Question 5
When rolling a certain unfair six-sided die with faces numbered $1,$ $2,$ $3,$ $4,$ $5,$ and $6,$ the probability of obtaining face $F$ is greater than $\frac{1}{6},$ the probability of obtaining the face opposite face $F$ is less than $\frac{1}{6},$ the probability of obtaining each of the other faces is $\frac{1}{6},$ and the sum of the numbers on each pair of opposite faces is $7.$ When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}.$ Given that the probability of obtaining face $F$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$