Probability
Conditional Probability (Bayes)
nta_pyq_2025_apr
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 $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to:
Step-by-Step Solution
Key Concept: Use Bayes' theorem: $P(A|B) = P(A \cap B)/P(B)$, where $A$ = first ball is black, $B$ = second ball is black.
$P(A|B) = \frac{(6/10)(5/9)}{(6/10)(5/9) + (4/10)(6/9)} = \frac{30}{30+24} = \frac{30}{54} = \frac{5}{9}$. $m + n = 5 + 9 = 14$.
Correct Answer: 14