices.

Calibration is not profit. If someone publishes fair odds, the question to ask is not whether they sound sensible, but how they have compared with the starting price over a large sample.

Common mistakes

• Forgetting the margin. Comparing your 20% with a price implying 20% ignores that the price includes the bookmaker's cut.
• Treating fair odds as the true odds. Stripping the margin gives the market's view without its cut — not the horse's real chance.
• Ignoring commission. On an exchange, 5% commission on winnings turns 5.0 into an effective 4.8.
• False precision. "Fair odds 4.34" is a number from arithmetic; the uncertainty around it is far wider than two decimal places.

What TRF shows — and does not

The Value board works out a margin-free price for each runner from the Betfair Exchange market, stripping the margin across the whole field and skipping races where too little of the field can be priced. It then lists bookmaker prices that are longer than that fair price. That is arithmetic between two markets — no prediction model is involved — and the list is published unvalidated.

TRF does not publish its own model probabilities or "AI fair odds". The AI Score on each runner is a 0–100 ranking of the runners in a race, not a probability, and the methodology page explains what goes into it. Our performance page shows how well the score ranks runners, band by band — and says plainly that it tracks the market rather than beating it.

For working out returns at any price, including each-way and Rule 4 deductions, use the bet calculator.