  We consider the Cauchy problem for a generalized KdV equation
\begin{eqnarray*}
  u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0,
  \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl.
Math.46(1993), 527-620)proved that the problem is globally well-posed in
H^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling critical
regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc.
53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the
time of existence T and the continuous dependence cannot be expressed in terms
of the size of the data in the H^{\frac{3}{14}}-norm. In this present paper, we
prove that almost sure local in time well-posedness holds in H^{s}(\R) with
s>\frac{17}{112}, whose lower bound is below \frac{3}{14}.
  The key ingredients are the Wiener randomization of the initial data and
probabilistic Strichartz estimates together with some important embedding
Theorems.
