  We study the distribution function $P(\omega)$ of the random variable $\omega
= \tau_1/(\tau_1 + ... + \tau_N)$, where $\tau_k$'s are the partial Wigner
delay times for chaotic scattering in a disordered system with $N$ independent,
statistically equivalent channels. In this case, $\tau_k$'s are i.i.d. random
variables with a distribution $\Psi(\tau)$ characterized by a "fat" power-law
intermediate tail $\sim 1/\tau^{1 + \mu}$, truncated by an exponential (or a
log-normal) function of $\tau$. For $N = 2$ and N=3, we observe a surprisingly
rich behavior of $P(\omega)$ revealing a breakdown of the symmetry between
identical independent channels. For N=2, numerical simulations of the quasi
one-dimensional Anderson model confirm our findings.
