Can Probability Theory Be Used to Refute Evolution? (Part One)
September 19, 2005
In a privately published volume entitled The Evolution of Man Scientifically Disproved, in Fifty Arguments, the Reverend William A. Williams, writing in 1925, offered the following thought:
The evolution theory, especially as applied to man, likewise is disproved by mathematics. The proof is overwhelming and decisive. Thus God makes the noble science of mathematics bear testimony in favor of the true theories and against the false theories.
This refrain has been a mainstay of creationist literature ever since.
Claims of mathematical disproof of evolution typically make use of probability theory. The idea is to show that it so improbable that a given complex biological structure, such as the vertebrate eye, could have evolved gradually that it is effectively impossible for it to have done so. Frequently this argument will include actual calculations purporting to place the assertion on a rigorous mathematical footing.
As a professional mathematician, I am well aware of how impressive such calculations can appear to people untrained in probability. Math is unique in its ability to bamboozle a lay audience, which helps explain why creationists find it so appealing. Happily, though, you do not need to slog through the details of such a calculation to know that it is not correct. Probability theory is a major branch of mathematics that finds countless applications in a variety of sciences, but it is not powerful enough to support the sweeping conclusions creationists are trying to draw. To understand why, let us consider the basic elements of the subject.
Mathematicians use fractions to describe the probability that a particular event will occur. The top of the fraction records the number of favorable outcomes, while the bottom records the number of possible outcomes (among mathematicians, incidentally, it is quite common to refer to the top and bottom of a fraction, as opposed to the numerator and denominator.) In this context, a “favorable” outcome is simply one that conforms to whatever pattern we are interested in. A few examples will make this idea clear.
Let us suppose we are tossing a coin and want to know the probability that the coin will land heads. We would reason that since there are two possible outcomes (heads and tails), and only one of those outcomes is a head, we have a probability of 1/2 of tossing a head. Similarly, the probability that the coin will land tails is also 1/2.
Now suppose we have a standard deck of cards from which we draw a single card. What is the probability that the card will be a spade? This time we would argue that since there are fifty-two cards in the deck, thirteen of which are spades, the probability of drawing a spade is 13/52, which is the same as 1/4.
These two examples were specifically chosen for their simplicity. However, determining the number of favorable outcomes and possible outcomes can often be a tricky business requiring considerable ingenuity. For example, suppose we asked for the probability of being dealt a flush in a game of poker. Answering the question would require first that we determine the total number of possible five-card hands. Having done that, we would next determine how many ways there are of being dealt a flush. Both of these numbers can be calculated, but doing so requires a level of mathematical sophistication I do not wish to explore in this essay.
Instead, let us discuss a few issues that arise from my two examples above. First, in stating that the probability of tossing a head is 1/2, I was tacitly assuming that we were using a fair coin tossed in a fair way. If the coin were loaded in some way, or if a sleight-of-hand artist did the tossing, then the probability would no longer be 1/2. Similarly, I simply assumed that each of the fifty-two cards in the deck was as likely to be chosen as any other. Such assumptions are often not justified in real-world situations. In other words, instead of each outcome being as likely as any other, we might find that some outcomes are far more likely to occur than others. As we shall see, this fact presents an insurmountable barrier to most of the arguments creationists make in this regard.
Perhaps you are wondering why mathematicians use the language of fractions in describing probability. There are two reasons. One is that there does seem to be something intuitive in saying that if you draw cards over and over again from a deck, each time placing the card you chose back in the deck, then about one quarter of the time you will draw a spade. We have all flipped coins before, and we know that when we do so we commonly find that we get heads around half the time. The second reason is a bit more complicated. Often we are concerned not with the probability of an individual event happening, but rather with a whole series of events happening simultaneously. Other times we seek the probability that at least one of a given collection of events occurs. In many cases these questions can be answered by performing standard arithmetic operations on the individual probabilities of the events in question. Thus, complicated questions in probability can often be reduced to simple problems in fraction arithmetic.
Now let us attempt to apply this reasoning to evolution. What is the probability that an eye could arise gradually via known evolutionary mechanisms? In biological terms we are asking for the probability of evolving the genes necessary for constructing the eye, which immediately presents a problem. Complex structures like eyes do not arise from the action of a well-defined set of genes. Instead, there are many genes that play a role in eye formation, many of which serve other purposes as well.
