Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find a product reliability given the failure rate. Here's our problem statement: Assume that there is an 11 percent rate of disk drive failure in a year. Part A: If all your computer data is stored on a hard disk drive with a copy stored on a second hard disk drive what is the probability that during a year you can avoid catastrophe with at least one working drive? Part B: If copies of all your computer data are stored on three independent hard disk drives what is the probability that during a year you can avoid catastrophe with at least one working drive?
OK, so here we have given a failure rate for a disk drive. We want to know what's the reliability of our system if we have two of these disk drives in place. So we have one disk drive and then a copy of all the data on a second disk drive. What's the reliability of our system? Well, the probability that we're looking for is that at least one of these two are going to be working, because if one of them fails, that's okay; we still have everything on the other drive.
So we want the probability that at least one of these two is going to work. That's going to be the same as 1 minus the probability that none of them works. Remember that when you have the probability of “at least one” the way we calculate that is by taking 1 minus the complement of “at least one.” And the complement of at least one is “none.” So what then is the probability that none of them work? Well, the probability that none of them work is going to be the probability that the first one fails and the probability that the second one fails. If none of them work, both of them have to fail.
Well, we can calculate this probability. We know the probability that the first one's going to fail. That's given to us in the problem statement — “The disk drive as a failure rate of 11%.” So the probably that none of them work is equal to 0.11 (that's the probability that the first one will fail) and translates into multiplication (when we're dealing with probability problems and you see the word and, you know you need to use multiplication) and the probability the second one will fail is the same as the probability of the first to fail because the disk drives are identical. So now we have 0.11 times 0.11. We multiply that out, and we get 0.121. This then is the probability that none of them work.
But remember we were looking for the probability that at least one of the two work, so we have to take the complement of this in order to get the probability that we're actually looking for. And so we go ahead and do that. The probability that at least one of the two work is 1 minus the probability that none of them works. We know that the probability that none of them works is 0.0121, so we substitute that in here. And we get, when we punch that out in our calculator, 0.9879. So this then is the probability that catastrophe can be avoided. That's the reliability of our system. We're asked to round to four decimal places, and that's what we have here. So I'm just going to put that number in my answer field. Nice work!
Now Part B says with three hard disk drives what is the probability that catastrophe can be avoided is . . . . So now we have three instead of two. Well, the problem is going to work pretty much the same way. The probability that at least one of the three work is 1 minus the probability that none of them work. But the probability that none of them work as we’ve seen from the previous problem statement is simply the failure rate of one, the failure rate of the second, and the failure rate of the third all multiplied together.
We can simplify this by saying 0.11 to the third power instead of 0.11 times 0.11 times 0.11. We see these types of problems where you have different parts of a system and they're all independent of each other. Typically, the way that we calculate the probability that none of them work is we just take the probability of one and raise it to the power of however many of these similar parts that we have in redundancy. So here we've got three of them together, so we're going to raise this to the third power.
You punch this out on our calculator — 0.11 to the third power is 0.001331. Subtract that from 1, and we get 0.99869. We're asked to round to six decimal places, and that's exactly how many we have here. So I'm just going to put that number here in my answer field. Excellent!
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