Consider the following events
P: The report comes positive that is the report tells that a person has a disease.
D: The person has the disease.
According to the given conditions in the problem we have P ( P | D' ) = 3 / 100 [ D' is the complement of D that is the person don't have any disease]
and P ( P' | D ) = 2/100 [P' is the complement of P that is the result is negative]
Here we have to find P (D | P) = P(D ∩ P) / P (P) = P (P ∩ D) / P (P) = ( P (P|D) * P(D) ) / P(P) ..................(1)
Now, it is given that P (P' | D) = 2/100
=> 1 - P (P|D) = 2/100
=> P (P|D) = 1- 2/100 = 98/100 ..................(2)
Again the probability that a person has a disease is 1/200 that is P(D) = 1/200 ..............(3)
Hence P(D') = 199/200
Now, P (P) = P (D) * P(P|D) + P (D') * P(P|D')
Hence P (P) = 1/200 * 98/100 + 199/200 * 3/100 = 695/ ( 2* 10^{4} ) .....................(4)
Now, putting the values from 2,3 and 4 in 1 we get
So, P(D|P) = (98/100 * 1/200 ) / ( 695 / ( 2* 10^{4} ) = 98/695.
Thus the probability Rahul has the disease is 98/695.