Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find the reliability for series and parallel configurations. Here's our problem statement: Refer to the figure below in which search protectors p and q are used to protect an expensive high definition 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 98% probability of working correctly when a voltage surge occurs. Complete parts (a) through (c) below.
We see in the figure they're showing us the difference between a series configuration and a parallel configuration. In a series configuration, both of the surge protectors are placed one after the other. This means that in order for a surge to come through and fry our TV, both of these surge protectors have to fail. If one of them works, we’re protected. Contrast that with the parallel configuration. In a parallel configuration, only one of these surge protectors needs to fail in order for a surge to come through and fry our TV.
Now let's read Part A. Part A says, “If the two surge protectors are arranged in series, what is the probability that a voltage surge will not damage the television?” Well, what we're looking for here is the series configuration, and we're trying to ascertain the probability that at least one of these works. The probability that a voltage surge will not damage the television is the probability that at least one of these works, because in the series configuration we only need one of them to work to be protected.
We know that the converse — or rather the complement — of “at least one” is none, so the probability of at least one is going to be one minus the probability that none of them work. This comes from the principle of complements. The complement of at least one is none, so in order to solve this probability, we simply find the complement.
What is the probability that none of them work? Well, here we use the fundamental counting rule to find that probability. The probability that none of them work is the probability that one of them doesn't work multiplied by the probability that the other one doesn't work. We have a 98% probability of working correctly. That means we have a 2% probability of failure, because the most any probability can be is 100%, or 1. So if there's a 98% probability of working right, there's a 2% probability of not working right, or a 2% probability of failure. So the probability that it doesn't work is 2%.
We raise that to the second power because there are two surge protectors in our series configuration. If we had three search protectors in series, then we would be raising this number to the third power. But we only have two, so we're going to raise it to the second power. This then is easy to punch out on your calculator. When we do that, we get 0.9996, almost perfect reliability. That's outstanding! So this is the number I'm going to put in here. Notice the instructions say, “Do not round.” Excellent!
Part B says, “If the two search protectors are arranged in parallel, what is the probability that a voltage surge will not damage the television?” Well, here we have the parallel configuration. Notice in the parallel configuration we only need one of these to fail in order for the surge to come through and fry our TV. So the probability that both of these are working is what we're looking for here, because both of these have to work in the parallel configuration in order for our TV to be protected.
Well, the probability that both of them are going to work is simply the probability that one of them works multiplied by the probability that the other one works. We have a 98% probability of working correctly, so therefore the probability that both of them work is simply 0.98 raised to the second power. Again, if we had three surge protectors in parallel, then we would raise this number to the third power. But we only have two surge protectors, so we raise it to the second power. This is also easy to punch out on a calculator. When we do, we get 0.9604, so that's the number I'm going to put here. Well done!
And now the last part, Part C, says, “Which arrangement should be used for better protection?” Well, if we come over here and look at our reliabilities, the one with the higher reliability is the one that's going to offer us better protection. And clearly, that's the series configuration because, while 96% reliability is pretty good, 99+% reliability is even better. So we want the series configuration because it gives us a higher probability of protection. And looking at the answer options, that's going to be this one here. Nice work!
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