Question

A certain article manufactured in our facilities may present three different types of defects: minor aesthetic, major aesthetic and functional. Of the last 1,000 manufactured items, 105 have a minor cosmetic defect, 40 have a major cosmetic defect, and 25 have a functional defect.

Only 5 of the 1000 present the three types of defect, while 155 present at least one of the aesthetic defects. Twenty articles with major cosmetic defect and functional defect were identified while there were 8 articles with minor cosmetic defect and functional defect.

Calculate the probability of:

a) Randomly select an item without defect

b) The probability of selecting an item with both types of aesthetic defects

c) If you select an item and it turns out to have a minor cosmetic defect, what is the probability that it also has a functional defect?

Answer 1

Let A, B and C denote the number of items with Minor aesthetic, Major aesthetic and Functional defects respectively

n = 1000

n(A) = 105, n(B) = 40, n(C) = 25,

n(A and B and C) = 5

n(A or B or C) = 155

n(B and C) = 20

n(A and C) = 8

(a) The required probability = 1 - P(A or B or C)

= 1 - 155/1000 = 0.845

(b) n(A and B) = n(A or B or C) - n(A and B and C) - n(A) - n(B) - n(C) + n(B and C) + n(A and C)

= 155 - 5 - 105 - 40 - 25 + 20 + 8 = 8

The required probability = 8/1000 = 0.008

(c) The required probability = P(C | A) = P(A and C)/P(A)

= 8/105 = 0.0762