Question 6. [6 Marks] An inspector working for a manufacturing company has a 97%

Question

Question 6. [6 Marks] An inspector working for a manufacturing company has a 97% chance of correctly identifying defective items and a 0.7% chance of incorrectly classifying a good item as defective. The company has evidence that 1.2% of its line products are defective. a) 2 marks] What is the probability that a randomly selected item for inspection is classified as defective? Page 2 of 3 Continued on next page... Stat 3502 A Assignment 1 Due In Tutorial is classified as good is indeed defective? c) 2 marks] Compute the probability that two randomly selected items for inspection are clas- sified as good.

Explanation / Answer

Event A: Product is defective: P(A) = 0.012
Event X: Identifying defective product
Given : P(X | A) = 0.97 and P(X | Ac) = 0.007 , where Ac is the event where product is good. P(Ac) = 0.988
a) Probability that a randomly selected item is classified as defective i.e. P(X)
Using total law of probability:
P(X) = P(A).P(X|A) + P(Ac).P(X|Ac)
P(X) = (0.012 x 0.97) + (0.988 x 0.007) = 0.018556

b)
Probability that a randomly selected item is classified as good = 1 - 0.018556 = 0.995276 = P(Xc)
We have to find: P(A | Xc)
Using conditional probability: P(A | Xc) = [P(A) . P(Xc | A)] / P(Xc)
We know P(X | A) = 0.97  , P(Xc | A) = 1 - P(X | A) = 1 - 0.97 = 0.03
P(A | Xc) = (0.012 x 0.03) / 0.995276 = 0.00036

c)
Probability that a randomly selected product is classified as good = P(Xc)
So Probability that two randomly selected products are classified as good = P(Xc)2 =  0.9952762 = 0.99

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