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.)

Solution:

Let ‘n’ be the number of stations.

Here success is detecting plane

∴ P(success) = P = 0.65

P(failure) = q = 1 - p = 1 - 0.65 = 0.35

Thus using the formula for the binomial distribution, we have

We have, P(X = x) = ⁿC_x . p^x . q^{n - x}

∴ P(X = x) = ⁿC_x . (0.65)^x . (0.35)^{n - x}

a) P(at least 1) ≥ 0.98

⇒ 1 - P(x =0) ≥ 0.98

⇒ 1 - ⁿC₀ (0.65)⁰ . (0.35)ⁿ ≥ 0.98

We know that, ⁿC₀ = 1 and (0.65)⁰ = 1

⇒ 1 - (0.35)ⁿ ≥ 0.98 

⇒ 1 - 0.98 ≥ (0.35)ⁿ 

⇒ 0.02 ≥ (0.35)ⁿ 

∴ by inspection, n = 4

b) The expected number of stations 

= Mean of distribution = np

Here, n = 7 and p = 0.65

∴ np = 7 × 0.65

np = 4.55 ≈ 4.6

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

Summary:

Given the probability that a single radar station will detect an enemy plane is 0.65, 

(a) n = 4 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, the expected number of stations that will detect an enemy plane is 4.6.