Question

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?

Answer 1

a

n=5

p=0.9

P(X=x)=$\binom{n}{x}\cdot p^x\cdot(1-p)^{n-x}$

P(X=4)=$\binom{5}{4}\cdot 0.9^4\cdot(1-0.9)^{5-4}$ =0.3281

P(X$\geq$ 1)=1-P(X=0)

P(X=0) $=\binom{5}{0}\cdot 0.9^0\cdot(1-p)^{5-0}$ =0.00001

P(X$\geq$ 1)=1-0.00001

P(X$\geq$ 1)=0.99999

b.

For a success rate of 0.999 ,at least 1 out of 1000 radar could detect a missile with probability 0.999.Thus all of the radars can fail with 0.001 probability

0.1ⁿ=0.001

n log 0.1=log 0.001

n=3