There are three identical boxes I, II and III, each containing two balls. In box I, both balls are red, in box II, both balls are blue and box III contains one blue ball and one red ball. A boy randomly chooses a box and takes out a ball at random from it. If the ball is red, then the probability that the other ball in the box is also red colour is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{2}{3}\)
The other ball is also red only when the box chosen is box I. So the required probability is P(box I | red ball drawn), found by Bayes’ theorem.
Let \(E_1, E_2, E_3\) be the events of choosing box I, II, III, each with probability \(\frac{1}{3}\), and let A be the event that the ball drawn is red:
\( \displaystyle P(A|E_1) = 1,\quad P(A|E_2) = 0,\quad P(A|E_3) = \frac{1}{2} \)
\( \displaystyle P(E_1|A) = \frac{\frac{1}{3}\times 1}{\frac{1}{3}\times 1 + \frac{1}{3}\times 0 + \frac{1}{3}\times\frac{1}{2}} \)
\( \displaystyle = \frac{1}{1 + \frac{1}{2}} = \frac{2}{3} \)
NCERT (2026-27 edition):
Bayes' Theorem
