Proteins with nearly the same structure and function (homologous proteins) are found in increasing numbers in phylogenetically different, even very distant taxa (e.g. hemoglobins in vertebrates, in some invertebrates, and even in certain plants). In discussing the origin of those proteins biologists hardly at all consider convergent evolution because the origin of proteins is held to be a random process, at least ultimately, since selection can work only what the random process delivers as having a minimum adaptive value. The repetition (...) of a random process with the same result is considered to be extremely unlikely. The supposed (un)likelihood, however, is almost never determined quantitatively. This paper attempts such a quantitative determination. It appears that the probability for the random origin of a definite protein is greater than what one would expect in view of the enormous number of equally possible nucleotide sequences in the corresponding gene since what is equally possible is not always equally likely. The probability, however, of the convergent evolution of two proteins with approximately the same structure and function is too low to be plausible, even when all possible circumstances are present which seem to heighten the likelihood of such a convergence. If this is so, then the plausibility of a random evolution of two or more different but functionally related proteins seems hardly greater. (shrink)