This is mostly going to be a reply to your blog post. And any quotes below will be quotes from that blog post.
I’m not Joshua. I probably cannot satisfactorily explain what I mean by “random”, and Joshua probably cannot explain what he means.
For me, as a mathematician, “random” is a term in the mathematical theory of probability. But the theory does not explicitly define “random”. Rather, “random” is implicitly defined by its role in the theory.
When we leave mathematics and get to the word “random” in ordinary speech, it is even more murky. Some people are determinists. They believe that everything that will happen in reality is already determined. And, for determinists, the word “random” could not strictly apply to anything in the real world. But even determinists find it useful to talk of random events, though they may insist that they are not strictly random. Or, to say that differently, many people use “random” quite loosely without needing a tight definition.
I should note that some Christians are deteriminists, though they prefer to use the term “predestination” rather than “determinism.”
An electrical engineer might well use the expression “white noise”. And that’s a term used to specify a particular pattern of randomness.
We often talk of probability distributions. There are some important ones, such as the Normal distribution, the binomial distribution, the exponential distribution, the Poisson distribution. These probability distributions are patterns of randomness.
That’s fairly close to how biologists are using “random” when they talk of random mutations. But I’ll note that I am a mathematician, not a biologist.
I’ll have to disagree with that one. In particular, I disagree with the “equal chances”. You would have equal chances with a uniform probability distribution, but not with other important probability distributions.
In biology, I would not expect “equal chances” to apply. Some genes mutate at a higher rate than other genes.
I suspect that your miscommunication with Joshua has more to do with “pattern” than with “random”. You are thinking of a sequence of random events, and not expecting a pattern in the particular sequence of events. And I think Joshua was looking at the overall distributing, and that was where he was suggesting there could be a pattern to the randomness.