Question

Suppose that a factory has two machines, Machine A and Machine B, both producing iPhone touch screens. Forty percent of their touch screens come from Machine A and 60% of their touch screens come from Machine B. Ten percent of the touch screens produced by Machine A are defective and five percent of the touch screens from Machine B are defective. If I randomly choose a touch screen from those produced by both machines and find that it is defective, what is the probability that it came from machine A and also find the probability that it came from machine B ? 

Answer 1

Let: $A$ touch screen produced by Machine A

$B$ touch screen produced by Machine B

$D$ touch screen is defective

Given

$$P(A)=0.4, P(B)=0.6,$$

$$P(D|A)=0.1, P(D|B)=0.05$$

By Baye's Theorem the probability that defective touch screen came from machine A is

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

$$=\dfrac{0.1(0.4)}{0.1(0.4)+0.05(0.6)}=\dfrac{4}{7}$$

The probability that defective touch screen came from machine A is

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

$$=\dfrac{0.05(0.6)}{0.1(0.4)+0.05(0.6)}=\dfrac{3}{7}$$