Is a turing machine with random number generator more powerful? Let's extend the Turing machine so that it can read from a stream of random number generators (in addition to an infinite tape to read and write). Certainly the TM with randomness can do whatever a classical TM do, but what about the converse?

One can argue that the classical TM will always generate the same result given the same input,  while the TM with randomness can behave randomly, it can do more. But, then random-valued functions are not really what we call computable. I am aware of randomized algorithms and BPP and what not, but is there an extension of computability that deals with these kind of questions?
See https://mathoverflow.net/questions/58060/can-randomness-add-computability