Logan Wyatt

2022-07-10

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?

trantegisis

Expert

Each of the horses has a certain return when they win, A is a $6×$ return, B is $5×$, C $4×$, and D $3×$. 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 ${B}_{i}{r}_{i}\ge B$, so you end up with at least as much as you started. To maximize your guaranteed winnings, you want to maximize the minimum ${B}_{i}{r}_{i}$, as ${B}_{i}{r}_{i}-B$ is your net profit. This will happen when each ${B}_{i}{r}_{i}$ is equal, which occurs when
${B}_{i}=\frac{\frac{B}{{r}_{i}}}{\sum \frac{1}{{r}_{i}}}$
$\frac{B}{\sum \frac{1}{{r}_{i}}}-B$
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!