Surge Protectors Refer to the accompanying figure showing surg...

Question

Surge Protectors Refer to the accompanying figure showing surge protectors p and q used to protect an expensive television. If there is a surge in the voltage, the surge protector reduces it to a safe level. Assume that each surge protector has a 0.985 probability of working correctly when a voltage surge occurs.

c. Which arrangement should be used for better protection?

c. Which arrangement should be used for better protection?

Accepted Answer

Welcome back, everyone. In this problem, a chemical storage tank uses two pressure relief valves, A and B, to protect it against dangerous pressure buildup. Each valve independently has a 0.975 probability of opening successfully during a pressure surge. Which arrangement, if any, provides better protection for the tank, a parallel arrangement or a series arrangement? Now, here, in this case, we want to know which one provides better protection, OK? And if we're trying to figure that out, then we need to find the probability that it happens in a parallel arrangement versus the probability it happens in a series arrangement. Now we already know that each valve works independently and has a probability of 0.975 of opening successfully. Now for our series arrangement, we can tell that pressure is relieved as long as at least one valve opens. If this valve opens, pressure is relieved, and then if this valve opens, pressure is relieved. So it just needs one. And if pressure is relieved, OK, let's write that if pressure is relieved. As long as one, as long as. At least one valve opens, OK. Then even if we don't have that probability exactly, or we can't get it off the top of our heads, we know this is going to be a complement of the event where both valve fails. Now what is the probability a valve fails? Well, if the probability it opens is 0.975, the probability it fails is 1 minus 0.975, which equals 0.025. Thus, that means the probability both fail. Is going to be equal to 0.025 multiplied by 0.025, which equals 0.000625. OK? And if that's the probability both fail, then the probability pressure is relieved at least one opens for our series arrangement will be the complement of that. 1 minus 0.000625, which is 0.999. 375, OK. Now, how does, how does this compare to our parallel arrangement? Well, in our parallel arrangement, we can tell here that each valve controls a separate outlet. So both must work to safely relieve the pressure. OK? So the probability. For our parallel arrangement will be equal to the probability that the A works multiplied by the probability that B works, that 0.975 multiplied by 0.975, which is equal to 0. 0950625, OK? Now when we compare both of these probabilities, we can tell that the probability for the series arrangement is greater than the probability for the parallel arrangement, which means the series arrangement offers better protection for the tank during a pressure surge. Thanks a lot for watching everyone. I hope this video helped.

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