TITLE: A random walk on random lines
QUESTION [16 upvotes]: I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $L_1$ through $p_0$ at a random orientation.
Now, from $p_0$ walk either right or left with equal probability along $L_1$ until
you hit the first intersection point $p_1$ of $L_1$ with $\{A,B,C\}$, or reach $\infty$.
Clearly there is a $\frac{1}{2}$ chance you stay finite.
Through $p_1$ generate a randomly oriented line $L_2$, and walk right or left to the
first intersection point $p_2$ of $L_2$ with $\{A,B,C,L_1\}$.  And so on.
On the one hand, there are always routes to $\infty$.  On the other hand,
as the line arrangement thickens, the number of steps to escape grows.
So if the process survives a few iterations, it becomes less likely it will escape.
Intuition is likely useless here, but it feels to me like this should remain finite. 
Can anyone see an argument?  
If it does remain finite, then other questions
suggest themselves, but perhaps I should start with the basic infinite/finite question.
Here is a hand-executed example (corrected from the original by Pablo):

     

Update. Here is a first attempt at an implementation:

     

I just stopped the iteration after $p_8$.
Addendum. Bill Thurston's convincing analysis coupled
with independent simulations suggest it remains
finite, perhaps with probability about $0.16$. Interesting questions
remain: What is the distribution of the number
of steps before shooting off to $\infty$?  When a path remains finite forever, does its total
length remain bounded?

REPLY [4 votes]: I don't know, how to find the probability analytically, but numerical simulation is actually simple. It suggests, that the process remains finite with probability $0.162\pm 0.003$. Here there is a simulation in Wolfram Mathematica 7.0.1. Typical examples of processes with 20+ steps are shown below. 20 steps are drawn. Initial triangle is always in the center, but on some pictures it is too small to see it.
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