Bag A contains 2 red and 3 white balls, Bag B contains 3 red and 2 white balls. If a ball is drawn at random and is found to be red, then the probability that it was drawn from bag B, is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\frac{3}{5}\)
Let \(E_1\): bag A is chosen, \(E_2\): bag B is chosen, and R: the ball drawn is red. The bags are equally likely:
\( \displaystyle P(E_1) = P(E_2) = \frac{1}{2} \)
\( \displaystyle P(R \mid E_1) = \frac{2}{5} \)
\( \displaystyle P(R \mid E_2) = \frac{3}{5} \)
By Bayes’ theorem,
\( \displaystyle P(E_2 \mid R) = \frac{P(E_2)\,P(R \mid E_2)}{P(E_1)\,P(R \mid E_1) + P(E_2)\,P(R \mid E_2)} \)
\( \displaystyle = \frac{\frac{1}{2} \times \frac{3}{5}}{\frac{1}{2} \times \frac{2}{5} + \frac{1}{2} \times \frac{3}{5}} = \frac{\frac{3}{10}}{\frac{5}{10}} \)
\( \displaystyle = \frac{3}{5} \)
Shortcut: The bags are equally likely and hold 5 balls each, so every one of the 10 balls is equally likely to be drawn. Of the 5 red balls, 3 are in bag B, giving \(\frac{3}{5}\).