Question

3) Consider the following hypothetical facts: “One percent of people in the world are rational. We have a test for rationality. If someone is rational, they have a 60% chance of passing the rationality test. If someone is irrational, they have a 40% chance of passing the rationality test. Adam was just given the test, and he passed.”

(a) Assume that Adam was drawn randomly from the world population. What is the probability that he is truly rational?

(b) Predict the responses of a population of naive subjects who are asked to estimate the probability of Adam’s rationality, given the information above. Justify your answer. Describe the kinds of errors that they are likely to make.

Answer 1

P(r) = 0.01 p(ir) = 0.99

p(p/r) = 0.6 p(P=p/ir) = 0.4

Here, r = rational population in worls; ir = irrational population or other than rational population in worls

p= positive test result that you are rational n = negative test result that you are rational i.e., you are irrational

a) Given that Adam has passed the test, probability that he is actually rational is denoted by p(r/p)

p(p) = = 0.01*0.6 + 0.4*0.99 = 0.402

p(r/p) = = 0.01*0.6/0.402 = 0.015

Therefore, the probability that Adam is actually rational = 0.015

b) Naive people would mistake that the probability that Adam is rational is 40% or 0.4 as conditional probability might not be considered, this is the error that can happen