You are above a pit of fire and the only way out is a $1000-$ rung escape ladder. You're on the first rung.

Every second a standard die rolls. If it's $1$ or $2$ you move down one rung. Otherwise you will move up by one rung. If you move down from the first rung you will fall into the fire! If you move up from the $1000th$ rung, you escape.

What is the probability that you escape?

• $0$, because you have no way to move up.
– f''
Jun 19 '16 at 5:18
• No, you can get out of that pit. What if u get $4$ on every roll of die ? Jun 19 '16 at 5:20
• Then nothing happens, according to the problem.
– Deusovi
Jun 19 '16 at 5:21
• Apologies. I have added that part. Jun 19 '16 at 5:23
• Probability of escaping from the $k$th rung: $p(k)=\frac{1}{6}\left(2p(k-1)+4p(k+1)\right)$; solution is $p(k)=\beta_1+\beta_2 2^{-k}$; boundary conditions are $p(0)=0$ and $p(n+1)=1$; solving gives $\beta_2=2^{n+1}/(2^{n+1}-1)$, $\beta_1=-\beta_2$; therefore the answer is $p(1)=\left(2-2^{-n}\right)^{-1}\approx 1/2$. Jun 19 '16 at 5:45

For the $n$th rung, assign a value of $2^{1001-n}$, with escape having a value of $1$ and death having a value of $2^{1001}$. Every second, you have $\frac23$ probability of halving your value and $\frac13$ probability of doubling it. $\frac23(\frac12)+\frac13(2)=1$, so your expected value stays the same.
You start with $2^{1000}$ value and escape with probability $p$. The expected value remains constant, so $2^{1000}=1(p)+2^{1001}(1-p)$. Solving for $p$ shows that it is $\frac{2^{1000}}{2^{1001}-1}$.