I'm not so sure.
Why not?
There seems to be a lot of difficulty concerning the use of the word random. It is a word that can be used in different ways and there can be different degrees of randomness. Random doesnt necessarily mean that anything is possible, or that everything is equally possible. It is possible to have a certain degree of randomness within certain parameters. It is also possible for some outcomes to be more likely than others and still call the process random.
As to the question of whether genetic mutations are random, I think Zeno has responded to that question better than I can when he talks about the non-random elements. I think we can say that mutation is not completely random. Some mutations are more likely than others. Mutations may be more likely to occur on one part of the chromosome rather than other parts. Mutations may even be more likely in some environments than in others. But despite all this I still feel comfortable in saying that genetic mutation is random. There is no evidence that it is guided by any conscious force, and although it obeys certain physical laws, these physical laws do not seem to be guiding mutation in any specific direction. So yes, within certain parameters mutation is random.
And here is another thought.
Dr. Kenneth Miller has suggested that it is possible that genetic mutations may be influenced by quantum fluctuations. The number of atom that make up a microscopic gene is although huge, very small when compared to the number of atoms that make up any macroscopic object. It is possible that the quantum fluctuations that are factored out on the level of objects that we can see and interact with may not be totally factored out down at the microscopic level of the gene. If this is the case then genetic mutation may be even more random than we realize.
At any rate I dont believe that we currently have the ability to predict what kind of mutations will happen and when.