Bayes Rule

1. The English Speaking Traveller (from Pao). Suppose you are visiting a country in which:
   • One out of 10 natives speaks English
   • One out of five persons is a tourist
   • One out of two tourists speak English.
   You meet another person and converse in English. Would you deduce that they are native? ( Use Bayes rule to compute the probability that they are native.)

2. In a city of a million inhabitants, a murder is committed. You are accused of the crime, although you are innocent. Some hair is found at the scene which the genetic fingerprinting expert claims matches your DNA, although you have never been there. During the trial, the prosecutor asks this expert, ``what is the possibility of getting this same DNA signal from someone other than the defendant?'' The expert replies, ``the probability of getting this DNA signal given that the hair is not from the defendant's head is one in a million.'' You turn to your lawyer who makes the motion of a knife across the throat with his finger. Express the information provided by the expert as a conditional probability. What is the conditional probability that you committed the crime, given his evidence? Use Bayes rule to compute it and get yourself acquitted.