In other words, given odds of $~$(x : y)$~$ we can scale it by any positive number $~$\\alpha$~$ to get equivalent odds of $~$(\\alpha x : \\alpha y).$~$ When there are only two terms $~$x$~$ and $~$y$~$ in a set of odds, the odds can be written as a ratio $~$\\frac{x}{y}.$~$ It is understood that the odds ratio $~$\\frac{x}{y}$~$ refers to the odds $~$(x : y),$~$ or, equivalently, $~$\\left(\\frac{x}{y} : 1\\right).$~$

The $~$x/y$~$ notation is confusing - these ratios aren't probabilities are they?