We use a reflection argument, introduced by Gessel and Zeilberger, to
count the number of k-step walks between two points which stay within
the chambers of a Weyl group. We apply this technique to walks in the
alcoves of the classical affine Weyl groups. In all cases, we get
determinant formulas for the number of k-step walks. One important
example is the region m>x_1>x_2>...>x_n>0, which is a rescaled alcove of
the affine Weyl group C_n. If each coordinate is considered to
be an independent particle, this models n non-colliding random walks
on the interval (0,m).