One of the reasons Intelligent Design / Intelligent Origin Theory manages to keep a foothold in society is that a lot of people really don't understand the terms used in science.
One term that causes all kinds of problems is "random". Colloquially, "random" is taken to mean completely unpredictable and chaotic. That's not quite what it means in science, and certainly not in mathematics.
Take a fair coin and flip it. Assuming the coin is, in fact fair (not weighted, not tampered with in any way) and given a fair flip (so as not to influence how it lands), it has a 50/50 chance of landing "heads" or "tails".
It will land on whichever side it lands at random. That means, there will be a 50% chance of the coin landing "heads" and a 50% of the coin landing "tails" at each and every toss. And that will be all you can say about how the coin will land until it actually lands.
Despite the coin toss being random, you can still make predictions about it. For example, you can bet any sum of money you care to name, the coin will never come up anything but "heads" or "tails".
Likewise, if you roll a six-sided die, the numbers 1 through 6 will each come up, at random, an average of once in six times. You'll never be able to tell which of the numbers is going to come up on the next roll, because it's always one chance in six at each roll. But you can predict, with 100% certainty, you'll never roll a 7 or higher. Ever.
When mathematicians use the word "random", they mean that a variable will show up in some piece of a probability space at a frequency proportional to the size of that piece. If your probability space is six boxes, numbered from 1 to 6, corresponding to the sides of a six-sided die, each box represents one-sixth of the probability space.
Each time you throw the die, you are generating a new value for your variable. It will land in the probability space – one of those six boxes – 100% of the time. Each box has a 16-2/3 % chance of being the box the variable appears in. In this case, the boxes in the probability space are equal in size. In many other cases – such as a weighted die, the boxes become unequal in size.
Darwin postulated that all living things would differ in small ways from their parents, and that these differences would be in random directions. If you measured height, some would be a little taller, others would be a little shorter, and some would be just about the same height as their parents. Likewise, if you measured any other feature, you'd find similar small changes between parent and offspring. If you measured the differences between parent and offspring, you'd very likely find these differences clustered around the parental value, in a bell-shaped curve.
There's no real magic to the bell curve (or Gaussian curve). It's simply the result of a mathematical operation called the convolution integral, applied to a large number of variables. And most of the time, the exact shape of the curve doesn't matter anyway.
In Darwin's day, no one knew much about how characteristics were inherited. Well, no one except one Gregor Mendel, and he wound up publishing in an obscure botanical journal, and Darwin never read the paper. The point is, Darwin didn't know what sort of constraints there were on how living things could change from generation to generation. He had no idea what the probability space looked like.
In the past 150 years, the science of genetics has advanced quite a bit, and we have a much better idea of what the probability space looks like. For example, a point mutation at any DNA base can produce one of four results. There are four nucleotide "letters" in the DNA "alphabet", and if you change one of them at random, you're rolling a four-sided die. There's a 75% chance that a random mutation of one "letter" will change it to one of the other three "letters" – 25% for each "letter" – and a 25% chance that it will be "changed" to the same "letter".
The nature of the change then depends on where in the DNA that "letter" appears. Some locations are not terribly critical, and you can make any change you like, with no effect on the organism at all. Others, such as a particular critical section of the DNA that codes for red blood cells, can make all the difference in the world.
The point, though, is that these changes remain random.
They are constrained by the chemistry of the system, so that a purine nucleotide base will never be replaced with a Buick, but within those constraints, there is no preferred direction of change. Changing adenine at any location in the DNA out for thymine is just as probable as changing it out for guanine. There is no force pushing the change in any particular direction.
That's what is meant by "random".
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