A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball drawn at random from one of the two boxes, then the probability of getting a black ball is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{19}{42}\)
Each box is chosen with probability \(\frac{1}{2}\). The first box has 2 black balls out of 6, the second 4 black out of 7. By the theorem of total probability:
\( \displaystyle P(\text{black}) = \frac{1}{2}\cdot\frac{2}{6} + \frac{1}{2}\cdot\frac{4}{7} \)
\( \displaystyle P(\text{black}) = \frac{1}{6} + \frac{2}{7} = \frac{7 + 12}{42} = \frac{19}{42} \)
Tip: Do not pool the two boxes into one: 6 black out of 13 balls gives \(\frac{6}{13}\), which is wrong because a ball in the smaller box is more likely to be drawn.