Nathan had an infection, and his doctor wanted him to take penicillin. Because Nathan’s father and paternal grandfather were all
ergic to penicillin, Nathan had a 75% chance of having the same allergy. The doctor performed a skin test to see whether Nathan would react to it. The test is 98% accurate. What is the probability that Nathan is allergic to penicillin and the test predicts it?
Let events A=Nathan has allergy ~A=Nathan does not have allergy T=Nathan tests positive ~T=Nathan does not test positive
We are given P(A)=0.75 [ probability that Nathan is allergic ] P(T|A)=0.98 [probability of testing positive given Nathan is allergic to Penicillin]
We want to calculate probability that Nathan is allergic AND tests positive P(T n A)
From definition of conditional probability, P(T|A)=P(T n A)/P(A) substitute known values, 0.98 = P(T n A) / 0.75 solving for P(T n A) P(T n A) = 0.75*0.98 = 0.735
