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A missile protection system consists of n radar sets operating independently, each with a probability of .9 of detecting a missile entering a zone that is covered by all of the units.
a If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect the missile? At least one set?
b How large must n be if we require that the probability of detecting a missile that enters the zone be .999?

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MechEngineer

Answer:

a. probability that exactly four sets detect the missile is 0.06561

probability that at least 1 set detect the missile is 0.99999

b. n = 3

Step-by-step explanation:

a. The probability that exactly 4 sets with probability of detection being 0.9 and 1 set fail with probability of 1 - 0.9 = 0.1 is

0.9*0.9*0.9*0.9*0.1 = 0.06561

The probability of having at least 1 set detect the missile is the inverse of the probability of having none of the set detecting the missile, which means all of the set fail to detect the missile, which is

0.1*0.1*0.1*0.1*0.1 = 0.00001

So the probability that at least 1 set detect the missile is

1 - 0.00001 = 0.99999

b. For the system to have a success rate of 0.999, this means at least 1 radar could detect the missile with probability of 0.999, which means all of them can fail with probability of 0.001. For this to happen:

[tex]0.1^n = 0.001[/tex]

[tex](10^{-1})^n = 10^{-3}[/tex]

[tex]10^{-1n} = 10^{-3}[/tex]

[tex]-n = -3[/tex]

[tex]n = 3[/tex]

You need 3 radars

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