Iteration of Möbius transformations and attractors on the real line

Let z0 be an arbitrary point in the complex plane. For each positive integer n we choose sn(z) to be −5/(1 + z) or −0.5/(1 + z) equal probability. We introduce the orbit (zn)∞0, where zn = sn(zn − 1) for n ≥ 1. We prove that with probability one the orbit is attracted to the real axis. In the proof, we have to do some calculations on a computer.