Question

) How many such stations are required to be 98% certain that an enemy plane flying over will be detected by at least one station

Answer 1

Complete Question

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

(a) How many such stations are required to be 98% certain that an enemy plane flying over will be detected by at least one station?

(b) If seven stations are in use, what is the expected number of stations that will detect an enemy plane? (Round your answer to one decimal place.)

Answer:

a

`n \approx 4`

b

Explanation:

From the question we are told that

The probability that a single radar station will detect an enemy plane is
`p =0.65`

Gnerally the probability that an enemy plane flying over will be detected by at least one station is mathematically represented as

`P(X \ge 1 ) = 0.98`

=>
`P(X \ge 1 ) = 1 - P(X < 1) = 0.98`

=>
`P(X = 0) = 1 - 0.98` Note
`P(X < 1) = P(X = 0)`

=>
`P(X = 0) = 1 - 0.98`

=>
`P(X = 0) = 0.02`

Generally from binomial probability distribution function

`P(X = 0) = ^nC_0 * p^(0) * (1- p)^(n- 0)`

Here C represents combination hence we will be making use of of combination functionality in our calculators

Generally any number combination 0 is 1

So

`P(X = 1) = 1 * 1* (1- 0.65)^(n- 0) = 0.02`

=>
`(1- 0.65)^(n- 1) = 0.02`

taking log of both sides

`log [(0.35)^(n- 1) ] = log (0.02)`

=>
`{n- 1}log[0.35] = -1.699`

=>
`{n- 1}* -0.4559 = -1.699`

=>
`n= 3.7264 + 1`

=>
`n= 4.7264`

=>
`n \approx 4`

Gnerally the expected number of stations that will detect an enemy plane is

`E(X) = 7 * 0.65`

=>
`E(X) \approx4.5`