Setting up Posterior Probability and Bayes Theorem

Question

I would just like to ask some questions with regards to Bayesian probabilities and an exercise I am trying to solve.

Here is the problem:

A footprint was found at the crime scene, which is known to belong to the thief. The footprint is found to match the suspect's shoes. Ignoring the footprint evidence, you judge that the chances of the suspect is being guilty is 1/200 within the local area. If you told that the probability of the foot print matching by chance (i.e. if the suspect were innocent) is 1 in 5000. What is the posterior probability that the suspect is guilty?

I know that: 1/200 is my prior probability and 1/5000 is my evidence(normalisation constant)

How should I set up this problem to find the likelihood function so I can eventually find my posterior probability?

Many thanks

Answer 1

Define your events clearly, and you cannot be led astray.

Let $G$ be the event of guilt. Let $M$ be the event that the print matches. The desired probability, then, is $$\Pr[G \mid M].$$ By Bayes theorem, we have $$\Pr[G \mid M] = \frac{\Pr[M \mid G]\Pr[G]}{\Pr[M]}.$$ We are given that $\Pr[G] = 1/200$. By the law of total probability, $$\Pr[M] = \Pr[M \mid G]\Pr[G] + \Pr[M \mid \bar G]\Pr[\bar G],$$ where $\bar G$ is the complementary event of innocence. We are told that $\Pr[M \mid \bar G]$, the probability of a match given that the suspect is innocent, is $1/5000$. What is $\Pr[M \mid G]$? Think about what it means.