Messing with big numbers: using probability badly

There are a lot of really bad arguments out there written by anti-evolutionists based on incompetent use of probability. A typical example is this one. This article is a great example of the mistakes that commonly get made with probability based arguments, because it makes so many of them. (In fact, it makes every single category of error that I list below!)

Tearing down probabilistic arguments takes a bit more time than tearing down the information theory arguments. In the IT arguments, there's really one error that they make, and it's fundamental and unconstestable: they base the whole argument on a fundamental erroneous definition. Once you point out that the definition is wrong, the whole argument collapses.

The probabilistic arguments are different. There isn't one mistake that runs through all the arguments. There's many possibly mistakes, and each argument typically stacks up multiple errors.

For the sake of clarity, I'm going to try to put together a list of the fundamental types of errors; then I'll go through several different articles, and show which errors they each make.

So first, the Good Math/Bad Math Taxonomy of Probability Errors:

Big Numbers

This is the easiest one. This consists of using our difficulty in really comprehending how huge numbers work to say that beyond a certain probability, things become impossible. You can always identify these argument, by the phrase "the probability is effectively zero."

You typically see people claiming things like "Anything with a probability of less than 1 in 10^60 is effectively impossible". It's often conflated with some other numbers, to try to push the idea of "too improbable to ever happen". For example, they'll often throw in something like "the number of particles in the entire universe is estimated to be 3x10^78, and the probability of blah happening is 1 in 10^100, so blah can't happen".

It's easy to disprove. Take two distinguishable decks of cards. Shuffle them together. The likelihood of the resulting deck of shuffled cards having the particular ordering that you just produced is roughly 1 in 10^166. Yeah, there's more possible unique shuffles of two decks of cards than there are particles in the entire universe.

It sure as heck seems like something that unlikely isn't possible. Our intuition says that any probability with a number that big in its denominator is just impossible. Our intuition is wrong - because we're quite bad at really grasping the meanings of big numbers.

Perspective Errors

A perspective error is a relative of big numbers error. It's part of an argument to try to say that the probability of something happening is just too small to be possible. The perspective error is taking the outcome of a random process - like the shuffling of cards that I mentioned above - and looking at the outcome after the fact, and calculating the likelihood of it happening.

Random processes typically have a huge number of possible outcomes. Anytime you run a random process, you have to wind up with one of the outcomes. They're each incredibly unlikely, but you need to wind up with one of them. The probability of getting an outcome is 100%. The probability of your being able to predict which outcome is terribly small. The error here is taking the outcome of a random process which has already happened, and treating it as if you were predicting it in advance.

Bad Combinations

Combining the probabilities of events can be very tricky, and easy to mess up. It's often not what you would expect. You can make things seem a lot less likely than they really are by making an easy to miss mistakes.

The classic example of this is one that almost every first-semester probability instructor tries in their class. In a class of 20 people, what's the probability of two people having the same birthday? Most of the time, you'll have someone say that the probability of any two people having the same birthday is 1/365^2; so the probability of that happening in a group of 20 is the number of possible pairs over 365^2, or 400/365^2, or about 1/3 of 1 percent.

That's the wrong way to derive it. There's more than one error there, but I've seen three introductory probability classes where that was the first guess. The correct answer is very close to 50%.

Fake Numbers

To figure out the probability of some complex event or sequence of events, you need to know some correct numbers for the basic events that you're using as building blocks. If you get those numbers wrong, then no matter how meticulous the rest of the probability calculation is, the result is garbage. If I say that in rolling a fair die, the odds of rolling a 6 is 1/6th the odds of rolling a one, then I'm not going to get any meaningful predictions of probability. This one is incredibly common in evolution arguments: the initial probability numbers are just pulled out of thin air, with no justification.

Misshapen Search Space

When you model a random process, one way of doing it is by modeling it as a random walk over a search space. Just like the fake numbers error, if your model of the search space has a different shape than the thing you're modeling, then you're not going to get correct results. This is another astoundingly common error in anti-evolution arguments. Evolution does have a directive force shaping the search space: you don't traverse all paths in the search space: survival prunes the space in particular ways. The "search space" of evolution is something like a bumpy surface with some big dents. Roll a marble across that surface, and it's going to move in a particular way with a probability map that's totally different from the probability map of a flat surface.

False Independence

If you want to make something appear less likely than it really is, or you're just not being careful, a common statistical mistake is to treat events as independent when they're not. If two events with probability p_1 and p_2 are independent, then the probability of both p_1 and p_2 is p_1*p_2. But if they're not independent, then you're going to get the wrong answer.

For example, take all of the spades from a deck of cards. Shuffle them, and them lay them out. What are the odds that you laid them out in numeric order? It's 1/13! = 1/6,227,020,800. That's a pretty ugly number. But if you wanted to make it look even worse, you could "forget" the fact that the draws are dependent, in which case the odds would be 1/13^13 - or 1/3x10^14 - about 50,000 times worse.

I'm sure that I'm missing some - I expect this is a post that I'll be coming back to update several times. If you can think of any that I missed, please chime in.