A pressure vessel has five pressure relief valves. The probability that a valve will operate on demand is 0.75. What is the probability that at least one valve will operate on demand? What is the...

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Let X denote number of valves that open in n = 5 independent valves, where
each has a probability p = 0.75
We denote this by:
X ∼ Binomial(n = 5, p = 0.75)
P (X ≥ 1), n = 5, p = 0.75
P (X ≥ 1) = 1 − P (X < 1)
We first calculate:
P (X < 1) = P (0 ≤ X < 1) = P (X = 0)
P (X = 0) =
(5
0
)
(0.75)0(1 − 0.75)5 = 1(0.75)0(0.25)5 ≈ 0.0009766
P (X = 0) = 0.0009766 ≈ 0.001
P (X < 1) = 0.000977
P (X ≥ 1) = 1 − P (X < 1) = 1 − 0.000977 = 0.999023
P (X ≥ 1) = 0.999
Using excel function 1 − BinomDist(0, 5,...