Fixed place odds are calculated from a runner's chance of finishing in the paid places, which depends on the whole field rather than on a set fraction of its win price. Each bookmaker prices its fixed place bets on its own estimate of that chance, with a margin added, so two bookmakers can quote different place prices on one runner. A tote place dividend works differently: the place pool, less commission, is shared among bets on the placegetters and declared after the result.
To check a place price, estimate the chance yourself. The Harville method builds it from every runner's win chance. In the illustrative eight-runner field below, a $5.00 runner has a 56.64% chance of a top-three finish, a fair place price of 1 / 0.5664 = $1.77.
What a place price has to pay out on
A place bet pays if the runner finishes in any paid place, first included. With three places paid, place chance = chance of first + chance of second + chance of third. The fair place price is 1 / place chance, the conversion how betting odds work uses for any price.
Every race fills its paid places, so a field's place chances add up to 300% when three are paid and 200% when two are. A bookmaker's place market adds up to more, and the excess is its margin, the same check market percentage runs on a win market.
Note: How many places pay depends on the rules your bet is settled under. The tote rules gazetted in WA in 2010 pay first to third with 8 or more runners at final scratchings and first and second with 5 to 7, and take no place bets with 4 or fewer. Racing NSW's Rules of Betting pay NSW on-course place bets on the same field sizes, counted when the bet is made (BR 18). Online bookmakers set their own place terms, and place betting sets the rules side by side.
The Harville method: place probability from win odds
A place odds calculation from win chances takes five steps:
- Start from win chances with the margin taken out. From a bookmaker's win market, divide each implied chance by the market's total: in a market totalling 115%, a $5.00 runner is 20% / 1.15 = 17.4%, not 20%. How to remove the bookmaker margin compares methods, and the fair odds calculator removes the margin from a whole field.
- Chance of first = the win chance.
- Chance of second = the sum, over every rival that could win, of the rival's win chance x your runner's chance / (1 - the rival's chance).
- Chance of third = the same over every ordered pair of rivals taking first and second: first's chance x second's chance / (1 - first's) x your runner's chance / (1 - first's - second's).
- Place chance = the paid placings added up, and fair place price = 1 / place chance.
The method is named after David Harville, a statistician who published it in 1973, and the Harville formula sets out its terms. It assumes that once a runner has won, the rest contest second in proportion to their win chances, and so on down the order.
Worked: a $5.00 runner's place chance in an 8-runner field
Take an illustrative eight-runner race with win chances that already add to 100%, from a busy exchange market or a de-vigged bookmaker market. Runner 2 is the $5.00 runner, a 20% chance. Its chance of second, rival by rival:
| If this rival wins | Rival's win chance | Runner 2's share of the rest | Chance of that, then Runner 2 second |
|---|---|---|---|
| Runner 1, $4.00 | 0.25 | 0.20 / 0.75 = 0.2667 | 0.25 x 0.2667 = 0.0667 |
| Runner 3, $6.25 | 0.16 | 0.20 / 0.84 = 0.2381 | 0.16 x 0.2381 = 0.0381 |
| Runner 4, $8.00 | 0.125 | 0.20 / 0.875 = 0.2286 | 0.125 x 0.2286 = 0.0286 |
| Runner 5, $10.00 | 0.10 | 0.20 / 0.90 = 0.2222 | 0.10 x 0.2222 = 0.0222 |
| Runner 6, $13.33 | 0.075 | 0.20 / 0.925 = 0.2162 | 0.075 x 0.2162 = 0.0162 |
| Runner 7, $20.00 | 0.05 | 0.20 / 0.95 = 0.2105 | 0.05 x 0.2105 = 0.0105 |
| Runner 8, $25.00 | 0.04 | 0.20 / 0.96 = 0.2083 | 0.04 x 0.2083 = 0.0083 |
| Chance of second | 0.1906 |
Third place runs the same sum over every ordered pair of rivals that could take first and second, 42 pairs in this field. One of them, Runner 1 first and Runner 3 second, gives 0.25 x (0.16 / 0.75) x (0.20 / 0.59) = 0.0181. Added over all 42 pairs in a spreadsheet, Runner 2's chance of third is 0.1758. So its place chance is 0.2000 + 0.1906 + 0.1758 = 0.5664, and its fair place price is 1 / 0.5664 = $1.77.
Every runner, worked the same way:
| Runner | Win price (chance) | Top-three chance | Fair place price |
|---|---|---|---|
| Runner 1 | $4.00 (25%) | 64.95% | $1.54 |
| Runner 2 | $5.00 (20%) | 56.64% | $1.77 |
| Runner 3 | $6.25 (16%) | 48.37% | $2.07 |
| Runner 4 | $8.00 (12.5%) | 39.82% | $2.51 |
| Runner 5 | $10.00 (10%) | 32.97% | $3.03 |
| Runner 6 | $13.33 (7.5%) | 25.53% | $3.92 |
| Runner 7 | $20.00 (5%) | 17.53% | $5.70 |
| Runner 8 | $25.00 (4%) | 14.18% | $7.05 |
The top-three chances add up to 300%, one full share for each place (299.99% as rounded here), which is the check that the sums are right.
