I need help for my homework.Would you please help me? 1) A serial system is comp

Question

I need help for my homework.Would you please help me? 1) A serial system is comprised of 5 components connected in series. In.order.for the serial system to work, all five components must function correctly. Assume that each component fails to function independently of the other components. The probability with which each component fails is 0.01 (a) What is the probability that the whole system functions correctly? (b) What is the probability that at most 3 of the 5 components fail? (c) Let X denote the number of components that fail to function. Find the probability mass function of X (d) Let Y be a Bernoulli random variable that is equal to 1 if at least 4 components function correctly. Find the probability mass function of Y

Explanation / Answer

The probability that a component fails = 0.01

=> The probability that a component does not fail = 0.99

(a) Probability that the whole system functions correctly = 0.995 = 0.9510.

(b) The probability that atmost 3 of the components fail = 1 - Probability that 4 or 5 fail

= 1 - (5 * 0.014 * 0.99 + 0.015)

= 1 - 0.0000000495 + 0.0000000001

= 0.94.

(c) P(1) = 5 * 0.994 * 0.01 = 0.0480298005

P(2) = 10 * 0.993 * 0.012 = 0.000970299

P(3) = 10 * 0.992 * 0.013 = 0.000009801

(d) P(Y = 1) = P(4) + P(5) = 0.0480298005 + 0.9510

= 0.9990298005

P(Y = 0) = 1 - 0.9990298005 =

The probability mass function of Y is

x P(X = x) 0 0.9510 1 0.0480298005 2 0.000970299 3 0.000009801 4 0.0000000495 5 0.0000000001

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