A communications system consists of n components, each of which will, independently, function with probability p. The total system will be able to operate effectively if at least two-thirds of its components function. (a) For what values of p is a 9-component system more likely to operate effectively than a 6-component system? 

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9 Component: 34.3.6 . * A least 6 working 0 () (P) (1-P) = p 9 (1) P* (1-7)'= 93" (1-7) 7 2 () P (rp) = 36P (1-7) (8) P6 (1-P) 3 = 84p (1 - p) 8 6 component: 2662 7 4 * At least 4 working (b) p (1-P) = p (6) P (1-P)' - 6 p (1-P) 4 (4) P^ (1-p) = 15 p (1-p)" 5 9 P + 9p (1-p) + 36 p (1-p) + 84p6 (imp)> p6+ 6p (1-P) + 15,5" (1-p) 2 9 Component: 34.3.6 . * A least 6 working 0 () (P) (1-P) = p 9 (1) P* (1-7)'= 93" (1-7) 7 2 () P (rp) = 36P (1-7) (8) P6 (1-P) 3 = 84p (1 - p) 8 6 component: 2662 7 4 * At least 4 working (b) p (1-P) = p (6) P (1-P)' - 6 p (1-P) 4 (4) P^ (1-p) = 15 p (1-p)" 5 9 P + 9p (1-p) + 36 p (1-p) + 84p6 (imp)> p6+ 6p (1-P) + 15,5" (1-p) 2

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