James Tauber

I'm going to go ahead and let you know that you blew my mind just now.

But I think that the answer has got to be C) You cannot answer the question. By indicating that the answer is not 0.5 and not indicating the direction, Colin has taken away both bits of information about the probability. At this point, all that you know is that the output will be binary--hardly enough data to create a probabilistic estimate.

Nice Gedankenexperiment.
I also wish they would start teaching Barry's probability theory in schools. It makes so much more sense. Given what we know :-)

What are the names - Boolean and Fuzzy?

I get the feeling I'm missing something but I think Barry is right: the potential biases average out unless we have a history or some insight into whether Colin was tempted to bias the coin one way or another.

I'll just stick to studying Greek Syntax...this isn't my field...

"What's the probability of the next toss being a tail?"

Zero (or some value close to it, damn you quantum physics and your border line possibilities) coins have heads and tails but they don't suddenly grow _a_ tail

btw, i'm not a native english speaker and it's the first time i've seen this enigma, just in case you're wondering :)

There is the question of whether the space of possible biases is uniformly distributed. Isn't that assumption material?

@chris, I'd assume it's "Bayesian" and "Frequentist"

I don't know as much as I'd like about Bayesian statistics, but isn't the core principle that you always base your probability on what you know now? Before I knew there was a bias, I would have said P(T) = 0.5. Now I would say P(T) = P(T | B_T) * P(B_T) + P(T | B_H) * P(B_H). (Where B_T is biased toward tails and B_H is biased towards heads, of course). But I still have no reason to believe B_T or B_H is more likely, so I will assume those probabilities to be 0.5, and I have no reason to believe that abs(P(T) - P(B_T)) and abs(P(T) - P(B_H)) differ, either, so my answer is still 0.5. I could refine that by flipping the coin a few times, by physical inspection of the coin, by interrogating Colin, etc.

Frank is talking about the probability I'd assign at a future future time when I have complete information, and in that sense he's right - that probability is not 0.5, by definition.

By the way, your syndication links from advogato are broken - your RSS feed must be using site-relative links that are intended to be used only on http://jtauber.com/.

0.5!
Reasoning: you could equally bias by nothing, but more importantly, if I go back to school when these types of question were thrown into the mix in exams, you got nothing for no answer, maybe something for a wrong answer with reasoning, and wasted time if you dwelt on it too long. I learned to bite the bullet and put something down - with my reasoning, and move on to the calculus which I could do :-)

- Paddy.

(That was not 0.5 factorial. It is 0.5 exclamation mark)

- Paddy.

Every time I think I understand the difference between the frequentist and Bayesian interpretations of probability, someone poses a question like this and confuses me again.

Wouldn't the frequentist also pick 0.5? (Although, you have cleverly not had the frequentist pick an answer, which may be the point...)

I say Barry is correct in this instance. No matter that there is bias, it is unknown and the bias will only become evident until a series of tosses can be evaluated and inform our next choices. The fact that we do know there is bias has no bearing on the present toss in question given our limited information. If Colin had asked what is the probability of tails coming up in a series of 100 tosses, then Frank would be correct.

@Stan, The question posed here is not at all very artificial or something. I would say, whenever someone throws a real world coin, you just *have* to assume that it is biase in one way or the other. No real coin is perfect.

I'm also not sure if a Frequentist would necessarily take Frank's position. Maybe he would see it as a two step process, where first the bias is determined in a stochastic process and second, coins are tossed. If you approach the limit of infinitely many differently biased coins tossed infinitely many times and look at the frequencies, they still peak at 0.5.

After all you have no reason to assume an asymmetric distribution of biased coins.

So maybe a Frequentist might arrive at the same answer, but he needs these ridicoulus infinitely large ensembles of virtual coins and tosses.

But then, the coin at hand is tossed once, and you get for example heads. What are the odds of heads or tails for the next toss?

A Bayesian (or Laplacian) approach on the other hand can deal gracefully and meaningfully with prior knowledge. And with mixed situations where prior knowlegde and very little experimental knowlegde exist.

Barry is right. Statistical analysis only matters when it allows us to make predictions based on incomplete information, if you can't analyse it until you know everything about the coin and the toss then probability is irrelevant because you would already know exactly what is goin to happen every second.

It sounds like an argument over the meaning of the word "probability". As a layman I couldn't tell you which is the "correct" definition, assuming there is one.

There's plenty of English words that have been turned into technical terms with precise meanings in a particular field, which is a natural source of confusion for introductory courses -- especially when two different scientific disciplines use the same English word in different ways!

Asking just for "the probability" is not enough, because the purpose of the answer and the social context determine the choice between the three sensible answers:

- 0.5 for lack of knowledge about the supposed bias;
- I don't know;
- I don't know, but not 0.5 because I believe Colin's unproven statement that the coin is biased.

0.5 is the right answer for betting on a subsequent coin flip (before adjusting the estimate according to experimental outcomes, of course).

Admitting ignorance is the right answer if the probability is treated as a physical description, a synthetic indicator of what we know about the bias of the coin.

The choice between being certain that the coin is biased or not depends on external factors, such as Colin being an imaginary character in a trick question (that is, a liar unless proved otherwise) or an actual trustworthy friend who likes philosophical conversation.

To my mind, since the bias is not known, the probability of it being biased towards tails is 1/2, therefore keeping the possibility of getting a tails in next flip again 1/2. Once again we don't know the degree of bias so we can't factor it in.

It is similar to betting in a race where you do know it is fixed but you still do not know which horse is going to win, so for you the information is useless.