Odds: Refresher

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):

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 $\frac{x}{y}.$