## Hot debate about probability on Richard Wiseman's blog

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macronencer
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### Hot debate about probability on Richard Wiseman's blog

Hi folks. Richard Wiseman's Friday Puzzle has stirred up a hornet's nest, as it sometimes does. This time it's roughly evenly polarised, which is quite interesting and relatively unusual.

Sorry if this counts as "done to death" - I'm not entirely sure it's not a variant of the Monty Hall problem, but giving it the benefit of the doubt, hopefully it's OK to post the link...

http://richardwiseman.wordpress.com/201 ... uzzle-131/

FYI I'm in the "1/5" camp. Can anyone here please help by either:
(1) telling me I'm wrong and why I'm wrong, or
(2) telling me I'm right and giving me an explanation that might convince those in disagreement (who claim the answer is 1/3)?

It's driving me a bit crazy. Well, a bit MORE crazy, I guess. It feels like the 0.99999... = 1 thing all over again, and I'm worn out with going over the same arguments again and again.

Thanks in advance!
I think those are crocodile tears: you must be in de Nile.

antonfire
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### Re: Hot debate about probability on Richard Wiseman's blog

It's a variant of the Monty Hall problem. The way this one is phrased is particularly ambiguous. There are two main interpretations of the problem, depending on just how Richard picks the blue stone out of his hand: (A) Richard looks into his hand, and if he sees a blue stone, he removes it. (B) Richard picks a random stone from his hand and removes it; that stone happens to be blue.

Under interpretation (A), the answer is 1/5, for the reason he gives. Under interpretation (B), the answer is 1/3.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

Aiwendil
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### Re: Hot debate about probability on Richard Wiseman's blog

I wouldn't say it's a variant of the Monty Hall problem. The key to the Monty Hall problem is that an omniscient host reveals an additional piece of information to the contestant, and reveals it in a predictable way. This problem has no such component.

There is a possible ambiguity in the statement of the problem, though. The two scenarios are:

A. The person looks at both the stones and sees that at least one of them is blue.

B. The person looks at just one of the stones, and sees that it is blue.

The set-up of the problem seems, to me at least, to clearly indicate situation B, but the answer he gives seems to assume situation A. To answer B, you'd have to keep track of the order of the stones, and enumerate the permutations instead of the combinations. Calling the white, yellow, and blue stones W, Y, B1, and B2, the permutations are:

W-Y
W-B1
W-B2
Y-W
Y-B1
Y-B2
B1-W
B1-Y
B1-B2
B2-W
B2-Y
B2-B1

So there are twelve permutations. If we see that the first stone is blue, then we are restricted to the last six of those permutations. Of those six, in two the second stone is also blue. So the probability that the second stone is blue is 1/3. Note that if we look at the permutations but with only the condition that either of the stones is blue (situation A), we get ten possibilities, two of which have both stones blue, so we recover the answer of 1/5 that he gives.

It's possible that I'm overlooking something here, as I've worked through this pretty quickly, but I think it's right.

skullturf
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### Re: Hot debate about probability on Richard Wiseman's blog

I agree that the question is at least a little ambiguous. And antonfire gave an excellent brief summary of the two answers -- I don't think there's much to add to those two answers.

Note that the wording in the original question is "I bring my hand out in a fist, look inside my fist and remove a blue stone."

I interpret "look inside my fist" as "see everything that's inside my fist". In other words, he opened his fist wide enough to see both stones, and decided to remove a blue one.

However, on my very first (rather quick) read-through, I read "look inside my fist" as "I open my fist just wide enough to see only one stone", which is the same as choosing a stone from his fist randomly. That's no longer my preferred way to read the question, but I can see how people could think it's ambiguously or confusingly stated.

macronencer
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### Re: Hot debate about probability on Richard Wiseman's blog

Thanks for the comments, all of you.

So you actually think that the procedure followed by Richard (the chooser of the blue stone) is relevant? I thought that conditional probability questions such as this one were supposed to be answered from the point of view of the observer. All the information we really have is that a blue stone was selected when the two stones were selected (presumably at random). In my view, the way it's selected is an unknown in the problem, and thus should be considered irrelevant. Or have I missed something here?

