# Guaranteed Winning Strategy on Horse Betting Odds

Suppose four horses - $$A, B, C$$, and $$D$$ - are entered in a race and the odds on them, respectively, are $$5$$ to $$1$$, $$4$$ to $$1$$, $$3$$ to $$1$$, and $$2$$ to $$1.$$ If you bet $$\1$$ on $$A$$, then you receive $$\6$$ if $$A$$ wins, or you realize a net gain of $$\5$$. You lose your dollar if $$A$$ loses. How should you bet your money to guarantee that you can always make money no matter which horse wins?

I found this question from Horse Betting Odds - But Guaranteed Win! and made some revision.

I think if I bet my money on A, B, C, and D under the following scenario, such as A:B:C:D = 20:15:12:10. I can always make money. These numbers come from the ratios 1/3 : 1/4 : 1/5 : 1/6 multiplied by 60， where 60 is the lowest common multiple for 3,4,5, and 6.

However, I wonder if there is a formal mathematical way to derive this solution. Is this question related to solving the linear inequality system or the probability?

• Oct 28 '20 at 18:41

Each of the horses has a certain return when they win, A is a $$6\times$$ return, B is $$5\times$$, C $$4\times$$, and D $$3\times$$. In general say you have a set of horses $$H_i$$ that each have a return $$r_i$$, and you bet some total amount $$B$$ with $$B_i$$ on each horse $$H_i$$. As long as $$B_i \ge \frac{B}{r_i}$$ for all $$i$$, you will not lose money. That is because one of the $$H_i$$ will win, and that bet will give you $$B_ir_i \ge B$$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum $$B_ir_i$$, as $$B_ir_i-B$$ is your net profit. This will happen when each $$B_ir_i$$ is equal, which occurs when $$B_i = \frac{\frac{B}{r_i}}{\sum \frac{1}{r_i}}$$ and your guaranteed profit is $$\frac{B}{\sum \frac{1}{r_i}} - B$$ This is why in a real horse race $$\sum \frac{1}{r_i}$$ will always be greater than $$1$$ (if for some reason it's not go make a big bet!). In this question it was only $$.95$$ allowing you to make a nice little profit!
• Thanks for your explanation. One more question is that why when each $B_ir_i$ is equal, then the profit is maximized? Jan 4 '19 at 10:51
• @BrattSwan If they are not equal, say $B_ir_i >B_jr_j$, then you make more money if $H_i$ wins and less if $H_j$ does. So to maximize your guaranteed winnings no matter what the outcome it, you would want to move some of that $B_i$ money into $B_j$. Jan 4 '19 at 11:07