The Harville formula estimates the chance of every finishing order in a race from each runner's chance of winning. It assumes that once the winner is known, the others fill second place in proportion to their own win chances, so for two runners i and j with win chances pi and pj: P(i first, j second) = pi x pj / (1 - pi).
Feed it fair win chances, with the bookmaker's margin taken out as how betting odds work explains.
The Harville formula, worked
With illustrative fair win chances for a five-runner race: A 40%, B 25%, C 18%, D 10% and E 7%.
| Order | Working | Chance | Fair price |
|---|---|---|---|
| A first, B second | 0.40 x 0.25 / (1 - 0.40) | 16.67% | $6.00 |
| B first, A second | 0.25 x 0.40 / (1 - 0.25) | 13.33% | $7.50 |
| A and B in either order | 16.67% + 13.33% | 30.00% | $3.33 |
| A, B then C | 0.40 x (0.25 / 0.60) x (0.18 / 0.35) | 8.57% | $11.67 |
For a third runner k, multiply the two-place chance by pk / (1 - pi - pj): here 0.18 / (1 - 0.40 - 0.25) = 0.18 / 0.35. Each fair price is 1 / chance.
Harville place probability
A runner's chance of running second sums pj x pi / (1 - pj) over every other runner j that could win. For A:
0.25 x 0.40 / 0.75 + 0.18 x 0.40 / 0.82 + 0.10 x 0.40 / 0.90 + 0.07 x 0.40 / 0.93 = 0.13333 + 0.08780 + 0.04444 + 0.03011 = 29.57%
If two places are paid, A's place chance is 40.00% + 29.57% = 69.57%, a fair place price of 1 / 0.6957 = $1.44. E's second-place sum, 0.40 x 0.07 / 0.60 + 0.25 x 0.07 / 0.75 + 0.18 x 0.07 / 0.82 + 0.10 x 0.07 / 0.90 = 9.31%, makes its place chance 7.00% + 9.31% = 16.31%, a fair $6.13.
With three places paid, add each runner's chance of running third the same way, and place odds explained works a full field.
Using Harville for place odds and trifecta pricing
Set a Harville fair price against a fixed place price, or a quinella, exacta or trifecta approximate on the tote. Approximates move until the pool closes and a dividend is declared only after the result.
Harville can price every combination an exotic covers, and trifecta strategy covers which to include; the flexi calculator counts the combinations in a box, standout or banker bet and turns your outlay into a flexi percentage.
Risk: Betting involves risk. A Harville fair price is an estimate built on another estimate, the win chances, and an exotic bet can lose every combination it covers. See responsible gambling for limits and support.
Harville's bias towards favourites in the minor places
Benter's 1994 report, on Hong Kong races run from September 1986 to June 1993, found that runners with a high win chance finish second and third less often than Harville predicts, and runners with a low one more often, with the gap larger for third.
So in the example, A's $1.44 place price is probably too short and E's $6.13 too long, and exotic orders with the favourite in second or third come out too likely.
Reading win prices on the Terminal
B337's Terminal shows prices from 40+ bookmakers across racing and sports, with racing win prices lined up runner by runner beside exchange back and lay. The board shows prices, not chances: convert each to a fair chance first. Prices are read on a repeating cycle, so a fast market can run ahead of the screen: check before you bet. A free account opens a limited view of the Terminal with live odds.
Risk: Betting involves risk. A price on the board is not a forecast, and a runner can lose at any price. See responsible gambling for limits and support.