I note that the official answer given is 1/5, although this doesn't necessarily mean much, as Richard has been wrong a few times in the past!
I think those are crocodile tears: you must be in de Nile.

Yakk
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### Re: Hot debate about probability on Richard Wiseman's blog

No, you need access to algorithms used by "actors" in the scene.

Here is a toy example.

There is a box. It contains one white and one black stone.

Bob opens the box, looks into it, and removes a stone. You don't get to see it.

What is the probability the remaining stone is white? What is the probability the remaining stone is black?

Any answer to this question implicitly includes a distribution on the algorithm that Bob uses. If we describe that algorithm in the problem, the question is easy to answer. If you don't, we need to make some assumption.

Suppose you say "well, the odds are 50:50!". After you make that claim, but before you look in the box. Bob offers you a bet: you give Bob 1000\$, and if the stone is white he'll give you a million dollars.

You could view this as information about Bob's algorithm (if you where a Bayesian), but the point is that (falsifiable) claims (with any certainty) about this situation depend on what Bob did when he looked into the box and took out a stone, as the bounds on the possible probabilities are anywhere between 0% and 100% as a function of the algorithm that Bob follows.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Aiwendil
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### Re: Hot debate about probability on Richard Wiseman's blog

I'd say that if the way the blue stone was selected is unknown, then the problem isn't well-defined.

If we don't know the rules that govern the selection, then from our point of view, the probability of a given outcome will depend not only on the explicit parameters of the problem (number of stones of each colour, how many are chosen, etc.) but also on "probability that the selection rule is X", "probability that the selection rule is Y", and so on. But since those probabilities are not defined, the problem becomes impossible to calculate (unless you believe in objective Bayesian priors, I suppose, in which case it just becomes impossible to calculate in practice).

skullturf
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### Re: Hot debate about probability on Richard Wiseman's blog

macronencer wrote:So you actually think that the procedure followed by Richard (the chooser of the blue stone) is relevant?

Call the four stones A,B,C,D, and say C and D are the two blue ones. Say Richard draws two stones from the box a large number of times -- say about 6000 times.

There are (4 choose 2) = 6 equally likely possibilities.

About 1000 of the 6000 times, he draws A and B. (Case i)
About 1000 of the 6000 times, he draws A and C. (Case ii)
About 1000 of the 6000 times, he draws A and D. (Case iii)
About 1000 of the 6000 times, he draws B and C. (Case iv)
About 1000 of the 6000 times, he draws B and D. (Case v)
About 1000 of the 6000 times, he draws C and D. (Case vi)

Scenario 1: Richard opens his fist wide enough to see everything, and checks whether the statement "There is at least one blue" is true.

Scenario 2: Richard opens his fist just wide enough to see one stone, effectively randomly picking one of the two stones in his hand.

In Scenario 1, we know we're not in Case i, and the five other cases are equally likely. The final answer is 1/5.

In Scenario 2, it's still true that we can be in any of Cases ii through vi. However, in Cases ii through v, in only half the trials (about 500 of them) will he draw the blue stone from his fist. In the approximately 6000 total trials, there will be:

about 500 trials where he drew A and C, and happened to see C when he opened his fist
about 500 trials where he drew A and D, and happened to see D when he opened his fist
about 500 trials where he drew B and C, and happened to see C when he opened his fist
about 500 trials where he drew B and D, and happened to see D when he opened his fist
about 1000 trials where he drew C and D -- in all 1000 of these, he sees a blue stone when he opens his fist.

The final answer in Scenario 2 is 1/3.

antonfire
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### Re: Hot debate about probability on Richard Wiseman's blog

Aiwendil wrote:I wouldn't say it's a variant of the Monty Hall problem. The key to the Monty Hall problem is that an omniscient host reveals an additional piece of information to the contestant, and reveals it in a predictable way. This problem has no such component.

The Monty Hall problem is this problem, without the yellow stone.

macronencer wrote:So you actually think that the procedure followed by Richard (the chooser of the blue stone) is relevant? I thought that conditional probability questions such as this one were supposed to be answered from the point of view of the observer.

