One of the motivations for the introduction of inflation was to explain the large entropy of the observed Universe. Penrose has recently pointed out that the entropy of the observable Universe is small when compared with the maximal possible entropy. The authors show that this phenomenon is also a natural consequence of inflation. Furthermore, inflation is generic in the expanding phase but exceedingly unlikely in the collapsing phase. This introduces an asymmetry in the evolution of the Universe which resembles to a large extent the asymmetry that appears in the growth of entropy and it might be related to the question of the arrow of time.