Question 31
The faces of a 12-sided die are numbered $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$, and $12$ such that the sum of the numbers on opposite faces is $13$. The die is meticulously carved so that it is biased: the probability of obtaining a particular face $F$ is greater than $1/12$, the probability of obtaining the face opposite $F$ is less than $1/12$ while the probability of obtaining any one of the other ten faces is $1/12$. When two such dice are rolled, the probability of obtaining a sum of $13$ is $29/384$. What is the probability of obtaining face $F$?