If we could reliably deal with probabilities this way, science would be a lot easier. From the Bayesian perspective, an observer's prior knowledge affects the probabilities that observer assigns to certain facts. If Carla happened to catch a glimpse of the second stone, for instance, she won't say the probability of it being blue is 1/5, or say that it's 1/3. She'll either say that it's 1, or say that it's 0. Similarly, if Alice caught a glimpse of Richard's script, and Bob somehow got the opposite idea about how Richard behaves, they will assign different probabilities to the second stone being blue upon seeing Richard pick a blue stone out of his hand.

That is, yes, in this case knowing how Richard behaves is a relevant piece of information. If you don't have any information about how he behaves, you can't sensibly assign a probability that the second stone is blue.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

Aiwendil
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### Re: Hot debate about probability on Richard Wiseman's blog

antonfire wrote:The Monty Hall problem is this problem, without the yellow stone.

Well, that's true if you interpret the problem as saying that the person follows a rule whereby he looks at both stones in his hand, chooses a blue one if there is one, and displays it. As has been discussed, it's not clear whether that's really the scenario the problem intends to depict. Certainly there's no suggestion in the problem as stated that the chooser was guaranteed to show a blue one if possible, even if one interprets it such that he looks at both stones. (Also, a Monty Hall-style problem would presumably ask whether the remaining stone in the hand is more or less likely to be non-blue than one of the remaining stones selected at random, rather than simply asking what the probability is that it would be blue).

antonfire
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### Re: Hot debate about probability on Richard Wiseman's blog

In case someone is still not convinced that how the stone is chosen matters, both answers say very concrete and testable things about repeated experiments. Actually running these experiments can be instructive.

I modified the code from this post to simulate the problem: http://codepad.org/PctYHLnZ#output. Hopefully, even if you don't know python, it's fairly readable. Richard behaves according to interpretation (A), Dick behaves according to interpretation (B).
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

macronencer
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### Re: Hot debate about probability on Richard Wiseman's blog

I want to thank you all for getting so involved in this - it's getting very interesting! I like the python script, very neat!

I understand the point about the algorithm of choice making a difference, of course. The point I'm having trouble with might be considered a philosophical one, and I can't remember enough from my degree to get this straight in my head (it was 25 years ago!)

As far as I'm concerned, probability always has to be contextual. In other words, if you knew every cause and effect involved in a situation, down to the level of individual quantum events, then there would be no uncertainty, and therefore the probability of anything you care to name would necessarily be 0 or 1. I'm temporarily discounting the question of quantum uncertainty and/or many worlds interpretations, and assuming a deterministic universe

OK, so from there, I reason as follows:
Since complete visibility of information implies a probability of 0 or 1, the hiding of information is the very essence of what probability means. So it makes sense to me that any information that is hidden from us must be discounted in any of our calculations. This becomes all the more obvious when a human being is making some kind of choice and using an algorithm unknown to us. How are we to estimate the probability of that person having used any particular algorithm? Since we cannot, we are forced to ignore it and base our calculations only on the information we are given.

In the problem as stated, the only thing we know for certain is that Richard picked at least one blue stone. We are not told how he decided to show it to us, so that seems irrelevant to me. Based on the available information, the answer is 1/5. I am finding it hard to understand where I've gone wrong in this reasoning...
I think those are crocodile tears: you must be in de Nile.

antonfire
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### Re: Hot debate about probability on Richard Wiseman's blog

macronencer wrote:How are we to estimate the probability of that person having used any particular algorithm? Since we cannot, we are forced to ignore it and base our calculations only on the information we are given.
Ignore it how? If you ignore it, you can't compute a probability of it at all! You are doing the opposite of ignoring it. You are assuming he behaves a certain way.

You can't get something from nothing; probability isn't a magic hammer that lets you make useful statements given no information whatsoever. It's a particular way of presenting information. You cannot make probabilistic statements without first being given some information about probabilities of certain events. In this case, if you start with no information (probabilistic or otherwise) about how Richard behaves, you simply cannot answer the question!

