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So we have this missile protection system that has $n$ radar sets that are all independent. Each have a probability of $0.9$ of detecting a missile.

How large must $n$ be if we want the probability of detecting a missile to be $0.999$?

My solutions:

So I did this three ways. I know$E(y)=np$ so if we solve $.999=(0.9)n$, we get that we need $2$ systems.

Also, if we do $1-(0.9)^n = 0.999$, we get than we need $66$ systems.

ALSO, if we do $1-(0.1)^n = 0.999$, we get that we need $3$ systems.

I am pretty sure its the middle one or last one...but I am not sure which...

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$3$ but that is assuming they don't fail $10$% of the time each for the same exact reason. The radar sets are all independent but they could still react to the same input in a similar way and thus all fail for the same reason and thus not giving you the $99.9$% assurance you are looking for. For example, if someone does a broad area radar jam and all 3 systems fail simultaneously, even though they are "independent". I am not really sure what you mean by independent anyway. Independent of each other? Stand alone functionality? "Independent" doesn't mean that one could get jammed but the others cannot. I am assuming they are identical radar systems just positioned differently.

This problem is similar to telling $3$ of your friends the same message and telling them to deliver that message to the same person. If they all go about it in their own way and each person is $90$% reliable (only $10$% chance of each person failing to deliver the intended message), then the chance of your message not getting thru is only $10$% * $10$% * $10$% = $0.1$% thus your message should get thru $99.9$% of the time which is $0.999$ reliable but if something happens that affects all 3 of them at once (like a local volcano erupts and kills all 3 of them), then your message aint gettin' thru and the 99.9% aint happenin'.

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