A factory runs three shifts. In a given day, 1% of the items produced by the fir

Question

A factory runs three shifts. In a given day, 1% of the items produced by the first shift are defective, 2% of the second shift's items are defective, and 5% of the third shift's items are defective. The shifts all have the same productivity.

Suppose two items, independently sampled, are each defective. What is the chance they both were produced by the third shift?

Suppose three items, independently sampled, are each defective. What is the chacne at least two of them were produced by the third shift?

Explanation / Answer

P(D|S1) = 0.01, P(D|S2) = 0.02, P(D|S3) = 0.05
P(S1) = P(S2) = P(S3) = 1/3

#1.
P(S3|D) = P(D|S3)*P(S3)/(P(D|S1)*P(S1) + P(D|S2)*P(S2) + P(D|S3)*P(S3))

= 0.05*0.3333/(0.01*0.3333 + 0.02*0.3333 + 0.05*0.3333)
= 0.625

Required probability = 0.625*0.625 = 0.3906

#2.
P(X>=2) = 3C2*0.625^2*(1-0.625) + 0.625^3
= 0.6836

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