Consider this: you are saying that, given no information about how Richard behaves, the probability that the second stone is blue is 1/5. This should translate into a concrete statement about some repeatable experiment. Try to write a script similar to the one I gave, which simulates the experiments given no information about how Richard behaves. What do you put in the loop?
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

skullturf
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### Re: Hot debate about probability on Richard Wiseman's blog

macronencer wrote:The point I'm having trouble with might be considered a philosophical one

That's fair enough. I agree that probability involves philosophical subtleties.
macronencer wrote:Since complete visibility of information implies a probability of 0 or 1, the hiding of information is the very essence of what probability means. So it makes sense to me that any information that is hidden from us must be discounted in any of our calculations. This becomes all the more obvious when a human being is making some kind of choice and using an algorithm unknown to us. How are we to estimate the probability of that person having used any particular algorithm? Since we cannot, we are forced to ignore it and base our calculations only on the information we are given.

In the problem as stated, the only thing we know for certain is that Richard picked at least one blue stone. We are not told how he decided to show it to us, so that seems irrelevant to me. Based on the available information, the answer is 1/5. I am finding it hard to understand where I've gone wrong in this reasoning...

In a general way, this seems to make sense. What we don't know, we must ignore. It would seem we must assume that we simply don't know what method or algorithm Richard is using.

Although... is that true?

Let's consider just the initial step of drawing the two stones from the box (before the problem even gets tricky). If we really don't know what method or algorithm Richard is using, then for all we know, Richard is playing a trick on us -- maybe he sneakily chose the blue stones to be a different texture, so he can distinguish them by touch, and he always draws the two blue stones. Or maybe he draws the two blue stones precisely 98% of the time.

Of course, in this problem, we are probably supposed to assume that when the two stones are initially drawn, all sets of two stones are equally probable.

But that means that in some sense, we are supposed to make an assumption about the method or algorithm that Richard is using. If his initial drawing of two stones is random, that's different from his initial drawing of two stones being informed and deliberate.

So with that in mind, maybe it makes more sense why we need to know, when Richard selects one stone from the two in his fist, whether that choice is random, or informed and deliberate.

EDIT: Ninja'd somewhat by antonfire's nice succinct answer.

macronencer
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### Re: Hot debate about probability on Richard Wiseman's blog

antonfire:
OK, I do understand where you're coming from now. Thank you for taking the time to explain.

Is this actually just the frequentist vs. Bayesian debate?
There's a really interesting article here about that...
http://lesswrong.com/lw/7ck/frequentist ... pretation/

If I were asked to write a script similar to yours that included Richard's behaviour, without knowing his behaviour I agree that I could not do so. I don't have any information about how Richard behaves.

But consider this: neither do I have any information about how the stones behave as they move around in the box when he selects them. Is there any reason I shouldn't regard Richard's mind as being just as unpredictable as the random movement of the stones before they are selected in the first step of the puzzle? If we decided that anything whose behaviour was hidden from us would prevent our formulating the probability of the event, then wouldn't it be impossible ever to use probability at all?

I confess that I still feel uncomfortable about the underlying philosophy of this discussion. It feels to me as if the human being is being considered to have free will, instead of being considered a deterministic system. You could debate that question of course, but surely within a philosophical, not a mathematical framework.

UPDATE: Just seen your post too, skullturf. Interesting that you drew attention to the possibility of Richard's "cheating" when drawing the initial stones - interesting, because this is, again, a reference to human intervention, i.e. a rational agent with a purpose...
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Yakk
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### Re: Hot debate about probability on Richard Wiseman's blog

Rather, it is within the bounds of a typical human agent model to be "cheating". It would not be considered dishonest if the human agent described by the word Richard was "cheating" in many senses (ie, he looked into his hand and removed a blue stone, instead of picking one at random that happens to be blue, or whatever). What the human agent did wasn't described very well.

It is not within the bounds of human experience for a typical colored stone to teleport, or change color. So generally you skip over this possibility.

Using an enthropic reasonableness standard (ie, given a model, we consider the observations to be reasonable if the observed events or "less extreme" events under some ordering chosen prior to the observations, would have a p% chance or greater of occurring), we don't rule out the teleporting stones or changing color stones, nor do we rule out a number of possible behaviors for Richard. We might then add in knowledge about how the rest of reality acts to rule out more possibilities (like teleporting stones), or just deal with some arbitrary subset of models we want to play with.

