Smart home technologies are a promising way to improve health safety of frail people living alone at home. They allow for example on-line recognition of Activities of Daily Living (ADLs) performed by a person, in order to detect dangerous or unusual behaviour. Since human behaviour is not deterministic, probabilistic approaches are often used for ADL recognition, despite difficulties encountered in model building and probabilistic indicators computing. In this paper, it is proposed an approach, based on a Probabilistic Finite State Automata, to detect which activity is being performed. For that a new indicator, called the normalised likelihood, is proposed. The robustness of this indicator to the size of the observed behaviour as well as its computational complexity are also addressed. Finally, the quality of the obtained results are discussed on the basis of an experiment performed in a living lab.