National security requires that defense technology be able...

National security requires that defense technology be able to detect incoming projectiles or missiles. To make the defense system successful, multiple radar screens are required. Suppose that three independent screens are to be operated and the probability that any one screen will detect an incoming missile is 0.8. Obviously, if no screens detect an incoming projectile, the system is unworthy and must be improved. (a) What is the probability that an incoming missile will not be detected by any of the three screens? (b) What is the probability that the missile will be detected by only one screen? (c) What is the probability that it will be detected by at least two out of three screens? (d) How many screens do you have to use to get 99.9% effectiveness?

Transcript

00:01 Alright, so we have this system for detecting missiles, and we know that the probability of any one screen detecting a missile is 0 .8.

00:15 So we have these three screens, right, they all have a probability of 0 .8 of detecting a missile.

00:23 So a asks what's the probability that it'll not be detected by any of the screens.

00:29 So the probability of not being detected by any one of the screens is 0 .2 since the success rate is 0 .8.

00:36 So then we would take 0 .2 to the third power, and that would get us the probability of failing all three times.

00:44 So that is 0 .008.

00:48 So that's my failure rate right now.

00:51 For b, it asks, what is the probability that the missile will be detected by only one screen? so this would be the first screen getting it times the second screen not getting it times the third screen not getting it.

01:10 But then we have to multiply this by three because there's really three different ways this could happen, right? the second one could be the one that gets it or the third screen could be the one that gets it.

01:20 So we have to multiply by three to account for all those probabilities.

01:24 So when i take point three times, sorry, three times point eight times point two times point two times point.

01:28 I get 0 .96, sorry, 0 .096 for the probability that only one of the screens gets a missile.

01:39 For c, whereas what's the probability that at least 2 of the screens gets it.

01:48 So we already found the probability that none of them do and the probability that 1 does, so the probability of at least 2 should be 1 minus each of those probabilities, because those are the only options is not getting it at all, only one getting it, or at least two getting it...

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