Maharashtra State BoardHSC Arts 12th Board Exam
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A lot of 100 items contain 10 defective items. Five items are selected at random from the lot and sent to the retail store. - Mathematics and Statistics

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Sum

A lot of 100 items contain 10 defective items. Five items are selected at random from the lot and sent to the retail store. What is the probability that the store will receive at most one defective item?

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Solution

Let X = number of defective items.

p = probability that item is defective

∴ p = `10/100 = 1/10`

∴ q = `1 - "p" = 1 - 1/10 = 9/10`

Given: n = 5

∴ X ~ B `(5, 1/10)`

The p.m.f. of X is given as:

P[X = x] = `"^nC_x  p^x  q^(n - x)`

i.e. p(x) = `"^5C_x (1/10)^x (9/10)^(5 - x)`

P (store will receive at most one defective item)

= P[X ≤ 1] = P[X = 0] + P[X = 1]

= p(0) + p(1)

`= ""^5C_0 (1/10)^0 (9/10)^(5 - 0) + "^5C_1 (1/10)^1 (9/10)^(5 - 1)`

`= 1 xx 1 xx (9/10)^5 + 5 xx 1/10 xx (9/10)^4`

`= (0.9)^5 + (0.05)(0.9)^4`

`= (0.9 + 0.5)(0.9)^4`

= (1.4)(0.9)4

Hence, the probability that the store will receive at most one defective item is (1.4)(0.9)4

Concept: Binomial Distribution
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 8 Binomial Distribution
Miscellaneous exercise 8 | Q 7 | Page 254
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