Suppose a company manufactures the following types of engine

Suppose a company manufactures the following types of engine valves:

A-Regular valves

B-Hard-faced valves for maximum wear

C-Aluminum coated valves

It is known that 60% of production is for regular valves; 28% is for hard-faced valves; and 12% is for aluminum valves. Assume that 50%of regular valves, 10% of hard-faced valves, and 14% of aluminum valves fail from fatigue. What is the probability that valve is hard-faced?

Solution

Let the total no. of valves manufactured by the company be x. As per the question,
The no. of regular valves fail from fatigue = 0.5x
The no. of hard-faced valves fail from fatigue = 0.1x
And the no. of aluminium valves fail from fatigue = 0.14x
So the total no. of valves fail from fatigue = 0.74x

Thus the probability that the observed fail valve is hard-faced =
(No.of hard-faced valves fail from fatigue)/(Total number of fail valves)
= 0.1x/0.74x = 100/74