# Difference between revisions of "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). | + | {{arbitallink|https://arbital.com/p/62c/|Bayes' Rule: Odds form}}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). |

<math>P(A|B) = P(B|A)\frac{P(A)}{P(B)}</math> | <math>P(A|B) = P(B|A)\frac{P(A)}{P(B)}</math> | ||

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<math>a:b = p:(1-p)</math> | <math>a:b = p:(1-p)</math> | ||

+ | |||

<math>p = \frac{a}{a+b}</math> | <math>p = \frac{a}{a+b}</math> | ||

− | Suppose you have a box that has a 5% chance of containing a diamond. You also have a diamond detector that beeps | + | 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 2 | + | 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%. |

==See also== | ==See also== |

## Revision as of 02:56, 14 October 2016

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%.