# 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 $$\frac{x}{y}.$$

Parents:

• Odds

Odds express a relative probability.