I think you misunderstand me. I know probability reasonably well.
You sound like you have never worked with random processes in any real world application.
There are many thing which appear, superficially, to be random, producing various types of distribution at a macro level.
I don't know what you mean. When you shuffle an ordered deck of cards, the information that was present in the ordering is lost. How is that only superficial?
But when you look more carefully at the detailed processes at work, they are not random. For instance, throwing a dice could be calculated with the mechanics of a bouncing cubic object;
If that were true then every toss would be the same in controlled experiments. The processes at work are random processes -- otherwise we would be living in a deterministic world.
and the weather can be predicted with more and more accuracy by using more and more powerful computers, eg. to calculate smaller and smaller moving parcels of air;
I can guess, with narrowing margin of error, the consecutive deals at a game of blackjack, merely by counting the cards. That doesn't change the fact that the cards are shuffled (randomized).
where, superficially, both of those things have the appearance of randomness.
Loss of information can't properly be called superficial. It's either there or it's not.
Throws of a dice will produce an even distribution,
The distribution tells you the probability of rolling a given number. It does not tell you what the next roll will be with certainty. This is what we mean by "random".
and the weather can be said to have a "chance" of rain, and so on.
But there is uncertainty. Hence, we call it random.
I think there are very few truly random processes.
That would be a very challenging PhD dissertation to defend.
But there are plenty which seem too complicated to work out mechanistically, and look random from a distance.
The problem with looking at a distance is that you lose the scope of what nature is actually doing. Consider, for example, the difference in outcomes from rolling one die or rolling two. That difference has nothing to do with the mutual effects of air turbulence. It has to do with the
nature of two die versus one. That nature has nothing to do with mechanics. In other words, there is more to Nature than mechanics. I will call this case "natural causes" although there is no causation here, merely the intrinsic nature of having two dice as opposed to one.
I think you'll find that it's not distance from nature, but a proximity to it, which affords the best vantage point for deciding how things work.
I am suggesting that - most likely (imo) - genetic mutations and combinations are like that - superficially random, but in fact very predictable, if only we knew the processes at work.
I think the processes at work are not that mysterious. Chemicals bond, but only probabilistically. At the tails of the distribution, no bonding occurs, or else the improbable bonding of the wrong nucleotide or amino acid occurs, and a mutation results. Crossover is more obviously random for reasons best described as systemic rather than chemical. Every crossover produces a new random assortment of alleles, whereas genetic mutations due to bonding errors are rare.
Did you do the fruit fly experiment, or at least read about it? I think the process of random mutation is well enough understood to address your doubts. Are you familiar with the evolution of the peppered moth? Anthropogenic soot was the cause of the change in niche pressure (from light coloration to dark). But random mutation provides the cause for spontaneous variations; either that or the random re-emergence of the recessive gene causes it, probably through the randomizing effect of chromosome crossover during meiosis.
Of course it was the pressure of predatory selection which caused the evolution to occur.
There you have two simple examples of random causes of mutations, the second of which lead to a remarkable shift in the dominant phenotype.
Incidentally, I think the best place to arrive at an understanding the stochastic nature of genetics is, well, genetics. If you aren't familiar with Mendel's pea plant experiments, it's the seminal work that explains randomization of the genes in crossover. You can't say with certainty what the color of the flower or pod will be, but control this by hybridizing, and you get a randomized result which follows the distributions Mendel discovered.