A boiler has 5 identical relief valves. The probability that any particular valv

Question

A boiler has 5 identical relief valves. The probability that any particular valve will open on demand is 0.74. An operator tries to open all five valves a.) Assuming independent operation of the valves, calculate the probability that at least one valve opens. Give answer to 4 places past decimal Tries 0/5 Tries 0/5 Tries 0/5 Submit Answer b.) Again, assuming independent operation of the valves, calculate the probability that at least one valve fails to open. Give answer to 3 places past decimal. c) What is the expected number of valves that fail? Give answer to 2 places past the decimal. d) What is the variance of the number of valves that fail? Give answer to 2 places past the decimal. Submit Answer Submit Answer Submit Answer Tries 0/5

Explanation / Answer

We use the binomial distribution pdf function to solve the problem:

We have been given params of binomial distribution , n = 5, p(open) = .74

a. P(X>=1) = 1-P(X<1) = 1- P(X=0) = 1- (5C0*.74^0 *.26^5) = .9988
b. P(X>=1 fails) = 1- P(X=0 ) = 1-5C0(0.26^0)*(0.74^5) = .778
c. Expected no. of valves that failed = np = 5*.74 = 3.70
d. Variance = ? = npq = np(1-p) = 5*.74*.26 = .96

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