Question

Matthew works as a production engineer in a manufacturing plant. There are three machines in this plant, namely, machine 1, machine 2, and machine 3. Thirty percent of all jobs are done through machine 1 while fifty percent of all jobs are done through machine 2. In an eight-hour time period, the machines are inspected twice (end of the third hour and end of the sixth hour) to indicate whether a job is being done correctly. Historically, machine 1 will be doing a job correctly 94% of the time. On the other hand, machine and machine 3 will be doing a job correctly 95% and 90% of the time, respectively. These percentages remain the same for the second inspection. For any job, the state (done correctly or done incorrectly) can change after the initial inspection.

(a) What is the probability that a job is being done correctly after the second inspection?

(b) Given that a job is being done correctly after the second inspection, what is the probability that the job is being handled by machine 1?

Answer 1

a)
probability that a job is being done correctly after the second inspection =P(machine 1 and job is being done correctly after the second inspection)+P(machine 2 and job is being done correctly after the second inspection)+P(machine 3 and job is being done correctly after the second inspection)

=0.30*0.94+0.5*0.95+0.2*0.9=0.937

b)

P(machine 1| being done correctly after the second inspection)=0.3*0.94/0.937=0.3010