Question

Brightly electrical Company has three factories (𝐴,𝐵, 𝐶) that manufacture desk lamps. A quality control management knows that 35% of lamps are produced by factory 𝐴, 35% by factory 𝐵 and the other 30% by factory 𝐶. He also knows that, generally, 1.5% of desk lamps from factory 𝐴 are defective while 1% 𝑓𝑟𝑜𝑚 𝐵 are defective and 2% from factory 𝐶 are defective.

i.What is the probability that a lamp chosen at random is defective?

ii. If a randomly selected lamp is defective, what is the probability that the lamp was manufactured in factory 𝐶? 

Answer 1

Let $A$ denotes the event that lamp is produced by factory $𝐴,$

$B$ denotes the event that lamp is produced by factory $B,$

$C$ denotes the event that lamp is produced by factory $C.$

Let $D$ denotes the event that lamp is defective.

Given $P(A)=0.35, P(B)=0.35, P(C)=0.3,$

$P(D|A)=0.015, P(D|B)=0.01, P(D|C)=0.02.$

i. By the Law of Total Probability

$$P(D)=P(A)P(D|A)+P(B)P(D|B)$$

$$+P(C)P(D|C)$$

$$=0.35(0.015)+0.35(0.01)+0.3(0.02)$$

$$=0.01475$$

ii. By the Bayes' Theorem

$$P(C|D)$$

$$=\dfrac{P(C)P(D|C)}{P(A)P(D|A)+P(B)P(D|B)+P(C)P(D|C)}$$

$$=\dfrac{0.3(0.02)}{0.35(0.015)+0.35(0.01)+0.3(0.02)}$$

$$=\dfrac{0.006}{0.01475}=\dfrac{24}{59}\approx0.40678$$