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):

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 \(\frac{x}{y}.\)

Parents:

  • Odds

    Odds ex­press a rel­a­tive prob­a­bil­ity.

    • Probability theory

      The logic of sci­ence; co­her­ence re­la­tions on quan­ti­ta­tive de­grees of be­lief.