A book you cannot lose
A race has three runners, and a bookmaker offers odds of evens on the first, 3 to 1 against the second and 4 to 1 against the third. At odds of to 1, a winning stake returns , stake included; a losing stake is lost.
You split a total stake of 1 dollar across the three runners so as to make your profit as large as possible whatever the result. What is that guaranteed profit?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Odds of to 1 correspond to an implied probability . Add up the three implied probabilities.
Choose the stakes so that every result returns the same amount.
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Idea. The implied probabilities add up to less than 1, so the odds are too generous overall. Staking in proportion to the implied probabilities makes every outcome pay the same, and that amount is more than the total staked.
Step 1: implied probabilities.
Step 2: equal returns. To return the same amount whatever happens, stake on each runner: , , . These add up to , and the total stake is 1, so The stakes are , and .
Step 3. Every result returns , so the guaranteed profit is dollars per dollar staked.
Why equal returns is best. The guaranteed profit is the smallest of the three returns minus 1. If one return were larger than the others, moving a little stake from that runner to the others would raise the smallest return. So at the optimum all three are equal.
In practice. Bookmakers make the implied probabilities add up to more than 1 (the excess is their margin), so a single bookmaker does not offer this. Mismatched prices across bookmakers can.