Bag I contains 3 red and 2 white balls, and Bag II contains 1 red and 3 white balls. One ball is drawn at random from each bag. The probability that the two balls drawn are of the same colour is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{9}{20}\)
The draws from the two bags are independent, so the probability of a pair of colours is the product of the two probabilities. The balls are of the same colour if both are red or both are white.
Both red:
\( \displaystyle \frac{3}{5}\times\frac{1}{4} = \frac{3}{20} \)
Both white:
\( \displaystyle \frac{2}{5}\times\frac{3}{4} = \frac{6}{20} \)
The two cases cannot happen together, so
\( \displaystyle P = \frac{3}{20}+\frac{6}{20} = \frac{9}{20} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{20}\) — Incorrect. This is the probability that both balls are red only.
Option (2) \(\displaystyle\frac{3}{10}\) — Incorrect. This is \(\frac{6}{20}\), the probability that both balls are white only.
Option (3) \(\displaystyle\frac{11}{20}\) — Incorrect. This is \(1-\frac{9}{20}\), the probability that the balls are of different colours.