2006 AIME II Problem 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 1 / 6, the probability of obtaining the
face opposite face F is less than 1 / 6, the probability of obtaining each of the other faces
is 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 47 / 288. Given that the probability of obtaining face F is m / n,
where m and n are relatively prime positive integers, find m+n.