2 a high performance airplane contains three identical...

2) A high-performance airplane contains three identical computers. Only one is used to operate the aircraft; the remaining two are spares that can be activated in case the primary system fails. During one hour of operation the probability of a failure of the primary computer (or any of the activated spare systems) is 0.0005. Assuming that each hour represents an independent trial: a) What is the average time for all three computers to fail? b) What is the probability that all three computers will fail in a 5-hour flight?

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Transcript

00:01 In this exercise, we have three identical computers.

00:06 In each hour, the probability of one of them failing is .0005.

00:13 And for part a, we were asked, what is the average time for all three computers to fail? so let's define a random variable x, which is the number of hours, for the three computers to fail.

00:37 Here, x is a negative binomial random variable.

00:51 So we have our probability of success is .005.

00:55 050.

00:57 The number of successes we're looking for is 3.

01:01 And we want to know how many one -hour trials are necessary so that all three computers have failed.

01:11 Now the mean of a negative binomial is given by r over p.

01:19 So that is 3 over 0 .005.

01:26 And that is 6 ,000 hours.

01:29 So on average, it will take 6 ,000 hours before all three engines have failed, or all three computers have failed.

01:40 And for b, we're asked for the probability that all three computers have failed in a five -hour flight.

01:46 So that is what is the probability that x is at most five hours? now we can solve this using software, which makes it a lot easier.

01:58 Let's use excel for this.

02:01 So in excel, if we start the computation with equals, then we start to enter negative binomial...

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