# There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item? - Mathematics and Statistics

Sum

There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?

#### Solution

Let X denote the number of defective items.

P(getting defective item) = p = (10)/(100) = 0.1     ...[Given]

∴ q = 1 – p = 1 – 0.1 = 0.9
Given, n = 4
∴ X ∼ B(4, 0.1)
The p.m.f. of X is given by
P(X = x) = ""^4"C"_x (0.1)^x (0.9)^(4 - x), x = 0, 1,...,4
P(sample will include not more than one defective item)
= P(X ≤ 1) = P(X = 0 or X = 1)
= P(X = 0) + P(X = 1)
= ""^4"C"_0 (0.1)^0 (0.9)^4 + ""^4"C"_1 (0.1)^1 (0.9)^3
= (0.9)4 + 4 x 0.1 x (0.9)3
= (0.9)3 (0.9 + 0.4)
= 1.3 x (0.9)3.

Concept: Random Variables and Its Probability Distributions
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Exercise 8.3 | Q 1.03 | Page 150