Probability
Total Probability Theorem
nta_pyq_2025_apr
Grade 12
Question:
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is $29/45$, then $n$ is equal to:
Step-by-Step Solution
Key Concept: Use the law of total probability, conditioning on whether the transferred ball is white or black.
$P(\text{white from Bag 2}) = \frac{4}{9}\cdot\frac{n+1}{n+4} + \frac{5}{9}\cdot\frac{n}{n+4} = \frac{4n+4+5n}{9(n+4)} = \frac{9n+4}{9(n+4)} = \frac{29}{45}$. Solving: $45(9n+4) = 29 \times 9(n+4) \Rightarrow n = 6$.
Correct Answer: 6