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Anomaly-free perturbations of loop quantum cosmology reveal a deformedspace-time structure, in which the signature changes when the energy density is$\rho=\rho_c/2$. Furthermore, in loop quantum cosmology, one can obtain aneffective 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 investigatethe horizon problem and the probability of inflation in the framework of loopquantum cosmology. Two models are considered: the quadratic inflation and thenatural inflation. We use the Liouville measure to calculate the probability ofinflation which solves the horizon problem, and find that, for the quadraticinflation model, the probability is very close to unity, while for the naturalinflation model, the probability is about $35\$.

Loop quantum cosmology: The horizon problem and the probability of inflation