Suppose you are concerned that you have a particular allergy. The allergy is rare – 1% of the population has the allergy – but to ease your worries you get tested for the allergy. The test is 98% reliable, which means two things: 98% of the people who have the allergy and get tested test positive, and 98% of the people who do not have the allergy and get tested test negative. The formal language we consider is built up from a vocabulary containing three sentence letters: a says that you have the allergy, t says that you get tested for the allergy, and p says that you test positive for the allergy. Based on the information above you think the probabilities are as follows

Pr(a)=.001, Pr(t)=1, Pr(p|a∧t)=.98, and Pr(¬p|¬a∧t)=.98

What is the conditional probability that you have the allergy given that you get

tested for the allergy and test positive, Pr(a|p∧t)?