Question

A random person has a probability of 0.36 of being a descendant of Ghengis Khan. A company advertises a blood test which can tell you if you are a descendant, and it is correct 99% of the time. If you take the test and it comes back negative, what is the probability you actually ARE descended from Ghengis?

Answer 1

We are given here that:
P( descendant ) = 0.36,

Also as the test is correct 99% of the time, therefore we have here:
P( + | descendant) = 0.99,
P( - | descendant) = 1 - 0.99 = 0.01

Also,
P( - | not descendant) = 0.99
P( + | descendant) = 1 - 0.99 = 0.01

Using law of total probability, we have here:
P( - ) = P( - | descendant) P(descendant) + P( - | not descendant) P(not descendant)

P(-) = 0.01*0.36 + 0.99*(1 - 0.36) = 0.6372

Using bayes theorem, we get here:
P(descendant | -) = P( - | descendant) P(descendant) / P(-)

= 0.01*0.36 / 0.6372

= 0.0056

Therefore 0.0056 is the required probability here.