  The shot noise in the electrical current through a ballistic chaotic quantum
dot with N-channel point contacts is suppressed for N --> infinity, because of
the transition from stochastic scattering of quantum wave packets to
deterministic dynamics of classical trajectories. The dynamics of the electron
spin remains quantum mechanical in this transition, and can affect the
electrical current via spin-orbit interaction. We explain how the role of the
channel number N in determining the shot noise is taken over by the ratio
l_{so}/lambda_F of spin precession length l_{so} and Fermi wave length
lambda_F, and present computer simulations in a two-dimensional billiard
geometry (Lyapunov exponent alpha, mean dwell time tau_{dwell}, point contact
width W) to demonstrate the scaling (lambda_F/l_{so})^{1/alpha tau_{dwell}} of
the shot noise in the regime lambda_F << l_{so} << W.
