5. A communication system consists of n components each of which will function indepen- dently with...

Q5) a) The probability that the system will operate efficiently for n = 3 is computed here as:
= Probability that 2 or 3 components work efficiently

This is the required expression for the probability here.

b) For n = 5, the same probability is computed as the probability that at least 3 components work efficiently . ( Computed using binomial distribution function )

= (5c3)*p³(1-p)² + 5*p⁴*(1 -p) + p⁵

= 10p³(1 + p² - 2p) + 5p⁴ - 5p⁵ + p⁵

= 6p⁵ - 15p⁴ + 10p³

This is the required expression for the probability here.

c) The probability p such that the 5 component system will be more likely to operate efficiently than the 3 component dystem is computed here as

6p⁵ - 15p⁴ + 10p³ > 3p² - 2p³

6p³ - 15p² + 10p > 3 - 2p

6p³ - 15p² + 12p - 3 > 0

2p³ - 5p² + 4p - 1 > 0

(p - 1)²(2p - 1) > 0

Now (p - 1)² is always greater than 0, therefore (2p - 1) > 0

p > 1/2

Therefore p > 0.5 is the required range of p here.