Dutching Calculator — Equal-Profit Stakes for Any Selection

The dutching calculator divides a single total stake across multiple competing selections so that whichever selection wins, you receive the same total return. This technique is called dutching and is widely used in horse racing, football, and any market with three or more outcomes.

To use the calculator:

1. Enter your total stake — the combined amount you want to spread across all selections.
2. Choose the number of selections (2, 3, or 4).
3. Enter the decimal odds for each selection. Use the odds available at the same bookmaker, or mix odds from different books for an arbitrage dutch.

The calculator shows the exact stake for each selection, the guaranteed return if any of those selections wins, and whether the combined implied probability is below 100% (which signals a true arbitrage opportunity where your return exceeds your stake).

All odds must be in decimal format (e.g. 3.50 rather than 5/2 fractional or +250 American). Odds of 2.00 represent even money; odds below 2.00 represent the favourite.

Dutching mathematics relies on the concept of implied probability — the probability implied by a set of odds. Here is the full derivation:

1. Implied probability of each selection:
   p_i = 1 / O_i
   where O_i is the decimal odds for selection i.
2. Total overround (implied probability sum):
   Overround = Σ p_i = Σ (1 / O_i)
   When using a single bookmaker, this number is typically above 1.0 (above 100%), reflecting the book's built-in margin. When shopping across books, you can sometimes find a combination below 1.0, creating an arbitrage opportunity.
3. Stake for each selection:
   Stake_i = S × (p_i / Overround) = S × (1/O_i) / Σ(1/O_j)
   This distributes your total stake S proportionally to each selection's implied probability.
4. Guaranteed return (constant for any winning selection):
   Return = Stake_i × O_i = S × (1/O_i / Overround) × O_i = S / Overround
   Notice the O_i cancels, confirming the return is identical regardless of which selection wins.
5. Guaranteed profit:
   Profit = Return − S = S / Overround − S = S × (1/Overround − 1)
   Positive when Overround < 1 (arb). Negative when Overround > 1 (book edge).

Example 1 — Three-Way Football Market (No Arb)

A bookmaker offers Home Win at 2.40, Draw at 3.40, and Away Win at 2.90. You want to dutch all three outcomes with a $100 total stake.

• Implied probs: 1/2.40 + 1/3.40 + 1/2.90 ≈ 0.4167 + 0.2941 + 0.3448 = 1.0556
• Overround: 105.56% — book has a 5.56% edge
• Home Win stake: $100 × (0.4167 / 1.0556) ≈ $39.48
• Draw stake: $100 × (0.2941 / 1.0556) ≈ $27.86
• Away Win stake: $100 × (0.3448 / 1.0556) ≈ $32.67
• Guaranteed return: $100 / 1.0556 ≈ $94.73 (a guaranteed $5.27 loss)

Even without an arb, dutching is useful here: you guarantee seeing any outcome return $94.73. You have covered the entire market and your maximum loss is capped at $5.27 — identical to the book's built-in margin of 5.27%.

Example 2 — Two-Selection Arbitrage Dutch (Arb)

Book A offers Team X to win at 2.10. Book B offers Team Y to win at 2.10 (in a two-outcome market). You dutch both with $200.

• Implied probs: 1/2.10 + 1/2.10 ≈ 0.476 + 0.476 = 0.952
• Overround: 95.24% — arb exists!
• Stake on each: $200 × (0.476 / 0.952) = $200 × 0.5 = $100 each
• Guaranteed return: $200 / 0.952 ≈ $210
• Guaranteed profit: +$10 (5% ROI)

This is a classic sports arbitrage: the same event is priced at 2.10 on both outcomes at two different books, creating a risk-free $10 profit on a $200 investment.

Example 3 — Horse Racing (4 Selections)

A horse race has four realistic contenders at odds: 3.50, 4.00, 5.00, and 6.00. Total stake: $120.

• Implied probs: 0.286 + 0.25 + 0.20 + 0.167 = 0.903
• Overround: 90.3% — genuine arb! The four contenders represent less than 100% probability
• Total return: $120 / 0.903 ≈ $132.90
• Guaranteed profit: +$12.90

This situation arises in larger fields where the remaining non-dutched runners create the overround. By covering just the top four contenders at the best available prices (across multiple bookmakers), you lock in a profit while accepting the risk that a longer-priced runner wins.