Nathan had an infection, and his doctor wanted him to take penicillin. because nathan’s father and paternal grandfather were allergic 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

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