Question

A missile protection system consist of $n$ radar sets operation independently,each with a probability of .9 of detecting a missile entering a zone that is covered by all 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

p=0.9, n=5

a) i)Exactly four sets

P(X=4)

$P(X=x) ={n\choose x} p^xq^{n-x}$

$={5 \choose 4}*0.9^4*0.1^1$

=0.3281

ii) Atleast one set

P(X$\ge$ 1)

$=\sum_1^5{5\choose x} *0.9^x*0.1^{5-x}$

=0.99999

b) Atleast one radar set should detect the existence of a missile.

$P(X\ge1) =1-P(X=0)$

$0.999=1-0.1^n$

$0.1^n=0.001$

n=3