Did not need repair find the probablility that it was produced by company C?

Companies A, B and C produce 20%, 50%, and 30%, repectively, of major appliances sold under the Kenmore name in Lea County. In Lea County 3% of company A appliances need repair within 1 year of service, 2% of company B appliances need repair within 1 year of service, and 1% of company C appliances need repair within 1 year of service.


If you bought a Kenmore appliance and it did not need repair within the first year, find the probability that it was produced by compancy C?

Did not need repair find the probablility that it was produced by company C?
You need to use Bayes' Theorem:


NR = No Repair and R = Repair


A, B or C = Company A B or C





P(C | NR ) = P(NR | C)*P(C)/[P(NR | A)*P(A) + P(NR | B)*P(B) + P(NR | C)*P(C) ]





P(NR | C) = 1-1% = 0.99 P(C) = 30%


P(NR | B) = 1- 2% = 0.98 P(B) = 50%


P(NR | A) = 1-3% = 0.97 P(A) = 20%





P(C | NR) = (30% * 0.99)/[(0.99*30%) + (0.98*50%) + (0.97*20%)]





= 0.297 / [ 0.2997 + 0.49 + 0.194] = 0.297 / 0.981 = .30275





So there is a 30.275% chance the item was produced by company C given there were no repairs the first year.

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