Let $P$ be the probability that the sum of the outcomes of rolling 10 fair dice is divisible by $7$. Then the value of $\left[\dfrac{1}{P}\right]$ (where $[\cdot]$ denotes GIF) is:
Step-by-Step Solution
Key Concept: Use the recurrence $P_n = \frac{1}{6}(1-P_{n-1})$ for the probability that the sum of $n$ dice is divisible by 7 (given 7 residues but only 6 faces). From $P_0=0,P_1=0$: $P_n=\frac{1}{7}\left(1+\left(-\frac{1}{6}\right)^{n-1}\cdot 6\right)$.
From the recursive formula, $P_{10} = \frac{1}{7}(1-\frac{6}{6^{10}\cdot...})$... More carefully: $p_n = \frac{1}{7}(1+(-\frac{1}{6})^{n-1}\cdot 6)$... From solution: $\lfloor 1/P_{10}\rfloor = \lfloor 6.999...\rfloor = \mathbf{6}$.
Correct Answer: 6