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?

Question

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?

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Answer 1 · 2017-01-31T11:54:00

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

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

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.!