# Odds

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Odds ratios are an alternate way of expressing probabilities, which simplifies the process of updating them with new evidence. The odds ratio of A is P(A)/P(¬A).

${\displaystyle P(A|B)=P(B|A){\frac {P(A)}{P(B)}}}$

${\displaystyle P(\neg A|B)=P(B|\neg A){\frac {P(\neg A)}{P(B)}}}$

${\displaystyle {\frac {P(A|B)}{P(\neg A|B)}}={\frac {P(B|A)}{P(B|\neg A)}}{\frac {P(A)}{P(\neg A)}}}$

Thus, in order to find the posterior odds ratio ${\displaystyle {\frac {P(A|B)}{P(\neg A|B)}}}$, one simply multiplies the prior odds ratio ${\displaystyle {\frac {P(A)}{P(\neg A)}}}$ by the likelihood ratio ${\displaystyle {\frac {P(B|A)}{P(B|\neg A)}}}$.

Odds ratios are commonly written as the ratio of two numbers separated by a colon. For example, if P(A) = 2/3, the odds ratio would be 2, but this would most likely be written as 2:1.

The relation between odds ratio, a:b, and probability, p is as follows:

${\displaystyle a:b=p:(1-p)}$ ${\displaystyle p={\frac {a}{a+b}}}$

Suppose you have a box that has a 5% chance of containing a diamond. You also have a diamond detector that beeps two thirds of the time if there is a diamond, and one third of the time if there is not. You wave the diamond detector over the box and it beeps.

The prior odds of the box containing a diamond are 1:19. The likelihood ratio of a beep is 2/3:1/3 = 2:1. The posterior odds are 1:19 * 2:1 = 2:19. This corresponds to about a probability of 2/21, which is about 0.095 or 9.5%.