A goldsmith has two bags of stones. The first bag contains 5 white and 10 black stones, and the second contains 4 white and 6 black stones. One of the two bags is chosen at random and a stone is drawn from it at random. What is the probability that the stone is white?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\frac{11}{30}\)
The white stone can come through either bag. Each bag is chosen with probability \(\frac{1}{2}\); multiply along each route and add the two routes.
First bag: \(\displaystyle \frac{1}{2}\times\frac{5}{15}=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}\)
Second bag: \(\displaystyle \frac{1}{2}\times\frac{4}{10}=\frac{1}{2}\times\frac{2}{5}=\frac{1}{5}\)
P(white) \(\displaystyle =\frac{1}{6}+\frac{1}{5}=\frac{5+6}{30}=\frac{11}{30}\)
Why the other options are incorrect
Option (1) \(\frac{1}{6}\) — Incorrect. This is only the route through the first bag.
Option (2) \(\frac{1}{5}\) — Incorrect. This is only the route through the second bag.
Option (4) \(\frac{11}{50}\) — Incorrect. The two routes add to \(\frac{1}{6}+\frac{1}{5}=\frac{11}{30}\), and \(\frac{11}{50}\) is smaller than \(\frac{11}{30}\).