Finding the reliability for redundant systems

Intro

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 redundant systems.  Here's our problem statement: The principle of redundancy is used when system reliability is improved through redundant or backup components.  Assume that a student's alarm clock has a 12.8% daily failure rate.  Complete Parts A through D below.

Part A

Part A says, “What is the probability that the student's alarm clock will not work on the morning of an important final exam?”  So what we're looking for is the probability that one alarm clock will not work.  This is the probability that the alarm clock will fail, and we're told in the problem statement “Assume that a student's alarm clock has a 12.8% daily failure rate.”  So guess what the probability of one clock not working is going to be.  If you guessed 12.8%, you win a prize!  I don't know what the prize is, but you win something.  So that's the answer that we're going to put here in our answer field.  Good job!

Part B

Part B says, “If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important final exam?”  So what we're looking for now is the probability that two alarm clocks do not work.  This is going to be the probability that the first alarm clock doesn't work multiplied by the probability that the second alarm clock doesn't work, because what we're looking for is the probability the first one doesn't work and the probability that the second one doesn't work.  That word and indicates we need to use multiplication, so we're going to multiply the probability that each alarm clock doesn't work together.  This is the same as taking the probability that one alarm clock doesn't work and squaring it — squaring it because we have two alarm clocks.   Punch that out in your calculator, and you get 0.016384.  However, the instructions tell us to round to five decimal places, so when we do that, we get 0.01638.  Excellent!

Part C

Part C says, “What is the probability of not being awakened if the student uses three independent alarm clocks?”  Well, again we are looking for the probability that three alarm clocks acting separately do not work.  I hope you can see a pattern here.  If we have one alarm clock, we just take the failure rate for one alarm clock.  If we have two alarm clocks, we take the failure rate for two alarm clocks and multiply them together.  So it stands to reason for three alarm clocks we're going to take the probability that each of three alarm clocks doesn't work and multiply them together.   That gives us 0.002097152.  Again, we are asked to round to five decimal places, so that brings us 0.00210.  Good job!

Part D

Part D says, “Do the second and third alarm clocks result in greatly improved reliability?”  Well, if we were to actually make a table of the results that we've just obtained and then convert those to reliability, we could then chart a graph to see what the effect is of adding alarm clocks to our clock system.

To get reliability out here, we have the failure rates that we calculated from the previous three parts of the problem.  Reliability is simply the complement of failure.  So something that you can rely on is something that works, and the probability something works is the complement of something not working, or something failing.  So we simply take 1 and subtract the probability of failure to get the reliability.  Notice the probability of failure is expressed in decimal form, but reliability is expressed in percent form.

We could then take these reliability numbers and plot them on a graph.  Notice there is a pretty significant jump by adding one extra alarm clock to the system.  So if we have one alarm clock and we add just one extra clock — so now we have two — that's a pretty sizable jump.  Adding one additional alarm clock after the second one to get three alarm clocks doesn't provide us with the same sort of jump or increase in reliability.  So having two clocks is much better than one, but having three clocks isn't all that much better than two.

Let's go back here to our problem and look at our answer options.  Answer Option A says, “Yes, because you can always be certain that at least one alarm clock will work.”  That may be a viable answer, but let's check the other answer options first.  Obviously Answer Options B and C aren't going to be correct because the graph clearly shows having an extra alarm clock is going to be helpful.  Answer Option D says, “Yes, because total malfunction would not be impossible but it would be unlikely.”

This is a much better answer than Answer Option A.  Answer Option A says, “because you can always be certain that at least one alarm clock will work.”  Technically, that's not true.  As this graph shows you can get increased reliability and get closer and closer to 100% reliability, which is certainty that you're going to have work what you want to have work.

However you'll never actually reach 100% as we saw from calculating these failure rates up above in the problem statement.  The numbers that we get by adding on additional clocks get smaller and smaller and smaller, but the only way you can get zero failure rate is if you multiply by zero.  And since you're always multiplying by a number that's greater than zero, this number will continue to get smaller and smaller but never actually reach zero.

Therefore, the reliability will get closer and closer to 100% but never actually reach there.  Certainty means 100%, so you can't always be certain; you'll always have some amount of probability of failure.  So the answer that we're going to select is Answer Option D.  Nice work!

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