A shop sells mobile phones of two brands, P and Q, in the ratio 3 : 1. Of the phones sold, \(\displaystyle\frac{1}{10}\) of brand P and \(\displaystyle\frac{1}{5}\) of brand Q are returned under warranty. A phone sold by the shop is chosen at random. The probability that it is returned under warranty is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{8}\)
Let \(E_1\) and \(E_2\) be the events that the phone is of brand P and of brand Q, and let R be the event that it is returned. The ratio 3 : 1 gives
\( \displaystyle P(E_1) = \frac{3}{4},\quad P(E_2) = \frac{1}{4} \)
By the theorem of total probability:
\( \displaystyle P(R) = \frac{3}{4}\times\frac{1}{10} + \frac{1}{4}\times\frac{1}{5} \)
\( \displaystyle P(R) = \frac{3}{40}+\frac{2}{40} = \frac{5}{40} = \frac{1}{8} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{3}{20}\) — Incorrect. This is the plain average of \(\frac{1}{10}\) and \(\frac{1}{5}\), which ignores that brand P sells three times as much.
Option (3) \(\displaystyle\frac{3}{10}\) — Incorrect. This adds the two return rates without weighting them.
Option (4) \(\displaystyle\frac{7}{40}\) — Incorrect. This pairs the shares the wrong way: \(\frac{3}{4}\times\frac{1}{5}+\frac{1}{4}\times\frac{1}{10}\).