Question

Components are selected from three machines A, B, and C to check for its quality. The results are summarized in the table below:

	Defective	Not Defective
Machine A	30	520
Machine B	20	480
Machine C	35	765

(a) From the given information, construct a fully labelled probability tree diagram.
(b) Given that a chosen component is defective, what is the probability that it is from machine B?
(c) Determine whether the events ‘Defective’ and ‘Machine B’ are independent?
(d) Determine whether the events ‘Not Defective’ and ‘Machine C’ are mutually exclusive?

Answer 1

a) Total number of machines = 1850

P(Machine A) = (30+520)/1850 = 0.2793

P(Machine B) = (20+480)/1850 = 0.2703

P(Machine C) = (35+765)/1850 = 0.4324

P(Defective | Machine A) = 30/550 = 0.0545

P(Defective | Machine B) = 20/500 = 0.04

P(Defective | Machine A) = 35/800 = 0.0438

Tree Diagram

b) Bayes' Theorem: P(A | B) = P(A & B) / P(B)

P(Machine B | Defective) = 20/(30+20+35)

= 0.2353

c) P(Machine B) P(Machine B | Defective)

Therefore, the events are not independent.

d) P(Not Defective and Machine C) 0

Therefore, the events are not mutually exclusive