The probability of the evolution of a character depends on two factors: the probability of moving from one character state to another character state and the probability of the new character state fixation. More the evolution of a character is probable more convergent evolution will be witnessed, consequently, convergent evolution could mean that the convergent character evolution result as a combination of these two factors. We investigate this phenomenon by studying the convergent evolution of biochemical functions. We use for the investigation the case of β-lactamases. β-lactamases hydrolyzes β-lactams which are antimicrobials able to block the DD-peptidases involved in bacterial cell wall synthesis. β-lactamase activity is present in two different superfamilies: the metallo-β-lactamase and the serine β-lactamase superfamily. The mechanism used to hydrolyze the β-lactam is different for the two superfamilies. We named this kind of evolution an allo-convergent evolution. We further show that the β-lactamase activity evolved several times within each superfamily, a convergent evolution type that we named iso-convergent evolution. Both types of convergent evolution can be explained by the two evolutionary mechanisms discussed above. The probability of moving from one state to another is explaining the promiscuous β-lactamase activity present in the ancestral sequences of each superfamily, while the probability of fixation is explained in part, by positive selection as the organisms having β-lactamase activity allows them to resist to organism secreting β-lactams. Indeed a mutation increasing the β-lactamases activity will be selected as the organisms having this activity will have an advantage over the others.