Odds and Chances in Horse Race Betting

Let there be $n$ horses in the race. If the better bets $\displaystyle \frac{a_i}{a_i+b_i}:\frac{b_i}{a_i+b_i},$ for $(i=1,2,\ldots,n),$ against the $i^{th}$ horse winning, then if the $k^{th}$ horse does win, the better wins $\displaystyle \frac{a_k}{a_k+b_k}$ and loses

$\displaystyle \sum_{i=1,i\ne k}^{n}\frac{b_i}{a_i+b_i}=\sum_{i=1}^{n}\frac{b_i}{a_i+b_i}-\frac{b_k}{a_k+b_k}.$

Therefore, the better's winnings are

$\displaystyle \frac{a_k}{a_k+b_k}-\left(\sum_{i=1}^{n}\frac{b_i}{a_i+b_i}-\frac{b_k}{a_k+b_k}\right)=1-\sum_{i=1}^{n}\frac{b_i}{a_i+b_i},$

which is positive and the same whatever horse wins.

1. J. G. McLoughlin et all, Jim Totten's Problems of the Week, World Scientific, 2013, problem #313.