The problem prompt asks for \(P^1_1 = \frac{\bar\gamma}{\sqrt{5}}\left(1 + \frac{1+\bar\gamma^2}{1 - \bar\gamma^2}\right) = \frac{1}{\sqrt{5}}\frac{2}{\sqrt{5}} = \boxed{\frac{2}{5}}\), the probability of succeeding if we start at \((1,1)\).
Similarly, we can also compute \(P_0^1 = \frac13 + \frac23 \frac25 = \boxed{\frac35}\) the probability of succeeding if we start at \((1,0)\). We can verify these values with some monte-carlos testing. For example, we can simulate a run using the following python code:
def simulate(x):
if not x: return 1
y = random.randint(-1,1)
if not y: return 0
return simulate(x+y)
def simulate1(x):
y = random.randint(-1,1)
if not y: return simulate(x)
return simulate1(x+y)
Doing a simple trial gives me statistics that converges to the values we’ve computed.