Probability
Probability
nta_pyq_2025_jan
Grade 12
Question:
Two balls are selected at random one by one without replacement from a bag containing $4$ white and $6$ black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\dfrac{m}{n}$ where $\gcd(m,n)=1$, then $m+n$ is equal to:
Step-by-Step Solution
Key Concept: $P(B_{1}|B_{2})=\dfrac{P(B_{1}\cap B_{2})}{P(B_{2})}.$ Compute $P(B_{2})$ via the law of total probability over the first draw.
$P(B_{1}\cap B_{2})=\dfrac{6}{10}\cdot\dfrac{5}{9}=\dfrac{1}{3}.$
$P(B_{2})=\dfrac{6}{10}\cdot\dfrac{5}{9}+\dfrac{4}{10}\cdot\dfrac{6}{9}=\dfrac{30+24}{90}=\dfrac{3}{5}.$
$P(B_{1}|B_{2})=\dfrac{1/3}{3/5}=\dfrac{5}{9}.$
$m+n=5+9=14.$
Correct Answer: 2