I'm answering questions from the 'hardest exam in the world' | Hacker News Reader

I'm answering questions from the 'hardest exam in the world'

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34 pointsnqureshi38 comments

Intuition 1 is incorrect.

The probability of the marble being under the box during that period of time was 1 because its state never changed. It was merely your incomplete knowledge that encouraged you to (incorrectly) assign a probability of 0.5. You've become confused over WHAT is the subject of your test.

If, however, you were to change the experiment such that there were two boxes, one with a marble and one without, you now have the ability to engage in real probability. The probability that a marble is present under each box is either 1 or 0. However, the probability that you will CHOOSE TO LIFT the box containing the marble is 0.5. This is not a probability of presence or absence of a marble; it's the probability that YOU will choose to look under a particular box.

Probability requires the possibility of change, and the probability of change in a marble's presence under a box (such as by teleporting from one box to another) is small indeed.

You're making assumptions when you say that "the probability that you will CHOOSE TO LIFT the box containing the marble is 0.5".

Consider that there may be 3 boxes, and the person setting up the experiment flipped a coin for each box. If the coin landed heads up, a marble was placed in the box. If the coin landed tails up, no marble was placed. This means there may be 0 - 3 marbles total. Until you know the actual state of the box, the probability remains 0.5.

Probability is temporal. You need to specify timeframe.

If you are talking about before the coins were flipped to decide whether a marble was to be placed into the box, then at that time BEFORE the coin was flipped, the probability would be 0.5 per box.

If you are talking about AFTER the coin has been flipped, the probability for each box goes to either 0 or 1. This is regardless of whether an outside observer is aware of the state or not.

If you were to now enter the room, having no knowledge of what boxes (if any) have had a marble placed in them, the probability of a marble being in a particular box has not changed. It is STILL either 1 or 0. There are only two ways to induce probability at this point: Pick a single, random box to open, in which case you are measuring the probability that you will choose a box with a marble in it (you are measuring yourself, not the marbles). The other is to guess whether a particular box has a marble in it, and then open it, in which case you are measuring whether your guess is correct or not (once again, measuring yourself, not the marble).

Most people go wrong here because they think they are measuring one thing but are actually measuring something else.

This would then go down to theoretical physics and whether or not it was already defined which box you would choose. I think something closer to the Schrödinger's cat experiment, and whether the cat will die. In this case, I think you could objectively say the probability is 50/50.

The Schrödinger's cat experiment is flawed, as pointed out by Niels Bohr and explained here: http://web.archive.org/web/20061130173850/http://www.ensmp.f...

The wave function collapses as you induce probabilistic action. And once again, to speak of probability one must speak temporally. Lack of observation of the change does not negate the change in probability.

I understand the question quite differently. The question of objective probabilities is a question of whether the world is deterministic.

If you believe the world is governed by physical laws and you further believe you understand those laws (physics), you consequently must believe that with sufficient information about the past you can predict the future (determinism). So if I was to throw a dart at a dartboard, with sufficient information about arm speed, humidity, wind etc you could perfectly predict where on the dartboard it would land. The problem is that the mathematical / computational complexity necessary to achieve that is still beyond our means. Enter Probability, which we use to reduce the computational burden of predicting the future. As probability will always be model dependant (ie subject to someone's view of the world and necessarily wrong some % of the time) - it follows that an objective probability cannot be consistent with a deterministic world.

So is the world deterministic?

Your premises are wrong: you can believe in physics, but the laws of physics could not deterministic.

To the best of our knowledge this is the case[1], and we have strong indications [2] that this is a fundamental feature of nature: the non deterministic aspects do not come from our lack of understanding and knowledge.

[1] http://en.wikipedia.org/wiki/Quantum_mechanics [2] http://en.wikipedia.org/wiki/EPR_paradox

Certain irony about saying that a LAW of physics is not deterministic. Conceded though that determinism does not hold up at the quantum level.

Disagree though that quantum non-determinism translates to non-determinism at the scale we experience [1]. Have never seen a Feynman diagram for a soccer ball. Which is why I feel comfortable saying that if you believe the macro world is deterministic then you cannot also believe there are objective probabilities (unless, as you indicated you are talking about the subatomic world).

