  Anomaly-free perturbations of loop quantum cosmology reveal a deformed
space-time structure, in which the signature changes when the energy density is
$\rho=\rho_c/2$. Furthermore, in loop quantum cosmology, one can obtain an
effective causal structure only for a low density region ($\rho\leq\rho_c/2$),
which gives a natural initial condition to consider the horizon problem.
Choosing the initial value at $\rho(0)=\rho_c/2$ in this paper, we investigate
the horizon problem and the probability of inflation in the framework of loop
quantum cosmology. Two models are considered: the quadratic inflation and the
natural inflation. We use the Liouville measure to calculate the probability of
inflation which solves the horizon problem, and find that, for the quadratic
inflation model, the probability is very close to unity, while for the natural
inflation model, the probability is about $35\$.
