Bayes' Theorem (odds)

This version rewrites Bayes’ Theorem for two conditional probability and to get the odds on the posterior hypothesis :

in which

• is the odds on the prior
• is the likelihood ratio between two likelihood functions

Likelihood < 1 odds of posterior < odds of prior hypothesis. In legal evidence where hypothesis represents innocence, it means that new evidence decreases the odds of innocence.

Example

The butler & the knife

Inspector Clouseau arrives at the scene of a crime where a victim lies dead in the kitchen alongside the possible murder weapon, a knife (K). The butler (B) is the inspector’s main suspect. The inspector’s prior criminal knowledge of the situation leads him to estimate the probability that the butler is the murderer is P(B) = 0.6. He also estimates the probability of observing this evidence (K = knife was used) given the butler committed the murder, P(K|B) = 0.5, and given the butler did not commit the murder, P(K|~B) = 0.3.

1. What is the marginal probability that the knife was used, P(K)? [0.42]
2. What is the probability that the butler is the murderer given the knife was used, P(B|K)? [0.714]
3. What are the odds on the prior hypothesis, P(B)/P(~B), and what does it imply for this situation? [1.5]
4. What is the likelihood ratio, P(K|B)/P(K|~B), and what does it imply for this situation? [1.67]
5. What are the odds on the posterior hypothesis, P(B|K)/P(~B|K), and what does it imply for this situation? [2.5]

Smoking & cancer

(Last problem) According to the CDC (Centers for Disease Control and Prevention), men who smoke are 2323 times more likely to develop lung cancer than men who don’t smoke. Also according to the CDC, 21.6%21.6% of men in the U.S. smoke. What is the probability that a man in the U.S. is a smoker, given that he develops lung cancer?