A two-way market quotes the favourite at decimal odds of 1.25 and the outsider at 4.5. Remove the overround proportionally and give the fair probability of the favourite as an exact fraction.

A two-way market quotes the favourite at decimal odds of 1.25 and the outsider at 4.5. Remove the overround proportionally and give the fair probability of the favourite as an exact fraction.

Approach: Invert each decimal price into a raw implied probability, add them to size the margin, then rescale both legs by the reciprocal of that total.

18/23. Decimal odds invert to implied probability, so 1/1.25 = 4/5 and 1/4.5 = 2/9. They sum to 36/45 + 10/45 = 46/45, which is 102.22%, so the overround is 2.22% and that excess is the operator margin. Removing it proportionally scales each leg by 45/46, so the fair probability of the favourite is (4/5) * (45/46) = 18/23. That is 78.26%, and the outsider takes the remaining 5/23, or 21.74%. Proportional removal is a convention rather than a fact about the book. A real two-way market usually loads more margin onto the outsider, so the true fair probability of the favourite sits a little below 78.26%.

Follow-up: The same book quotes a third outcome at 26.0. What is the overround now, and what is the fair probability of the favourite?

Key concepts: decimal odds, implied probability, overround.