Question

5. A communication system consists of n components each of which will function indepen- dently with probability p. The total system will operate effectively if at least half of its components function. (a) What is the probability that the total system will operate effectively if n = 3? (b) What is the probability that the total system will operate effectively if n = 5? (c) For what values of p will & 5-component system be more likely to operate effectively than a 3-component system?

Answer 1

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.