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.

Answer 1

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