Saturday, April 17, 2004

What is Probability?

There has been a log of blogbuzz lately about Intelligent Design (ID), probability, and Evolution.  Moreover, these three things tend to be mentioned together.  I started to write a piece about linguistics and the Sapir-Whorf-Korzybski Hypothesis, after being reminded of this concept by a post  on Evangelical Outpost (EVO).   I got sidetracked when I saw some other posts on EVO that pertain to Intelligent Design. 

In the interest of brevity, I won't review the definitions of ID and evolution.  Do your own Dogpile  search if you want.

Anyway, Mr. Carter at EV writes  that:

We could predict, post hoc, what type of universe would be required to produce human life, but we'd be unable to test the theory (we aren't able to repeat the Big Bang). We could, however, determine the likelihood that the event could have occurred by pure chance. Since the probability of such a series of events occurring by coincidence would be close to zero, we would be lead, by evidence and experience, to the conclusion that the universe was “designed.”

I have news for Mr. Carter.  We do have absolute knowledge of the probability that the universe would turn out exactly as it did.  The probability is not near zero.  It is exactly 1, or in common parlance, 100%.  Exactly.  Not "close" to 100%,  but exactly 100%.  Mr. Carter can be forgiven for his error.  He uses the commonplace definition of the word "probability."  His usage is not scientifically correct.  He should use the word "plausibility."   Here is the correct definition (from Wikipedia) of the word "probability:"

The general idea of probability is often divided into two related concepts:

  • Aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon. This concept can be further divided into physical phenomena that are predictable, in principle, with sufficient information, and phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel, and an example of the second kind is radioactive decay.
  • Epistemic probability, which represents our uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are to assign a probability to the proposition that a proposed law of physics is true, and to determine how "probable" it is that a suspect committed a crime, based on the evidence presented.
When "probability" is used in a mathematical or scientific context, it is used in the first sense: aleatory probability.  The author signifies this by enclosing the word in quotes under the section on epistemic probability.  The concept of aleatory probability makes sense only when referring to future events.  This being the case, there is -- by definition -- no sense to the idea that one can calculate the probability of something that has already happened.  You cannot write a formula for the probability of a past event, because there is no definition for it. 

To illustrate: let's say we find a coin that is truly fair, meaning that future coin flips are equally likely to be heads or tails.  Flip the coin, catch it in the palm of your right hand, and cover it with you left hand.  Now, what is the probability that the coin turned up heads?  In fact, you do not know.  You do know the average (arithmetic mean) probability: 0.5, or 50%.  The coin is either heads-up or tails-up.  Averaging 1 and zero gives you 0.5.  But, it is not correct to say that the probability of the heads condition is 50%.  It is either 1 or zero. 

Mr. Carter's post, and the related ones that followed, generated a total of 170 comments as of this writing.  There is a reply on Panda's Thumb that gave rise to 96 comments. 

In the text on Panda's Thumb:

I'll take issue with Joe's claim that we can determine the likelihood of the big bang, or the so-called anthropic coincidences, occurring by "pure chance", and I'll challenge him to produce such a calculation. We hear this argument over and over again, but it's never accompanied by an actual probability equation. If you think we can calculate the probability of either of those two things, let's see the probability equation. It should also be noted here that even if such a probability equation were possible, it wouldn't tell us anything meaningful about whether the event could have occurred naturally or supernaturally.

This is illustrated by the following example:

Go outside and pick up a small rock. The probability of that rock being on that spot on the earth *by chance alone* is roughly the area of the stone divided by the surface area of the earth, or about one chance in 10 to the 18th power (one followed by 18 zeros). If picking up the stone took one second, the probability of such an event occurring at this precise moment over the lifetime of the universe is now even smaller by another factor 10 to the 18th power! This simple event is so incredibly unlikely (essentially zero probability) that one wonders how it could be accomplished!

Mr. Carter's reply:

U= The known universe
N= Occurrence of an event necessary for human life to exist

Prob(U)= prob(N) + prob(N1) + prob (N2) + prob (N3)…

The perfect illustration of this is Marshall Berman’s example of the rock in the backyard.

This is picked up on Letters of Marque (another Ann Arbor blog), and there are 13 comments.  Excerpt follows:

Where even to begin with this? First of all, that's not exactly a calculation of the probability, since it skips the hard steps (deciding what events were necessary for human life to exist and determining the probability that they would happen), which is what was called for.

And I dislike seeing N1 and N2 and all that in there without N1 and N2 being defined, although in this case Joey probably means something like let N1, N2, ... Nn be events required for human life to exist. Then...

There's the fact that to the best of my knowledge, probabilities don't look like this. The only time adding probabilities makes sense is, say, if you're discussing mutually exclusive events. Like, what's the probability that I roll this dice and roll either a 1 or a 2? Well, it's the probability that I roll a 1 plus the probability that I roll a 2. Now, do we think that events N1 and N2 are mutually exclusive? Not unless he's imagining n mutually exclusive ways to get human life, which is rather odd. Generally, one multiplies probabilities. And one does that if the probabilities are independent, which is precisely what we don't know.

At any rate, this guy was apparently going off on how Leiter didn't understand logic or science. Uh huh. Joe, you seriously need to get yourself some Jaynes.

Dr. Myers at Pharyngula sums it up nicely:

[...] This is what we're dealing with all the time in the interface between biology and American culture. People outside the field are trying to dictate what biological theory should say, and damn, but are they loud and insistent that biologists are all wrong. Yet whenever we actually sit down to discuss the issues with these people, what do we find? Deep twisty mazes of nonsense and ignorance, and that despite their absolute certainty, they really know little, and that little is typically wrong. And it's not just Joe—the whole corrupt cabal nesting at the Discovery Institute is just as wrong on this subject. They're just better practiced at keeping their mouths shut.

There are 26 comments on this post at Pharyngula.  They branch off into a discussion of the old philosophical issue of the mind-body problem.  Which I might write about, someday...

 I did not read all of the comments from all of the posts, but scanning them, I did not see anyone go back and look at the exact definition of "probability." 

Yesterday, my wife spent two hours looking for a particular key1.  It happened to be in plain sight, less than three feet from us at the time.  She found it right after telling me about how frustrating it was to not be able to find it.  This kind of thing can happen to anyone.  The key here: start with the obvious. 

1 She was looking for the key to the gate for the horse's pasture.  We used to keep it on the pegboard in the barn, but Scotch kept reaching his head over the gate and knocking things off the pegboard, so we put the key inside the house.  For extra credit: What was the probability of finding the key in the barn?