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).
Thus, in order to find the posterior odds ratio , one simply multiplies the prior odds ratio by the likelihood ratio .
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:
Suppose you have a box that has a 5% chance of containing a diamond. You also have a diamond detector that beeps half of the time if there is a diamond, and one fourth 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 1/2:1/4 = 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%.