Question

A manufacturing plant makes radios that each contain an integrated circuit (IC) supplied by 3 different companies A, B and C. The probability that the IC in a radio came from one of the sources is 1 3 , i.e. the same for all the companies. ICs are known to be defective with probabilities 0.001, 0.003 and 0.002 for sources A, B and C, respectively. a) What is the probability that any given radio will contain a defective IC? b) If a radio contains a defective IC, find the probability that it came from company A.

Answer 1

Answer: (a) 0.002 (b)
`(1)/(6)`

Explanation:

Given : The probability that the IC in a radio came from one of the sources =
`P(A)=P(B)=P(C)=(1)/(3)`

`P(D|A)=0.001,\ \ P(D|B)=0.003,\ \ P(D|C)=0.002`

By using the law of total probability :-

`P(D)=P(A)\cdot P(D|A)+P(B)\cdot P(D|B)+P(C)\cdot P(D|C)\\\\\Rightarrow\ P(D)=(1)/(3)\cdot0.001+(1)/(3)\cdot0.003+(1)/(3)\cdot0.002\\\\\Rightarrow\ P(D)=0.002`

Hence, the probability that any given radio will contain a defective IC : 0.002

By using Bayes theorem , we have

`P(A|D)=(P(A)\cdot P(D|A))/(P(D))\\\\\Rightarrow\ P(A|D)=((1)/(3)\cdot0.001)/(0.002)\\\\\Rightarrow\ P(A|D)=(1)/(6)`

Hence, If a radio contains a defective IC, find the probability that it came from company A :
`(1)/(6)`