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A production system has two production lines; each production line performs a two-part process, and each process is completed by a different machine. Thus, there are four machines, which we can identify as two first-level machines and two second-level machines. Each of the first-level machines works properly \(98 \%\) of the time, and each of the second-level machines works properly \(96 \%\) of the time. All four machines are independent in regard to working properly or breaking down. Two products enter this production system, one in each production line. a. Find the probability that both products successfully complete the two-part process (i.e., all four machines are working properly). b. Find the probability that neither product successfully completes the two- part process (i.e., at least one of the machines in each production line is not working properly).

Step by step solution

01

Compute Probability of All Machines Working Properly

To find the probability that both products successfully complete the two-part process, we can multiply the individual probabilities of each machine working properly. Since each of the two first-level machines works properly 0.98 of the time, and each of the two second-level machines works properly 0.96 of the time, the probability that all four machines are working properly is \(0.98^2 * 0.96^2\).

02

Compute Probability of At Least One Machine Not Working Properly

To find the probability that neither product successfully complete the two-part process (meaning at least one of the machines in each production line is not working properly), we have to compute the probability of at least one machine in the production line not working. Since the machines work independently, the probability that a first-level machine does not work properly is \(1 - 0.98 = 0.02\), and for a second-level machine is \(1 - 0.96 = 0.04\). Hence, the probability that at least one machine in each line is not working is 1 - the probability that both machines in one line work, which would be the same for both lines. Thus, this probability is \(1 - 0.98 * 0.96\), and for two lines it would be \([1 - 0.98 * 0.96]^2\).

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