Question

A shop selling electronic items sells smartphones of only three reputed companies A, B, and C because chances of their manufacturing a defective smartphone are only 5%, 4%, and 2% respectively. In his inventory, he has 25% smartphones from company A, 35% smartphones from company B, and 40% smartphones from company C.
A person buys a smartphone from this shop

(i) Find the probability that it was defective.

Answer 1

The following events are defined: \( D \): A smartphone is defective. \( A \): The smartphone is from company A. \( B \): The smartphone is from company B. \( C \): The smartphone is from company C. The given probabilities are: \( P(A) = 0.25 \), \( P(B) = 0.35 \), \( P(C) = 0.40 \). Conditional probabilities of a smartphone being defective given its company are: \( P(D|A) = 0.05 \), \( P(D|B) = 0.04 \), \( P(D|C) = 0.02 \). The objective is to calculate \( P(D) \), the overall probability that a smartphone is defective. Applying the law of total probability: \[ P(D) = P(A) \cdot P(D|A) + P(B) \cdot P(D|B) + P(C) \cdot P(D|C) \] Substituting the provided values: \[ P(D) = 0.25 \times 0.05 + 0.35 \times 0.04 + 0.40 \times 0.02 \] \[ P(D) = 0.0125 + 0.014 + 0.008 \] \[ P(D) = 0.0345 \] Consequently, the probability of a smartphone being defective is \( P(D) = 0.0345 \), which is equivalent to 3.45%.