# 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.