Let $X_n$ be the "random Fibonacci sequence," defined as follows:

$X_0 = 0, X_1 = 1$;

$X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips.

It is known that $|X_n|$ almost surely grows exponentially by a (much more general) theorem of Furstenberg and Kesten about random matrix products: the base of the exponent was determined explicitly by Viswanath to be $1.13\ldots$

I am not too proud to say that I learned all this from Wikipedia:

http://en.wikipedia.org/wiki/Random_Fibonacci_sequence

What I did not learn from Wikipedia, or any of the references I gathered therefrom, was:

Question: what, if anything, do we know about the probability that $X_n = 0$, as a function of $n$?