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.
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.
Solution and Explanation
- \( A \): The event that the smartphone is from company A.
- \( B \): The event that the smartphone is from company B.
- \( C \): The event that the smartphone is from company C.
We are given: - \( P(A) = 0.25 \), \( P(B) = 0.35 \), \( P(C) = 0.40 \)
- \( P(D|A) = 0.05 \), \( P(D|B) = 0.04 \), \( P(D|C) = 0.02 \)
We need to find \( P(D) \), the total probability that a smartphone is defective.
Using the law of total probability: \[ P(D) = P(A) \cdot P(D|A) + P(B) \cdot P(D|B) + P(C) \cdot P(D|C) \] Substitute the values: \[ P(D) = 0.25 \times 0.05 + 0.35 \times 0.04 + 0.40 \times 0.02 \] \[ P(D) = 0.0125 + 0.014 + 0.008 \] \[ P(D) = 0.0345 \] Thus, the probability that the smartphone is defective is \( P(D) = 0.0345 \) or 3.45%.
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