The probability that a radar station will detect an enemy pl

The probability that a radar station will detect an enemy plane is 0.65.

a. If four stations are in use when an enemy plane flies overhead, what is the probability that the plane will not be detected?

b. If four stations are in use when an enemy plane flies overhead, what is the expected number of stations that will detect the enemy plane?

c. How many such stations are required to be 95% certain that an enemy plane flying overhead will be detected by at least one station?

Solution

P(detecting planes)=0.65

P(not detecting planes)=1-0.65=0.35

here we use the bnomial distribution.

a) we assume that the four stations work independently.

P(plane not detected)=0.35⁴=0.0150

b) n=4,p=0.65

for binomial distribution E(X)=np=4*0.65=2.6

c) let P(x=0)=z

1-z is the probability of atleast one station detecting the plane.

1-z>0.95

that is 1-0.35ⁿ>0.95

0.05<0.35ⁿ

Take log on both sides

log(0.05)<nlog0.35

log0.05/log0.35<n

we get 2.85 whihc can be approximated to 3.

So we need minimum 3 stations.