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 $P\left(A\right)+P\left(B\right)+P\left(C\right)$ 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 $(\frac{x}{y}:1).$ Odds of $(x:y)$ are sometimes called odds ratios, where it is understood that the actual ratio is $\frac{x}{y}.$