Behe botches basic probability...how likely is that?

In The Edge of Evolution, Michael Behe presents arguments against the role of random mutation in large-scale evolutionary change, using probability calculations that are so utterly mishandled that they call into question his scientific credibility and integrity. To present evolutionary genetics in this way, one must be possessed of both ignorance and arrogance (a nasty combination) and/or of the kind of compromised scientific integrity that gives rise to folk science. The issue I raise here is independent of Behe's many errors regarding the biology of genetics and adaptation. It's all about misusing probability, and it looks suspiciously like a bait-and-switch. Here is one of many places in EoE where Behe makes a very basic mistake in the presentation of probabilities:

Recall that the odds against getting two necessary, independent mutations are the multiplied odds for getting each mutation individually. What if a problem arose during the course of life on earth that required a cluster of mutations that was twice as complex as a CCC? (Let's call it a double CCC.) For example, what if instead of the several amino acid changes needed for chloroquine resistance in malaria, twice that number were needed? In that case the odds would be that for a CCC times itself. Instead of 10²⁰ cells to solve the evolutionary problem, we would need 10⁴⁰ cells. (pp. 62-63)

What Behe is saying is this: if event A has probability a, and event B has probability b, then the probability of both events happening is a times b. But that is only true if the events must happen simultaneously. That's the only time you multiply two probabilities. And to make matters worse, Behe is confusing two very different probabilities: the probability that the event will happen in any given attempt, and the probability that it will occur at all.

Consider the following example to see how these mistakes (if that's what they are) are so tremendously misleading. Suppose I hand you a pair of dice and ask you: "What is the probability that you will roll snake eyes?" You might immediately ask: "In how many rolls?" And that is quite an important question. The probability that you will roll snake eyes on any given roll is 1/36, which is the probability of getting a 1 on the first die (1/6) times the probability that you will get a 1 on the second die. Because the two events must happen together, we multiply the probabilities of the separate events. And that probability, 1/36, is pretty small.

But what is the probability that you will get snake eyes in, say, 12 rolls? It is in considering the second question that you can begin to see how Behe went badly wrong in his arguments in EoE. When Behe reports on the likelihood of occurrence of a "double CCC," he reports its probability of occurring in a single attempt. That's what the simple multiplication assumes, and Behe knows that this is nonsense.

The real question, then, is this one: what is the likelihood that a certain event will occur given a certain number of attempts? Behe is happy to calculate probabilities based on crude estimates of certain events in a certain organism, but he never takes a whack at the only interesting question: assuming a certain mutation rate, and a certain number of effective generations, what are the probabilities involved in particular mutational trajectories that have led to adaptation? What, for that matter, is the probability of a particular pair of mutations occurring in a human gene over a certain number of generations?

Back to our example of dice rolling. Behe's suggestion that we multiply the probabilities of "independent mutations" assumes that the two mutations occur simultaneously. Using his assumption of 1 in 100 million (10⁸) as the probability of occurrence of a given point mutation, then we would conclude that the probability of a double mutant is 10⁸ times 10⁸, or 10¹⁶. But how many attempts do we get? It matters a lot. If you have 12 tries to get snake eyes, your odds improve dramatically – the probability is now 0.29. Not bad, but things get even better when we correct Behe's other colossal error, which was assuming that the mutations must happen at the same time in the same organism. We know that's not true – even Behe knows that's not true (p. 111). This makes the game entirely different. Now I give you the dice and say, "How many rolls do you think it will take till you have a 1 on each die? The 1's don't have to appear together." Well, I did the calculations for one example scenario:

• You're trying to get two 1's, either together or sequentially.
• You get 6 rolls, if necessary, to get the first 1.
• After you get a 1, you get 6 rolls (if necessary) of the remaining die to get the second 1.

The probability that you will get snake eyes in this scenario is 0.60, and note that 12 is the maximum number of rolls here; many of the successes come in far fewer attempts.

It's hard for me to understand why Behe is so careless with such an important aspect of his argument. His whole case depends on probability, and yet the two weakest aspects of his story are his ludicrous extrapolations and his mishandling of probability. Perhaps he's just not a very careful thinker. Or perhaps we're seeing a sophisticated version of old-fashioned folk science, which typically depends on the kind of obfuscation that the EoE bait-and-switch suggests.