# odds ratio: the purpose and interpretation

Could someone please explain the purpose of the odds ratio and how it could be interpreted (i.e., the origin of their usage is in primary interest for me)? Why don't people simply use the difference between two proportions instead of odds ratio? Reference to the proper mathematical book will be great.

Odds ratio for a success rate of $p$ is defined as $\frac{p}{1-p}$. A nice feature of this ratio is the plain English explanation you can do with it in a betting scenario. As an example, assume $p = 0.8$. Your odds ratio is $\frac{0.8}{1 - 0.8} = 4$. So you could say you are four times more likely to win than loose or vice versa. Lots of horse race betting (and their pay off) also works in the same way.