There are three mathematical ideas that one needs to come to terms with in order to get the full flavor of how natural selection works.

1. One is the rate at which favorable mutations occur in organisms. These do occur by chance and the question is whether the frequency of such occurrences is sufficient to explain evolution.
2. The second is the rate at which favorable mutations become more numerous in the population. It is not enough to produce a single favorable organism. The population of varieties with advantageous properties has to eventually grow to sufficiently high numbers that it dominates the population and can form the basis for yet further mutations.
3. The third is whether the rate at which repeated small and favorable mutations build on each other is sufficient to produce major changes in complex systems (the eye, ear, and other organs for example) and even entirely new species.

It is only the very first item that works by pure chance. The other two are highly directed processes, not because there is an external intelligence at work but because they are subject to the pressures of natural selection, which considerably reduces the contributions of chance to the outcome.

Now it is undoubtedly true that the chance of producing a favorable mutation is small. Most mutations are deleterious to the organism. The chance of a favorable mutation, once produced, taking hold and becoming widespread in a species population is also small. And the chance of favorable mutations building on each other to produce complex organisms is also small. So if we leave things at this high level of generality, skeptics of natural selection can (and do) argue that the complexity of life as we know it is too unlikely to have occurred and that therefore some intelligence must be behind it. To get beyond that superficial argument and appreciate the power of the theory, one has to actually do the calculations.

Darwin himself was well aware of these difficulties but also had the intuitive sense that even events with very small individual probabilities have a good chance of occurring if you wait long enough and have large enough populations. Although he could not quantify it, Darwin knew that he needed a very long time for his theory to work, which is why he viewed with such interest research on the age of the Earth. All three processes listed above must be able to fit within the timeline allowed by the age of the Earth, which is why research in geology and physics have had important implications for the theory of evolution. But since the time scales involved are well beyond our own lifetimes, people have a hard time comprehending the workings of evolution.

As an example of this, take the lottery. The chance of buying one ticket and selecting six numbers from 1 to 49 that match the winning numbers is incredibly small (to be precise 1 in 13,983,816). But you can greatly improve your chances if you buy many tickets and plan to play week after week. The greater the number of tickets you buy, the shorter the time in which you can expect to hit the jackpot. Of course, even if you live long enough and invest enough, the total amount you spend on your tickets will almost always be much more than the amount you win but that is because the organizers of the lottery have pegged the prize money that way so that they can make a profit.

Only the first of the three items listed above for natural selection (the occurrence of favorable mutations) works the same way as the lottery, except that nature hasn’t rigged the system against you. Nature just doesn’t care. And this means that if there are large enough populations and long enough times available, natural selection will repeatedly hit the jackpot and produce the wonderful complexity we see.

One of the fundamental features of the theory is that mutations, or changes in organisms, occur at random. Most of these mutations are either fatal or sufficiently harmful to the organism so that the mutated variety dies away. After all, if you make random changes in anything (say the wiring of your computer or even your toaster) there is a much greater chance of making it worse than making it better. But on rare occasions, a beneficial mutation will occur that results in that new variety flourishing because it is better adapted to succeed in its current environment.

We now know something that Darwin did not, that these mutations occur at the level of the genes. Although the work that led to the discovery of the genetic laws of inheritance was done by Gregor Mendel at roughly the same time as Darwin and provided the material basis for understanding inheritance, Darwin was not aware of that cloistered monk’s research, although Mendel was aware of Darwin’s work. Mendel published his seminal paper in 1865 (Darwin’s On the Origins of Species appeared in 1859) but it went largely unnoticed until 1900 when several biologists who had been working on the problem of inheritance, independently came across Mendel’s work.

The synthesis of Mendel’s work on genetics with Darwin’s theory of natural selection is one of the great advances in modern science and the next post in this series will discuss that relationship.

Next in the series: The effect of Mendel’s work on Darwin’s theory

POST SCRIPT: Onion parody on evolution

The nice thing about this parody is that it captures very well the central problem with the arguments of intelligent design creationists and other religious believers who want to preserve a role for god by carving out a little niche for god to intervene in evolution.