Question

Q#2

a-

The probability that a doctor correctly diagnoses a particular illness is 0.7. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnosis and the patient sues?

b-

The probability that an automobile being filled with gasoline also needs an oil change is 0.25; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and the filter need changing is 0.14.(a) If the oil has to be changed, what is the probability that a new oil filter is needed?(b) If a new oil filter is needed, what is the probability that the oil has to be changed?

Answer 1

a) We are given here that:
P( correct diagnosis ) = 0.7, Therefore, P( incorrect diagnosis ) = 1 - 0.7 = 0.3
P( lawsuit | incorrect diagnosis ) = 0.9

Using Bayes theorem, we have here:
P( incorrect diagnosis and patient sues ) = P( lawsuit | incorrect diagnosis )P( incorrect diagnosis )

= 0.9*0.3

= 0.27

Therefore 0.27 is the required probability here.

b) We are given here that:
P( gasoline needs oil change ) = 0.25,
P( needs a new oil filter ) = 0.4,
P( both ) = 0.14

a) Given that the oil has to be changed the probability that a new oil filter is needed is computed using Bayes theorem as:
= P(both) / P( gasoline needs oil change )  
= 0.14 / 0.25

= 0.56

Therefore 0.56 is the required probability here.

b) If a new oil filter is needed, the probability that oil has to be changed is computed using Bayes theorem as:
= P(both ) / P( needs an oil filter )

= 0.14 / 0.4

= 0.35

Therefore 0.35 is the required probability here.