Determinism does hold up at the quantum level. Quantum mechanics is completely deterministic. Collapsing the wave function to a classic state, is the probabilistic aspect, if any.

I think that quantom mechanics makes us think differently about probability and determinism by giving the mathematical (human created) concept of probability a physical meaning. Similar to the example in intuition 2, a coin toss could be considered determinstic even if we do not try to predict the result. It is pre-determined that given a large enough number of coin tosses, very close to 50% of them would come out heads. Qunatom mechanics makes us consider probability as a physical property inherent in all physical objects. This way of thinking helps reasoning about genetic predisposition as well (which is always probabilistic). Just like any coin is "free" to make a choice, and yet it is certain that they would (almost) precisely split down the middle, so too can genetic predisposition can be said to be both constraining and not constraining at all at the same time.

Admittedly beyond at the limits of what this armchair quantum mechanic can say with confidence but to borrow from Hawkings... probabilities in quantum mechanics "reflect a fundamental randomness in nature" [The Grand Design]. So while quantum theories are testable, they are not always predictive (necessary for determinism).

Chaotic macroscopic system + non-determinism at atomic scale = non-determinism at macroscopic scale.

So, e.g., the temperature at the tip of the Eiffel tower on Jan 1, 2014 1pm is non-deterministic.

From a practical approach, the physical world can seem fairly deterministic. If you drop something in a vacuum and nothing hits it, you can say with a great deal of accuracy how long it will take to hit the ground. I don't think that this practical determinism translates to a philosophical determinism.

There are alternative uses of probability. For example, you might measure the weight of 50 people eating one of two diets, and determine an average weight for each group. You want to know if the different diets affect weight. It is highly unlikely that the average weight will be exactly identical for each group, so now you use statistical methods to determine the probability that the diet caused the observed difference in weight. You now assign a "probability" to the hypothesis that diet influenced weight -- is that probability objective?

>To sum up: probabilities are (a) true or false, (b) regardless of what I believe about this probabilities. Estimated probabilities (c) depend on the prior knowledge of the mind in question. Moreover, (d) the fact that the probability is X is a fact about my mind.

I'm not sure why you then go on to say that probability is neither objective nor subjective, and that probability has traits of both (which seems somewhat like a contradiction), rather than conclude that there are two different kinds of probability: objective probability and subjective probability ('probabilities' and 'Estimated probabilities' as you even differentiate yourself). (And then there is the other interpretation of the question of whether objective probability even exists at all or wether the world is deterministic.)

On "Intuition 2":

> At time t, if it is true that the probability of the marble being under a given box is 0.5, then if I guessed randomly, I would be correct one out of two times on average.

> The fact that the probability is 0.5 is completely independent of whether or not any particular mind believes that it is 0.5. But this is the definition of an objective truth. Therefore, probability must be objective.

This is begging the question. The two things combined is like saying "At time t, if it is true that probability is objective... therefore probability must be objective."

Saying p=0.5 necessarily implies the 1/2 thing. Neither implies any stance on whether it's subjective/objective so far. The 'objective' conclusion there requires the further fact of mind-independence, which is based on 'you can't change the odds that a fair coin comes up heads just by thinking about it' type arguments. So it's not circular/question-begging, IMO.

When a fair coin is tossed it will either come up heads or it wont. Put another way, the probability of getting heads is 1 or 0, unless you believe that something magical happens to the coin while it is in mid-air that somehow makes it impossible to predict what will happen when it lands (given a complete set of data).

So the outcome of the coin-toss is knowable, it just happens to be complex enough that it is not known (to us). To bridge the gap between the knowable and the not known we introduce a margin of error in our calculations, called Probability. Probability then, only exists as man-made mathematical spackle - and is by definition subjective.

To prove me wrong find an event that has an outcome that cannot be known in advance. That's why this question is on a philosophy exam.

Quantum mechanics?

Heisenberg uncertainty principle and the entire field of quantum mechanics?

Well played and conceded. So do we need two sets of rules? One for 'our world' and one for the subatomic world?

Good question. My best guess is that the law of large numbers averages out the weirdness so that you never spontaneously tunnel through the floor or whatever, but that's only a guess. And there's a lot of weirdness to deal with.

