Odds: Refresher


by Nate Soares Jul 6 2016 updated Oct 13 2016

A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).

Let's say that, in a certain forest, there are 2 sick trees for every 3 healthy trees. We can then say that the odds of a tree being sick (as opposed to healthy) are $~$(2 : 3).$~$

Odds express relative chances. Saying "There's 2 sick trees for every 3 healthy trees" is the same as saying "There's 10 sick trees for every 15 healthy trees." If the original odds are $~$(x : y)$~$ we can multiply by a positive number $~$\alpha$~$ and get a set of equivalent odds $~$(\alpha x : \alpha y).$~$

If there's 2 sick trees for every 3 healthy trees, and every tree is either sick or healthy, then the probability of randomly picking a sick tree from among all trees is 2/(2+3):

Odds v probabilities

If the set of possibilities $~$A, B, C$~$ are mutually exclusive and exhaustive, then the probabilities $~$\mathbb P(A) + \mathbb P(B) + \mathbb P(C)$~$ should sum to $~$1.$~$ If there's no further possibilities $~$d,$~$ we can convert the relative odds $~$(a : b : c)$~$ into the probabilities $~$(\frac{a}{a + b + c} : \frac{b}{a + b + c} : \frac{c}{a + b + c}).$~$ The process of dividing each term by the sum of terms, to turn a set of proportional odds into probabilities that sum to 1, is called normalization.

When there are only two terms $~$x$~$ and $~$y$~$ in the odds, they can be expressed as a single ratio $~$\frac{x}{y}.$~$ An odds ratio of $~$\frac{x}{y}$~$ refers to odds of $~$(x : y),$~$ or, equivalently, odds of $~$\left(\frac{x}{y} : 1\right).$~$ Odds of $~$(x : y)$~$ are sometimes called odds ratios, where it is understood that the actual ratio is $~$\frac{x}{y}.$~$