Harville's bias: too much of second and third for favourites
Harville's weak spot is the minor placings. Runners with little chance of winning still fight out third, so Harville tends to give favourites too much of second and third and long shots too little.
To see the size of it, flatten the win chances before sharing out second and third. Raise each to the power 0.75 (an illustrative figure) and use those, rescaled among the runners left, for second and third only.
| Runner | Top-three chance, Harville | Flattened (illustrative) | Fair place price, Harville vs flattened |
|---|---|---|---|
| Runner 1, $4.00 | 64.95% | 60.29% | $1.54 vs $1.66 |
| Runner 2, $5.00 | 56.64% | 53.35% | $1.77 vs $1.87 |
| Runner 8, $25.00 | 14.18% | 17.43% | $7.05 vs $5.74 |
The right amount of flattening has to be measured on race results, so the second column shows the direction, not a corrected price. Treat Harville as a first check, least reliable at the ends of the market, and give more weight to a place market with real money in it. Any price built from win chances alone, such as a same race multi priced that way, shares the assumption.
Fixed place prices against the exchange place market
Three kinds of place price sit side by side:
- A fixed place price is the bookmaker's own estimate plus its margin, locked in when the bet is accepted.
- An exchange place market lets punters back and lay each runner to place, for a set number of places, with commission on net winnings. A busy one is a direct estimate with no model in it; a quiet one can show a wide gap between back and lay.
- A tote place dividend is declared after the result, so it cannot be checked in advance, as tote dividends explains.
For Runner 2, with illustrative prices, check a fixed place price against every estimate:
| Estimate | Place price | Implied place chance |
|---|---|---|
| Bookmaker's fixed place price | $1.70 | 1 / 1.70 = 58.82% |
| Harville | $1.77 | 56.64% |
| Flattened Harville | $1.87 | 53.35% |
| Exchange 3-place market, back $1.84 and lay $1.90 | Midpoint $1.87 | 1 / 1.87 = 53.48% |
The fixed $1.70 implies a higher place chance than any estimate gives Runner 2, so it is the poorer price. At the exchange's 53.48% it returns 0.5348 x 1.70 = 0.909 per dollar, about 9 cents below fair. Commission trims the exchange price too, so at an illustrative 5% rate a back bet at $1.84 pays like 1 + 0.84 x 0.95 = $1.80. How those markets pay is the subject of exchange place markets.
Risk: Betting involves risk. Every place estimate here is a model's or a market's view, not a known chance, and a place bet that misses the places by a head still loses all of its stake. See responsible gambling for limits and support.
The same $5.00 runner in three different fields
The same $5.00 runner, a 20% chance, gets a different place chance in each field. With illustrative fields and Harville:
| Field | Places paid | Runner's place chance | Fair place price | Fraction of the win odds |
|---|---|---|---|---|
| Eight runners, the worked example | 3 | 56.64% | $1.77 | A quarter: $2.00 |
| Eight runners with a $2.00 favourite | 3 | 69.49% | $1.44 | A quarter: $2.00 |
| Seven runners | 2 | 39.38% | $2.54 | A third: $2.33 |
The field with a $2.00 favourite has win chances of 50%, 20%, 10%, 7%, 5%, 4%, 2.5% and 1.5%, and the seven-runner field 26%, 20%, 17%, 13%, 11%, 8% and 5%. Their fair place prices are 1 / 0.6949 = $1.44 and 1 / 0.3938 = $2.54.
A fixed fraction ignores who else is running. For NSW on-course bookmakers, BR 17 of Racing NSW's Rules of Betting sets each way terms. With 8 or more starters when the bet is made, the place half pays a quarter of the win odds on the first three: 1 + (5.00 - 1) / 4 = $2.00. With 5 to 7 at that time it pays a third of the odds on the first two, $2.33. That place half is worth a different amount in each race, and each way betting strategy covers when that matters.
Mistakes when working out place odds
| Mistake | What it costs (illustrative) | The fix |
|---|---|---|
| Feeding Harville a bookmaker's raw implied chances | They add to more than 100%, so every place chance comes out too high | Divide by the market total first, then check the place chances add to 300% or 200% |
| Taking a quarter of the win odds as the fair place price | $2.00 for Runner 2 in both eight-runner fields, against fair prices of $1.44 and $1.77 | Work the place chance from the field |
| Trusting Harville at the long end | Runner 8 is $7.05 by Harville and $5.74 flattened, so a fixed $6.50 looks short by one and generous by the other | Lean on a busy exchange place market for long shots |
Win prices for Harville on B337's Terminal
Harville needs a sound win chance for every runner, not just yours. The Terminal shows prices from 40+ bookmakers across racing and sports, runner by runner, with the best available price picked out and exchange back and lay prices beside them. For racing, its EV overlays judge value against the exchange's own market, as an estimate.
The Terminal reads prices on a repeating cycle, so check a price in your bookmaker account before you bet; a screen price can trail the bookmaker's own. A free account opens a limited view of the Terminal with live odds.
Risk: Betting involves risk. A fair place price is an estimate built on other estimates, prices move before a bet is accepted, and there is no guarantee of profit. See responsible gambling for limits and support.