Summary: In a recent paper [Part I, J. Complexity 24, No. 5-6, 582‒605 (2008; Zbl 1166.65021)] we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number $κ(f)$ for the input system $f$. In this paper we look at $κ(f)$ as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail $\Bbb P\{κ(f)>a\}$ and the expected value $\Bbb E(\log κ(f))$. Part II, see J. Fixed Point Theory Appl. 6, No. 2, 285‒294 (2009; Zbl 1215.65218).