Question

A company has 2 types of machines that produce the same product, one recently new and another older. Based on past data, the older machine produces 12% defective products while the newer machine produces 8% defective products. Due to capacity needs, the company must use both machines to meet demand. In addition, the newer machine produces 3 times as many products as the older machine.  

1. Setup a probability table depicting the machine type and defective outcome
2. Given a randomly selected product was tested and found to be defective, what is the probability it was produced by the new machine?
3. What is the probability that any selected product is defective (regardless of machine type)?
4. What is the probability that any selected product is non-defective and produced by the older machine?     

Answer 1

(a)

Let p shows the probability that product is produce by old machine so the probability that product is produce by new machine. is 3p. So there is only 2 machines so we have

p+3p = 1

p = 0.25

Let N shows the event that product is produce by new machine and O shows the event that product is produce by old machine. So we have

P(N) = 3 *0.25 = 0.75

P(O) = 0.25

Let D shows the event that product is defective so

P(D |N) = 0.08

P(D | O) = 0.12

By the complement rule,

P(D' | N) = 1 - P(D | N) = 0.92

P(D' | O) = 1 - P(D | O) = 0.88

Following is completed table:

	O	N	Total
D	0.12 *0.25=0.03	0.08*0.75=0.06	0.09
D'	0.88*0.25=0.22	0.92*0.75=0.69	0.91
Total	0.25	0.75	1

(b)

P(N | D) = P(N and D) / P(D) = 0.06 / 0.09 = 0.6667

(c)

The  probability that any selected product is defective is

P(D) = 0.09

(d)

The probability that any selected product is non-defective and produced by the older machine is

P(D' and O) = 0.22