# Bet to place for profit

Suppose there are $$3$$ runners $$R1, R2$$ and $$R3$$. The betting odds paid on each runner are $$r1:1$$, $$r2:1$$ and $$r3:1$$. This means that if, for example, you bet $$\2$$ for runner 1 winning, you will make $$2((r1+1)/1)-2$$ if runner 1 ends up winning.

The actual probabilities of each runner winning are unspecified. How much should you bet on each runner so that you always end up with a profit?

Is it simply a matter of solving the following system of inequalities (let $$a,b$$ and $$c$$ be the amount you bet for each of the runners) for $$a,b$$ and $$c$$:

$$a(r1+1)-a-b-c >0 \\ b(r2+1)-b-a-c >0 \\ c(r3+1)-c-b-a >0$$

In other words, you should bet an amount on each choice proportional to $$\frac{1}{r_i+1}$$, and you now guarantee the exact payout amount (per original unit of bets) of $$\frac{1}{\frac{1}{r_1+1}+\frac{1}{r_2+1}+\frac{1}{r_3+1}}$$ regardless of which actual choice wins.
Arbitrage exists when the above is greater than 1, that is, when the $$r_i+1$$s average more than 3, where the harmonic mean is appropriate.