## Answers

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)*p^{3}(1-p)^{2} + 5*p^{4}*(1 -p) + p^{5}

= 10p^{3}(1 + p^{2} - 2p) + 5p^{4} - 5p^{5} + p^{5}

= 6p^{5} - 15p^{4} + 10p^{3}

**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

6p^{5} - 15p^{4} + 10p^{3} > 3p^{2} - 2p^{3}

6p^{3} - 15p^{2} + 10p > 3 - 2p

6p^{3} - 15p^{2} + 12p - 3 > 0

2p^{3} - 5p^{2} + 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.**