But this objection is not yet fatal to the argument. While we may not be able to say specifically which genes are responsible for eye formation, we can reasonably assume there are quite a lot of them. Recall that genes are made from the four nucleotides adenine, thymine, cytosine and guanine (which we will abbreviate by A, T, C and G). Consequently, a gene can be modeled as a sequence whose elements are these four letters. As a conservative estimate, let us suppose that a gene one hundred letters in length is necessary to construct an eye. The actual number is surely far larger than this.
Therefore, the total number of possible outcomes in this case is simply the number of sequences of A’s, T’s, C’s and G’s that are one hundred letters long. This number is obtained by multiplying four by itself one hundred times, which is a very large number indeed. Only one of those sequences codes for the eye, as we know it. There are surely a fair number of trivial changes we could make in the precise gene sequence that will also produce the eye. Therefore, the number of favorable outcomes in this case will surely be greater than one. However, we can assert with some confidence that the number of favorable outcomes will be far smaller than the number of possible outcomes.
This seems to show that, while we may not be able to calculate precisely the probability of evolving the genes necessary for eye formation, we can still assert that the probability is very, very small.
Have we done it? Can we conclude that it is effectively impossible for evolution to have produced an eye? Many creationists would say that we could. You will find the argument described in the previous paragraph, presented in varying levels of detail, in a great many creationist outlets. Sadly, their analysis overlooks several crucial points.
Perhaps you have already spotted the flaw in this argument. In carrying out our calculation, we simply assumed that every hundred-letter gene sequence was as likely as any other. This assumption is completely unwarranted, for two reasons.
First, keep in mind that evolution works its magic by modifying preexisting structures. Consequently, the particular gene sequences likely to occur in a given generation are those attainable from preexisting sequences via known genetic mechanisms. As an example, suppose that in some organism we find the gene sequence ACGATCT. One source of genetic variation is the point mutation, in which an individual nucleotide is replaced in the next generation with a different nucleotide. Thus, it is perfectly reasonable to suppose that the offspring of our hypothetical organism will possess the gene sequence ATGATCT. By contrast, it is highly unlikely that we will encounter the sequence TGATAAG.
Second, we have ignored the action of natural selection in our reasoning thus far. Most of the hundred-letter gene sequences we could write down would lead to a badly defective organism were they to be found in nature. So even if the odd macromutation caused one of these sequences to appear in some unfortunate organism, natural selection would ensure that the gene was quickly flushed from the population in subsequent generations.
In summary, our proposed calculation of the probability of evolving an eye over a period of millions of years runs into two major obstacles. First, the fact that evolution works by modifying preexisting structure means that certain gene sequences are far more likely to occur than others. Second, the sieving action of natural selection guarantees that defective gene sequences will not linger for long in nature.
These obstacles are insurmountable, and they are fatal to any attempt to rule out evolution via probability theory alone. It would require almost God-like knowledge of natural history and the physiologies of long-extinct organisms to produce a meaningful probability calculation for any complex biological system.
Actually, though, the situation is even worse than that. For suppose that somehow we did manage to carry out such a calculation and suppose we found that it really is terribly improbable than our eye evolved by natural means. What would we learn from such a result?
Almost nothing. Improbable things happen all the time, you see, and the fact that something is improbable does not mean that it cannot happen. As a simple example, the next time you drive somewhere think about how improbable it was that all of the drivers in cars near you would be on the same road at the same time. There is an old adage that million-to-one odds happen eight times a day in New York City. Merely discovering, after the fact, that something terribly improbable has occurred gives us no warrant for seeking an extraordinary explanation.
But perhaps the situation is not as simple as I am suggesting. Let us suppose that I flip a coin ten times and obtain the following sequence of heads and tails:
It is fairly straightforward to show that there are 1,024 possible outcomes that could arise from such a series of coin flips. Since exactly one of those sequences matches the one above, we conclude that the probability of obtaining this particular sequence is 1/1024.
Now consider this sequence:
We can reason as before to conclude that the probability of this sequence is also 1/1024.
The fact is that any particular sequence of heads and tails is as unlikely as any other. Many people find this hard to accept. The second sequence just looks more improbable than the first. Why is that?
What strikes us about the second sequence is that it matches an easily identifiable pattern. The first sequence, by contrast, looks like every other jumble of heads and tails we have ever obtained by flipping a coin multiple times. This suggests that, while improbability by itself does not suggest anything extraordinary, the combination of improbability with a recognizable pattern does require a special explanation. Could we use this strategy to revive our probabilistic critique of evolution?
Many proponents of Intelligent Design believe that we can. In the second part of this essay, I will show why they are wrong. We will also consider some ways in which evolutionary biologists make legitimate use of probability theory in their work.