An item is defective (independently of other items) with prob 0.3. You have a me

Question

An item is defective (independently of other items) with prob 0.3. You have a method of testing whether the item is defective, but it does not always give you correct ansewr. If the tested item is defective, the method detects the defect with probability 0.9. If the tested item is good, then the method says it is defective with prob 0.2. a) If an item is tested and the test detects no defects, what is the chance that it is not defective? b) A box contains 4 items. You have tested all of them and the tests detect no defects. What is the probability that none of the four items is defective? c) 100 items are tested. Let X = the number of items that are tested are defective. Compute EX and VarX.

Explanation / Answer

a)

Let

D = defective
P = test positive for defects

a)

As

P(P') = P(D) P(P'|D) + P(D') P(P'|D') = 0.3*(1-0.9) + (1-0.3)*0.8 = 0.59

Hence,

P(D'|P') = P(D') P(P'|D') = (1-0.3)*0.8/0.59 = 0.949152542 [ANSWER]

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b)

As

P(D'|P') = 0.949152542

Then the probability of 4 of these happening is

P = 0.949152542^4 = 0.811603779 [ANSWER]

********************

c)

As P(P') = 0.59, then

P(P) = 1 - P(P') = 0.41.

Hence, as n = 100

E(x) = n p = 100*0.41 = 41 [ANSWER]

Var(x) = n p (1-p) = 100*0.41*(1-0.41) = 24.19 [ANSWER]

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