Chance News 38

Quotations

A mathematician is a device for turning coffee into theorems.

Paul Erdős

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An improbable event and a coincidence

I have an example of an improbable event and a coincidence; it shows the difference between them. At Forrest's graduation last night, all of the seniors marched, in alphabetical order, to the stage to receive their diplomas. The women were wearing gray gowns and the men were wearing black gowns. I was careful to note any siblings (as far as I could tell, there were none). GREAT! So now we have a random sequence of coin tosses of length about 310, and the coin is pretty close to fair. The longest sequence of consecutive men I observed was 9; this is somewhat longer than the expected length of the longest run of heads, which is about 7, and somewhat longer than the expected length of the longest run of either heads or tails, which is about 8. So I observed a fairly unusual event. The coincidence is that Forrest was in the longest run of men.

An email from Charles Grinstead to Laurie Snell about his son's graduation.

Irreligion

Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up John Paulos.

John suggested that Chance News readers might enjoy some of the argments he used in this book that relie on probability concepts. You can see more of these arguments [ http://thesciencenetwork.org/BeyondBelief2/watch/paulos.php in a talk] he gave at the recent conference

A bit more precisely a common creationist argument goes roughly like the following. A very long sequence of individually improbable mutations must occur in order for a species or a biological process to evolve. If we assume these are independent events, then the probability of all of them occurring and occurring in the right order is the product of their respective probabilities, which is always a tiny number. Thus, for example, the probability of getting a 3, 2, 6, 2, and 5 when rolling a single die five times is 1/6 x 1/6 x 1/6 x 1/6 x 1/6 or 1/7,776 - one chance in 7,776. The much longer sequences of fortuitous events necessary for a new species or a new process to evolve leads to the minuscule probabilities that creationists argue prove that evolution is so wildly improbable as to be essentially impossible.

This line of argument, however, is deeply flawed. Leaving aside the issue of independent events, I note that there are always a fantastically huge number of evolutionary paths that might be taken by an organism (or a process), but there is only one that actually will be taken. So if, after the fact, we observe the particular evolutionary path actually taken and then calculate the a priori probability of its being taken, we will get the minuscule probability that creationists mistakenly attach to the process as a whole.

Here's another example. We have a deck of cards before us. There are almost 1068 - a one with 68 zeroes after it - orderings of the 52 cards in the deck. Any of the 52 cards might be first, any of the remaining 51 second, any of the remianing 50 third, and so on. This is a humongous number, but it's not hard to devise even everyday situations that give rise to much larger numbers. Now if we shuffle this deck of cards for a long time and then examine the particular ordering of the cards that happens to result, we would be justified in concluding that the probability of this particular ordering of the cards having occurred is approximately 1 chance in 1068. This certainly qualifies as minuscule.

Still, we would not be justified in concluding that the shuffles could not have possibly resulted in this particular ordering because its a priori probability is so very tiny. Some ordering had to result from the shuffling, and this one did. Nor, of course, would we be justified in concluding that the whole process of moving from one ordering to another via shuffles is so wildly improbable as to be practically impossible.

The actual result of the shufflings will always have a minuscule probability of occuring, but, unless you're a creationist, that doesn't mean the process of obtaining the result is at all dubious.

A related creationist argument is supplied Michael Behe, a key supporter of intelligent design. Behe likens what he terms the "irreducible complexity" of phenomena such as the clotting of blood to the irreducible complexity of a mousetrap. If just one of the trap's pieces is missing -- whether it be the spring, the metal platform, or the board -- the trap is useless. The implicit suggestion is that all the parts of a mousetrap would have had to come into being at once, an impossibility unless there were an intelligent designer. Design proponents argue that what's true for the mousetrap is all the more true for vastly more complex biological phenomena. If any of the 20 or so proteins involved in blood clotting is absent, for example, clotting doesn't occur, and so, the creationist argument goes, these proteins must have all been brought into being at once by a designer.

But the theory of evolution does explain the evolution of complex biological organisms and phenomena, and the Paley argument from design has been decisively refuted. Natural selection acting on the genetic variation created by random mutation and genetic drift results in those organisms with more adaptive traits differentially surviving and reproducing. (Interestingly, that we and all life have evolved from simpler forms by natural selection disturbs fundamentalists who are completely unphased by the Biblical claim that we come from dirt.) Further rehashing of defenses of Darwin or refutations of Paley is not my goal, however. Those who reject evolution are usually immune to such arguments anyway. Rather, my intention here is to develop some loose analogies between these biological issues and related economic ones and, secondarily, to show that these analogies point to a surprising crossing of political lines.