Probability
Sampling with replacement, ratio of probabilities
nta_pyq_2023_jan
Grade 12
Question:
A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$. If $p : q = m : n$, where $m$ and $n$ are coprime, then $m + n$ is equal to ___.
Step-by-Step Solution
Key Concept: With replacement from 6 distinct colours: $p = 6 \times (1/6)^2 = 1/6$; $q = {}^6C_1 \times {}^5C_1 \times 4 \times (1/6)^3(5/6)$
$p = 6/36 = 1/6$. For q: choose colour (6 ways), choose different colour (5 ways), arrange (4 ways for which position is different colour) $(1/6)^3 \times (1/6) \times 4\binom{4}{1} = {}^6C_1 \times {}^5C_1 \times 4 \times (1/6)^4 = 6\times5\times4/1296 = 120/1296 = 5/54$. $p:q = (1/6):(5/54) = 9:5$. $m+n = 14$. Answer: 14
Correct Answer: 14