But she refuses to acknowledge that it is the correct definiton of "random" and insists that it is "essentially meaningless" and doesn't described evolution because Dawkins and Sagan say otherwise.
There is "no" correct definition. Your OP becomes "what is the evidence (peer reviewed) that evolution is not related to or described by a probability chart"?
There is no such evidence.
If you want to understand why biologists say that natural selection is the opposite of random or why they think it is misleading calling it that, you need to use the word as they define it. If you define random has having an equal probability of occurring then mutations are relatively random, while natural selection (what survives) is not.
I'm not saying the way you say it is meaningless because of anything Dawkins and Sagans say (their words are carefully chosen so as not to fit the creationist strawman, however), I'm saying it's meaningless because I don't think any processes or the evolution of anything can't be described as random given the loose definition you've given to random. Your definition doesn't do anything to distinguish itself from the creationist strawman. Evolution is random if you define random a lot of ways. Evolution is purposeless...not planned...influenced by relatively random occurrences.
You seem to have this need to prove Dawkins wrong--he's not wrong, he's just more careful and explanatory in his words...the Berkeley site even has a glossary for the word random. I don't know whether you find it useful or meaningful to describe the evolution of this thread as random, but I wouldn't. If someone specifically asked what the non-random aspects were, I would have a lot more to say then there are none.
You've defined random so vaguely that to say that evolution is or isn't random ends up meaning nothing at all...and it's a statement indistinguishable from the claim that makes creationists sum up the scientific understanding of evolution as a tornado in a junkyard building a 747.
There isn't anything to argue about. You've turned this into a semantic game. There is no single "correct" meaning for random. And there is no single correct way to define evolution. The biologists are going out of their way to explain preferential survival (natural selection) so as not to make evolution sound like the chance driven impossibility that creationists try to dismiss it as.
There is something about evolution which makes it very different than the tornado in a junkyard analogy,--due to people like Deepak Chopra et. al. we are forced to go out of our way and be extra clear so that people can understand what that difference is. Your wording doesn't do it. It's a simple concept, but due to this endless obfuscating under the pretense of looking for understanding or pretending to know what the hell they are talking about, biologists are forced to define their terms before hand and use their wording carefully. They are not incorrect. They just aren't defining random as loosely as you are. Moreover, they don't think that having random components necessarily makes an entire process random...because then ALL processes, would in essence, be random. Your definition does nothing to distinguish the relative randomness of mutation from the far less random inputs that affect selection--
that which brings "order" to the randomness.
Remember, evolution is a fact. The words don't change the facts. Words can be used to understand the facts or obfuscate the facts. There is nothing to win or lose when it comes to the FACTS. They are the same for everybody no matter what words you use.