Three manufacturing plants, say I, II and III, produce 20, 30 and 50 percent of a company’s output respectively. What is the probability that a randomly-chosen item from the company’s warehouse is defective?

The manager of plant III is very quality conscious and only 1% of the items from that plant are defective. Plants I and II have defective rates of 3% and 5% respectively.

An item is selected at random from the company’s warehouse and found to be defective. What is the probability it was manufactured in plant III?

1 Answer
Jan 31, 2017

(1):"The Reqd. Prob.="3.6%.

(2):"The Reqd. Prob.="25/36~~69.44%.

Explanation:

Let M_1=the Event that the Unit is manufactured in Plant I.

M_2 and M_3 denote Events similar to M_1.

Clearly, P(M_1)=20%=2/10, P(M_2)=3/10, &, P(M_3)=5/10.

Suppose that, D denotes the Event that the chosen Unit is

Defective.

In the usual Notation of Conditional Prob. , then, we are given that,

P(D/M_1)=1%=1/100, P(D/M_2)=3/100, and, P(D/M_3)=5/100.

"Now, (1) : The Reqd. Prob.="P(D)=sum_(j=1)^(j=3)P(M_j)P(D/M_j)

=(2/10)(1/100)+(3/10)(3/100)+(5/10)(5/100)

=36/1000=3.6%.

"Next, (2) : The Reqd. Prob.="P(M_3/D)

={P(M_3nnD)}/(P(D))={P(D/M_3)P(M_3)}/(P(D))

={(5/100)(5/10)}/(36/1000)=25/36~~69.44%.

Enjoy Maths.!