Nothing in mathematics, philosophy or statistics guarantees that you can assign probabilities to events given this kind of information. Some positions insist that there must be a "true" value to a probability, but no reasonable philosophy will guarantee you can find it.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

antonfire
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### Re: Hot debate about probability on Richard Wiseman's blog

There's nothing special about agents with a purpose in this case.

Strictly speaking, the problem statement gives you no information about the probability that a certain pair is the one that comes out of the box in Richard's hand, but the traditional way to interpret this is to assume that any pair of stones is equally likely. This is the only possible interpretation if the stones are indistinguishable while in the box. That (implicit) assumption tells us exactly the probability of choosing any given pair of stones.

Why do we make this assumption about how Richard pulls the stones out of the box, but don't make an assumption about how Richard behaves after that? Guess what, most people do. Everyone assumes each pair of stones is equally likely to come out of the box. Roughly half the people then go on to assume that Richard shows us a blue stone if he can, and the other half assumes that he picks picks a stone uniformly at random from his hand and it happens to be blue. Then you get a flame war because nobody knows what implicit assumptions the other people are making.

And no, this is not the Bayesian/frequentist debate. Both Bayesians and frequentists can interpret the problem both ways. Here I try to give the Bayesian perspective on it, and of course this is the frequentist one.
Last edited by antonfire on Tue Nov 22, 2011 12:29 am UTC, edited 1 time in total.
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### Re: Hot debate about probability on Richard Wiseman's blog

@macronencer : Firstly, we can apply probabilistic methods where there is some ignorance regarding the outcome of a process, but without some knowledge of the process we can't sensibly apply any reasoning, probabilistic or not. When we talk about rolling a dice, we know that it can come up with a whole number, 1, 2, 3, 4, 5, or 6, and we assume that the dice can't turn into a chicken as we throw it, or roll around indefinitely, never coming to rest. If we couldn't rule out those last outcomes, then the situation is too irregular to apply any particular reasoning to (imo, at least).

Where we have a finite number of possible outcomes, our initial probability distribution, before we condition it on other facts given, comes from an idea called the principle of indifference. We consider that the dice is not particularly made to come up with any particular number, the procedure is indifferent to whether we get a 2 or a 5, or so we assume (in fact, slight of hand can be used to control the outcomes of dice rolls and coin flips, unless there is care taken to remove such possibilities ie dice cups or letting a coin fall on the ground rather than caught after being tossed). From this assumed indifference, we can say that all events that the procedure is indifferent towards must have the same probability. So, for the dice, this means that we can say that a 2 will happen one sixth of the time.

Applying this to the Wiseman experiment, we can assume that all the pairs or stones have an equal chance to be drawn, so that drawing both blue stones should have the same probability as drawing the white an yellow stone. Similarly, a white and a yellow should be half as likely as a white and blue: we aren't indifferent to these possibilities, but if we had two hues of blue the reasoning becomes easy, and should apply to the circumstance when the blue stones are identical.

No such principle can be reasonably applied to the mind of Wiseman. We could come up with a guess as to the procedure used, with P("declaring blue if you see a blue")=p, and come up with similar labels and probabilities for all the different ways that Wiseman might act. But its not sensible to apply the principle of indifference here: with a dice or the stones, we see certain symmetries and infer that probabilities are likewise symmetric, but there are no such symmetries regarding all these different strategies.
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macronencer
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### Re: Hot debate about probability on Richard Wiseman's blog

This has been an interesting journey for me and I've come to realise a couple of things:

(1) Probability has a greater philosophical component than I realised (even though I knew it was there).
(2) Many puzzles of this form have apparent official 'correct' answers, and it seems that this may be misleading.

I am going to keep thinking about this until I am satisfied I have explored all the implications. I still have some uncertainties, but that's a good thing: without uncertainties, thought is rather dull.

Thanks again to you all for your contributions, and thank you for a refreshingly civilised discussion, in stark contrast to those taking place on most of the web. I should come here more often
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