Just because relativity showed Newtonian physics to be wrong about how the universe worked doesn't mean we can't use the latter as a decent approximation under appropriate constraints. Similarly, we don't really need to calculate the probability of the Sun teleporting over Wall Street when valuing Facebook's IPO.

Upvote, because I didn't know the 'hardes exam in the world' existed, and I now think it's awesome. Good luck.

Thanks. Did you check out the link to the general paper? One of my favourite questions from there: "Does the moral character of an orgy change when the participants are wearing Nazi uniforms?"

Yes. What does it say about my 'moral character' that the question made me smile?

Engaging in an orgy while clothed is certainly more perverted than doing so while naked.

Q: Does the moral character of an orgy change when the participants are wearing Nazi uniforms?

A: Yes

Did I pass the exam?

The question is inherently crippled, because there have to be objective probabilities, otherwise quantum mechanics wouldn't work.

I guess the question really is "if the universe operated according to classical principles, would there be objective probabilities?" And that is an interesting question, but it's a hypothetical alternate-universe one.

I parse this question differently. Most commenters seem to be answering 'can ALL priors be objective' when the question is only asking if any exist.

For people not sitting this exam, the probability of passing it is zero.

The goal of the exam is not to determine what knowledge you have assimilated, but how likely you are to generate new knowledge. Hence the very open nature of the question.

Scanning the exam's questions, I really can't see them as "hard" in the conventional sense.

Hard would be a question in math, physics or some other field which has a very difficult to discover objective answer. There's no limit to hard here but these essay questions are different. "Should wealth be inheritable?" just doesn't have an objective answer and one's answer could only be judged based on whatever the exam grader thinks is a good answer (well written, clever, whatever). In fact, considering one can choose three out of twenty seven, the exam doesn't seem much harder than some essay exams I've taken that were supposed to be extremely easy (CBest, for example). Some of the essays would be require to marshal facts in a given field (classics or economics) but we can hope the writers have some expertise in something.

I think the only way the questions could be considered hard would be if someone taking it thought they needed to settle one or another of these debatable questions. There you have might the situation where geeks and humanities majors each think the other has "really hard exams"...

Edit: And that's "are there objective probabilities" - an English major could take half an hour to answer without worrying about whether they'd really objectively established their answer. We geeks will stop and aim for an "objective" answer. But while one can establish whatever model one wants in the realm of math, one isn't going to pin down the concept of "objective" and "probability" as used by English speaking humans today.

It sounds like people are [mis-]using the word "hard" to mean "selective".

Some of the most pressing problems in society don't have answers that fit into the mindset of the more mathematically-minded among us.

They are problems nonetheless, for example: should free speech exist? You can't answer that question using science, yet we need people to think about that kind of thing.

For what it's worth, the reputation of All Souls College (which does not have students) is strongest in the humanities.

Saying the exam wasn't "hard" as such wasn't intended to slight All Souls College or the humanities.

It seems like what makes a question part of the humanities is that many if not most humans have some sort of answer to the question. For that reason, I imagine the humanities might be better served not conjuring up images of super-hard exams only a few can pass.

From the perspective of All Souls, they do want exams that only a few can pass. The goal of this kind of exam is to identify original and insightful thinkers.

I presume that it is hoped that those who enter All Souls will make original and long-lasting contributions to their field of endeavor.

In the natural sciences, a similar effort is supported by the Howard Hughes Foundation at Janelia Farms, as described here: http://www.nature.com/news/research-at-janelia-life-on-the-f...

Borrowing from the physics nomenclature, I guess you'd say Probability has "Subjective Objective Duality"

I would say yes, there are objective probabilities. For instance a path integral in quantum mechanics sums all possible paths for a particle along with each path's probability; if the calculations are done correctly, the numbers for a particular calculation will come out the same every time, even if different people perform the calculations. Since the facts and outcomes are observably equal for different people, I would say this fits the definition of objective knowledge. It is what makes quantum mechanics testable and useful.

There are plenty of other examples, like the Monty Hall problem, where many different people have confirmed the objective truth of a probability through direct testing.

The problem I see with this question is that it is couched as a philosophy question, not math or physics. That's what makes it "hard".

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