In tossing two unbiased dice, success is defined as an outcome of a doublet. If \(\displaystyle\frac{5^a}{2^b3^c}\) is the probability that the first success occurs at the 5th throw of a pair of dice, then a + 2b + 3c is equal to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 29
A doublet is one of the 6 outcomes (1, 1), (2, 2), …, (6, 6) out of 36. So on each throw of the pair:
\( \displaystyle p = \frac{6}{36} = \frac{1}{6},\quad q = 1 - p = \frac{5}{6} \)
The first success at the 5th throw means four failures followed by a success. The throws are independent, so
\( \displaystyle P = q^4\,p = \left(\frac{5}{6}\right)^4\times\frac{1}{6} \)
\( \displaystyle = \frac{5^4}{6^5} = \frac{5^4}{2^5\,3^5} \)
Comparing with \(\frac{5^a}{2^b3^c}\):
\(a = 4,\ b = 5,\ c = 5\)
\(a + 2b + 3c = 4 + 10 + 15 = 29\)