probability puzzle with dice The game board has 12 spaces. A goose starts on space 7, and a hunter on space 1. On each game turn a 6-sided die is rolled. On a result of 1 to 4, the goose moves that many spaces forward. On a result of 5 or 6, the hunter moves that many spaces forward. The goose wins if it reaches space 12 (the final roll does not have to be exact, moving past space 12 is ok). The hunter wins if she catches the goose, in other words reaches the same or a higher space. What are the probabilities of winning for the goose and the hunter?

Every position is a pair $(h,g)$, the coordinates of the hunter and goose. Let $P(h,g)$ the probability the goose wins starting from the position $P(h,g)$. Then $$ P(h,g)=\frac16\bigg(P(h,g+1)+P(h,g+2)+P(h,g+3)+P(h,g+4)+P(h+5,g)+P(h+6,g)\bigg) $$ This, combined with the base cases $$ P(h,g)=1\qquad\text{when }g\ge 12 $$ $$ P(h,g)=0\qquad\text{when }h\ge g $$ allows you to calculate $P(h,g)$ for all values of $(h,g)$, starting with the simpler cases (the ones closer to the end of the game), and working backward to the starting position $P(1,7)$. Otherwise, this is just brute force. Not that much brute force, though, since there are less than $20$ positions (the goose can only ever be on spaces $7$ through $11$, the hunter on $1,6,7,11$).