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Question
The probability that a machine will produce all bolts in a production run within specification is 0.998. A sample of 8 machines is taken at random. Calculate the probability that at most 6 machines will produce all bolts within specification.
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Solution
Let X = number of machines that produce the bolts within specification.
p = probability that a machine produce bolts within specification
p = 0.998 and q = 1 − p = 1 − 0.998 = 0.002
Given: n = 8
∴ X ~ B (8, 0.998)
The p.m.f. of X is given by
P(X = x) = `"^nC_x p^x q^(n - x)`
i.e. p(x) = `"^8C_x (0.998)^x (0.002)^(8 - x)`, x = 0, 1, 2, ..., 8
P(at most 6 machines will produce all bolts with specification) = P[X ≤ 6]
= 1 − P[x > 6]
= 1 − [P (X = 7) + P(X = 8)]
= 1 − [P(7) + P(8)]
= 1 − (1.014)(0.998)7
Hence, the probability that at most 6 machines will produce all bolts with specification = 1 − (1.014)(0.998)7.
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