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प्रश्न
Solve the following problem :
The probability that a machine will produce all bolts in a production run within the specification is 0.9. A sample of 3 machines is taken at random. Calculate the probability that all machines will produce all bolts in a production run within the specification.
उत्तर
Let X denote the number of machines that run within specification.
P(a machine will produce all bolts in production run within the specification) = p = 0.9
∴ q = 1 – p = 1 – 0.9 = 0.1
Given, n = 3
∴ X ~ B(3, 0.9)
The p.m.f. of X is given by
P(X = x) = `""^3"C"_x (0.9)^x (0.1)^(3 - x),x` = 0, 1, 2, 3.
P(all machines will produce all bolts in a production run within the specification)
= P(X = 3)
= `""^3"C"_3 (0.9)^3 (0.1)^0`
= 0.729
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Solution:
Here, n = 4
p = probability of defective device = 10% = `10/100 = square`
∴ q = 1 - p = 1 - 0.1 = `square`
X ∼ B(4, 0.1)
`P(X=x)=""^n"C"_x p^x q^(n-x)= ""^4"C"_x (0.1)^x (0.9)^(4 - x)`
P[At most one defective device] = P[X ≤ 1]
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∴ P[X ≤ 1] = `square`
