Chance News 38

Quotations

A mathematician is a device for turning coffee into theorems.

Paul Erdős

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Irreligion

Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up . By John Allen Paulos. 158 pp. Hill & Wang. $20.

John suggested that Chance News readers might enjoy some of the arguments that he used in his book that rely on probability concepts . We give a sample below and you can see more of his probability arguments in a talk he gave at the recent Conference "Beyond Belief Enlightenment 2.0" sponsored by the science network.

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.

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.

Paul Alper suggested that readers might enjoy the following:

Paulos often writes about unlikely events and how quickly the public tends to assume something supernatural is taking place. On page 52 of Irreligion he muses on numerological coincidences involving 9/11. He starts with 9/11 being "the telephone code for emergencies." The digits 9 + 1 + 1 sum to 11 and September 11 is the 254th day of the year so that 2 + 5 + 4 sum to 11. Further, there are another 111 days to the end of the year. The first plane to crash into the towers was flight number 11. The Pentagon, Afghanistan and New York City each have 11 letters. Moreover, any three-digit number when multiplied by 91 and 11 results in a six-digit number where digits four, five and six repeat digits one, two and three, respectively; in particular, starting with 911 results in 911,911. A few pages later he notes that on September 11, 2002 "the New York State lottery numbers were 911." The day before that,"the closing value of the September S&P 500 futures contracts" was 911. And to cinch it all, Johnny Unitas, the number one quarterback ever, died on September 11 and wore 19 on his jersey.

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.

The Drunkars's walk:How Randomness Rules Our Lives

Leonard Mlodinow
Pantheon Books, New York, 2008

There are not many writers who can successfully write about mathematics for the general public. but Leonard Mlodinow is one of these. He is a physicists who has written a number of successful books on physics and mathematics for the general public. He has also been an editor for Star Treck.

The Drunkar's walk is his most recent article. In this book wants to show we all have a hard time understanding probability and yet it plays and important role in our daily lives. To show that we are not wired to understand probability he has only to show them well known problems such as birthday problem, the Monte Hall problem, the two sisters problems, the study showing that there are no "hot hands" in basketball. More serious problems includes the The Prosecutors paradox, false positives etc.

But to show how probability affects our lives requires better understanding of probability. Mlodinow does this as he describes the history of probability. This includes the introduction of sample space by Cardano and his solution of dice problems and the problem of points solved by Pascal and Fermat. As he does this he points out similar problems that occur in everyday life. He continues with Bernoulli, deMere, and Bayes and along the way explains their contributions including the law of large numbers, the central limit theory and condition problems. A the same time showing how Bernoulli’s law of large number and deMere Central limit theory help us understand real life problems such as opinion polls.

What makes this book so interesting to read is the authors ability to discuss the history of a problem, the solution of the problem, and its application to the real word all at the same time.

For example we all like to show students that we can tell whether they physically tossing a coin or mentally tossing a coin. We ask have the students to toss a coin 200 times and the other half to write down sequences that would be equivalent to tossing a coin. We find that most of the sequences that are the results of tossing a coin have a sequence of 5 or more heads while very few of those who do it mentally have such a sequence.