Question

An assembly line produces a large number of products, of which 1% are faulty

in average. A quality control test correctly identies 98% of the faulty products,

and 95% of the awless products. For every product that is identied as faulty, the

test is run a second time, independently.

(a) Suppose that a product was identied as faulty in both tests. What is the

probability that it is, indeed, faulty?

(b) What if the quality control test is only performed once?

Answer 1

a)

P(identified as faulty in both test)=P(faulty and identified as faulty in both test)+P(not faulty and identified as faulty in both test)

=0.01*0.98*0.98+(1-0.01)*(1-0.95)*(1-0.95)=0.012079

therefore P( faulty given identified as faulty in both test)

=P(faulty and identified as faulty in both test)/P(identified as faulty in both test)

=0.01*0.98*0.98/0.012079=0.795099

b)

P(identified as faulty in a test)=P(faulty and identified as faulty in a test)+P(not faulty and identified as faulty in a test)

=0.01*0.98+(1-0.01)*(1-0.95)=0.0593

hence

P( faulty given identified as faulty in a test)

=P(faulty and identified as faulty in a test)/P(identified as faulty in a test)

=0.01*0.98/0.0593 =0.165261