We introduce a model of a {\em randomized branching program}
in a natural way similar to the definition of a randomized circuit.
We exhibit an explicit boolean function
$f_{n}:\{0,1\}^{n}\to\{0,1\}$ for which we prove that:

1) $f_{n}$ can be computed by a polynomial size randomized
ordered read-once branching program with a small one-sided error.

2) $f_{n}$ cannot be computed in polynomial size by any
nondeterministic ordered $read$-$k$-$times$ branching program
for any $k=o(n/\log n)$. The required nondeterministic size is
$2^{\Omega (n/k)}$.