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(i) Probability that the smartphone was defective:
Let D be the event that the smartphone is defective.
P(A) = 0.25, P(B) = 0.35, P(C) = 0.40
P(D|A) = 0.05, P(D|B) = 0.04, P(D|C) = 0.02
Using the law of total probability:
P(D) = P(A) * P(D|A) + P(B) * P(D|B) + P(C) * P(D|C)
P(D) = (0.25 * 0.05) + (0.35 * 0.04) + (0.40 * 0.02)
P(D) = 0.0125 + 0.014 + 0.008
P(D) = 0.0345
(ii) Probability that the defective smartphone was manufactured by company B:
We need to find P(B|D), which is the probability that the smartphone was manufactured by company B given that it is defective.
Using Bayes' theorem:
P(B|D) = [P(D|B) * P(B)] / P(D)
P(B|D) = (0.04 * 0.35) / 0.0345
P(B|D) = 0.014 / 0.0345
P(B|D) = 0.4058 (approximately)
Correct Answer: (i) 0.0345, (ii) 0.4058