A moose and a hunter on a board The game board has 12 spaces. A moose 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 moose moves that many spaces forward. On a result of 5 or 6, the hunter moves that many spaces forward. The moose 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 he catches the moose, in other words reaches the same or a higher space.

What are the probabilities of winning for the moose and the hunter?

 A: 
What are the ways in which the hunter can catch the moose?

If the hunter does catch the moose he takes at most two moves, since all his possible positions after two moves (11, 12, 13) are far enough for him to catch a moose still on the board. If he takes only one move, he must take six steps on the very first roll and land on the deer's initial position at 7; this scenario has probability $\frac16$.
All other scenarios where the hunter wins fall into two classes.


*

*Hunter moves 5 steps first, moose moves, hunter captures. There are 15 ways the moose can move while staying on the board:
1,2,3,4
11,12,21,13,22,31
111,112,121,211
1111
For each case where the moose takes $n$ moves, the associated probability is $\frac2{6^{n+2}}$: there are $n+2$ rolls in total and the hunter's last move can be either 5 or 6.

*Moose moves first, hunter moves twice subsequently to capture. The moose's moves can be any of the same 15 sequences presented above, but here both hunter's moves are irrelevant. If the moose makes $n$ moves the associated probability is $\frac{4n}{6^{n+2}}$: $n+2$ rolls, 4 choices for the hunter's moves and $n$ choices for the position of the hunter's first move among the moose's moves.


In summary, the probability that the hunter catches the moose in two moves works out as
n     prob     in base 6
1 4 ×  6/6^3 = 0.040
2 6 × 10/6^4 = 0.0140
3 4 × 14/6^5 = 0.00132
4 1 × 18/6^6 = 0.000030
         sum = 0.055350

When the $\frac16$ chance of the moose getting captured with one hunter's move is added, the final probability is
$$0.15535_6=\frac{2579}{7776}=0.331661\dots_{10}$$
Since only one party can win, the probability of the moose winning is
$$\frac{5197}{7776}=0.668338\dots$$
