Dutching Calculator

The Dutching Calculator splits a total stake across two or three selections so that every covered runner returns the same payout. Defaults: Odds 1 = 3.20, Odds 2 = 4.10, Odds 3 (optional) = 5.50, Total Stake $100. Inverse sum inv = 1/3.20 + 1/4.10 + 1/5.50 = 0.312500 + 0.243902 + 0.181818 = 0.738221. Stake i = (total / odds_i) / inv. That produces $42.33 on 3.20, $33.04 on 4.10, $24.63 on 5.50. Any of those three winning pays $135.46, a $35.46 equal profit, because payout = total / inv = 100 / 0.738221.

The four result rows are Equal Profit $35.46, Stake 1 $42.33, Stake 2 $33.04, Stake 3 $24.63. If you do not want a third runner, set Odds 3 to 0 (or blank it to a non-positive value). Then inv = 1/3.20 + 1/4.10 = 0.556402, stakes $56.16 and $43.84, payout $179.73, equal profit $79.73. The two-way Dutch is more profitable on paper because you are covering less of the market; it also loses whenever a third outcome you ignored wins. The optional 5.50 default is included so the page demonstrates the three-runner case that the fields advertise.

Dutching is not a hedge of an existing ticket. You do not already hold a 3.20 back; you are building a portfolio from cash. If inv ≥ 1 the book is overround across your covered set and equal “profit” goes non-positive: payout ≤ stake. The default 0.738 < 1, so the three prices Dutch. That usually means you left other runners uncovered (a five-horse race, a 1X2 plus other markets, a player-prop field). The $35.46 exists only if one of the three listed prices wins.

Proportionality: the shortest price always takes the largest slice ($42.33 of $100). Raising Odds 1 from 3.20 to 3.50 with the other two fixed lowers inv and lifts the equal profit, but you are then covering a weaker favorite. The calculator does not care which runners they are. It only sees three decimals and a bankroll slice. Type live prices, not morning prices; a 5.50 that is now 4.80 changes s3 and the whole table.

Work the defaults once more as a bookmaker-margin check: implied probabilities 31.25%, 24.39%, 18.18% sum to 73.82%, leaving 26.18% of the probability mass unpriced. If the true chance that none of the three wins is anywhere near 26%, the $35.46 is a statement about the other runners, not a gift. Educational split of $100, not a recommendation to Dutch three horses or three football results.

Dutching is equal-profit covering: several backs, one total stake, the same return whichever covered outcome lands. The name is usually traced to nineteenth-century “Dutch” bookmakers and to the broader Dutch-book argument in probability — a set of prices that can be bet so the book cannot lose. A related popular attribution in racing lore is a layer called Armand who laid off by backing several at weights inverse to their odds. The algebraic object is older than the nickname: stakes proportional to 1/odds, normalized to the budget.

It is the opposite of a parlay (parlay needs every leg) and the cousin of arbitrage (arbitrage covers a complete partition with inv < 1). Most retail Dutches are incomplete partitions: three horses in an eight-horse race, home and draw but not away, two quarterbacks in a large MVP field. Incomplete Dutches can print a large equal profit and still be −EV if the uncovered mass is more likely than the leftover 1 − inv. The calculator will not estimate that mass. It will tell you the stakes that equalize the covered set.

Bookmakers dislike efficient Dutches on the same account because they look like arbing. Splitting $42.33 / $33.04 / $24.63 across three books is the usual workaround and the usual way limits arrive. Pari-mutuel pools (Joseph Oller, 1860s) already implement a market-wide Dutch: the pool is split among winning-ticket holders in proportion to stake, which is inverse to starting-price odds after takeout. Fixed-odds Dutching is you playing Oller with a calculator against posted decimals.

On the default three-way, $35.46 / $100 is a 35.46% return if and only if one of 3.20, 4.10, 5.50 wins. That is a huge “if.” A two-way on 3.20 and 4.10 jumps the paper return to 79.73% and the uncovered mass from 26.18% to 44.36%. There is no free lunch hiding in the optional third field. Adding 5.50 spends $24.63 of the $100 to buy out part of that uncovered mass. Whether that purchase is cheap depends on the true probability of the 5.50, which you do not enter.

Use this page to split a planned $100, not to hedge a ticket you already hold. Decimal odds only. Place terms, rule-4, and dead-heats break equal profit because the book no longer pays the typed decimal in full. The output is a stake vector and a covered-set profit — not a tip, not a system, and not advice to back three correlated football selections that can all lose together.

Equal-profit covering splits a budget across two or three selections in proportion to the inverse of their decimal odds. Sum those inverses. Each stake is then (total stake over that selection's odds) divided by the inverse sum. If a covered runner wins, payout equals total stake over the inverse sum, identical for every included price. Equal profit is that payout minus the total stake. Defaults are odds 3.20, 4.10, and 5.50 with a 100 dollar budget.

Inverse of 3.20 is 0.3125. Inverse of 4.10 is about 0.243902. Inverse of 5.50 is about 0.181818. Sum is about 0.738221. Stake on 3.20 is 31.25 over 0.738221, about 42.331, displayed as 42.33 dollars. Stake on 4.10 is 24.390 over 0.738221, about 33.04 dollars. Stake on 5.50 is 18.182 over 0.738221, about 24.63 dollars. Payout is 100 over 0.738221, about 135.46 dollars. Equal profit is 35.46 dollars. The three stakes sum to 100.00 within rounding. Implied probabilities 31.25 percent, 24.39 percent, and 18.18 percent sum to 73.82 percent, leaving 26.18 percent of the mass unpriced.

Omit the third price and the inverse sum falls to about 0.556402. Stakes become about 56.16 and 43.84 dollars, payout 179.73, equal profit 79.73. The two-way looks richer because you cover less of the market; it also loses whenever an ignored outcome wins. Adding 5.50 spends 24.63 of the 100 to buy out part of that uncovered mass. If the inverse sum is at least one across a complete partition, equal profit is non-positive: you are paying the overround. The default 0.738 sits well below one because other runners remain uncovered.

Nineteenth-century laying-off already used weights inverse to odds; pari-mutuel pools after Oller implement a market-wide version of the same split. Ramsey and de Finetti later named a Dutch book as a set of prices that guarantee a loss to the price-maker. Racing lore attached the nickname Dutching to the retail stake vector. None of that changes proportionality to one over odds. The shortest price always takes the largest slice, here 42.33 of 100.

Assumptions: exactly one of the included selections wins, or any other winner is a total loss of the 100; each bet is a win-or-lose back at the typed decimal; no place terms, no dead-heat, no reduced-odds deduction. If two Dutch legs can win at once you have a middle, and equal profit understates the both-win column. If all three can lose together, the 35.46 dollars is irrelevant because you lose 100. The page prints the covered-set profit; it does not estimate true chances, so it cannot say whether 35.46 is a positive-expectation purchase.