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?
 A: 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 $1,6,7,11$).
A: The following Python code calculates the probability of the hunter to win:
def func(hunter,goose,num):
    if hunter >= goose:
        return 1.0/6**num
    if goose >= 12:
        return 0.0
    probability = 0.0
    for val in [1,2,3,4]:
        probability += func(hunter,goose+val,num+1)
    for val in [5,6]:
        probability += func(hunter+val,goose,num+1)
    return probability

print func(1,7,0)

Yields the following outcome:
0.387217078189

So the probability of the hunter to win is approximately $38.72\%$.
A: I ran a computer simulation of $10$ million decisions once and got the following results:
Goose winning probability:  $61.25637$%
Hunter winning probability: $38.74363$%
Easy simulation program to write in any general purpose computer language having a random number generator.  No point is showing the code here cuz since it is so simple and the person who asked the original question did not ask to see it.  I just grab a random integer number from $1$ to $6$ and "move" that many spaces, checking for a winner and until I get a winner.  I repeat that decision $10$ million times which is an arbitrary number but chosen as a balance of speed and accuracy.  I could have chosen $100,000$ or $1$ billion but there is not much advantage to doing so.
Also note that an interesting variation of this question is if you start the goose at position $6$ instead of position $7$, the probability of the goose winning becomes very close to a coin flip ($50$/$50$).  I get about $50.3$% in favor of the goose ($49.7$% for the hunter).
