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).
$P(A|B) = P(B|A)\frac{P(A)}{P(B)}$
$P(\neg A|B) = P(B|\neg A)\frac{P(\neg A)}{P(B)}$
$\frac{P(A|B)}{P(\neg A|B)} = \frac{P(B|A)}{P(B|\neg A)}\frac{P(A)}{P(\neg A)}$
Thus, in order to find the posterior odds ratio $\frac{P(A|B)}{P(\neg A|B)}$, one simply multiplies the prior odds ratio $\frac{P(A)}{P(\neg A)}$ by the likelihood ratio $\frac{P(B|A)}{P(B|\neg A)}$.
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:
a : b = p : (1 − p) $p = \frac{a}{a+b}$
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%.