## Tuesday, February 28, 2006

### Just flip a coin

I received some feedback on the fact that I have not written anything for a while. You mean those pictures on the left don't make enough 1000 words to make up for it? There are a lot of pictures over on the sidebar.

Fine. Here, consider this.

Your friend holds up a coin and asks you, "if I flip this coin, what's the chance it will come up heads?"

Being a rational human being, you reply "50 percent." If your middle school fractions teacher was a stickler for this kind of thing, you might add, "which can also be expressed as one-half, or zero-point-five." But essentially, it's a 50-50 chance. All right. Got it. Okay.

Now your friend flips a coin, catches it, puts it on her palm and covers it with her hand before you can see what the outcome was. Now she asks you, "what's the chance this coin is heads?"

What do you say?

No, really, what would you say? What are the chances?

I could just wait for people to comment, but that would be boring because who would bother? I just gave the answer, right?

WRONG!

The chance that the coin that has already been flipped is heads is either 1 or 0. It cannot be 0.5, because it is either heads, or it is not.

How is this different from asking before you flip the coin? It's either heads or it's not then, too, isn't it?

No, because then it hasn't happened yet. At that time, it could still be either one. You don't know, and the coin doesn't know, either. After you flip the coin, though, even though you don't know, the flip is already determined. It has happened. So the probability is either 1 or 0, and nothing in between.

Interesting, is it not?

I argue that this same reasoning is applicable to Newcomb's paradox. Some people disagree with me, but then again I bet many people would disagree with the argument above. That is why we have such controversies as the Monty Hall problem.

Other than the obvious perk of having all of my classes and professors in the same building, this is one of the best parts of being a math and psychology double major, because all these problems that are so hard in psychology are so simple if you blind yourself to everything but the math.

Alexander Woo said...

You're touching on an area of analytic philosophy that legions of philosophers are working on. (Well, probably a couple dozen, but given the number of philosophers, that qualifies as "legions".)

The basic question is "What does the statement 'X has a p% chance of happening' mean?"

Does it say something about the results of an implied experiment? (such as flipping a coin?) - this would be an OBJECTIVE interpretation of probabilistic statements.

But the problem with that is that we use probabilistic statements in many situations when experiments are impossible or even nonsensical.

Perhaps instead a probabilistic statement says something about the speaker's confidence in the statement? - this would be a SUBJECTIVE interpretation of probabilistic statements.

There are of course problems with this way of thinking also, since the probability of a coin turning up heads doesn't depend on what I think!

Really interested? This comment comes pretty close to exhausting what I know about this subject. Talk to Prof. Gerrard, who isn't an expert on this either, but i'm sure knows more than me.

Diana said...

Thank you for your carefully considered comment. I really appreciate it. I didn't realize that so many philosophers were working on this.

One thing I didn't mention is that the key thing here is that the original 50% -- "if I flip this coin, what are the probability it will be heads?" -- is based on long-run probability.

So you're right, the basic question of what the statement means is an important one. Usually, it means the long-run probability: If I flipped this coin many times, what is the fraction of the time it would come up heads? The long-run probability has no applicability, though, to something that has already occurred.

Philosophers may be working on the theoretical aspects, but I think mathematicians are done with it. Of course, this isn't really a question of mathematics, but of statistics. It has to do with whether you believe in Bayesian probability, which is a frequentist framework, or not.