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Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the first flip, you win $\$1$. If you flip heads on the second flip, you get $\$2$, on the third flip $\$4$, and so on. Now, if a casino were to host this games, how much should they make their buy-in?

Intuition says not much, but mathematically they should make it as high as they want. Why? Because the payout is infinite. The probability of flipping heads on the first flip is $\frac 12$, which gives a $\$0.50$ average payout. The probability that you flip heads on the second flip (which also means heads on the first flip) is $\frac 12\times\frac 12=\frac 14$, which also pays out $\$0.50$ on average. Continuing like this gives you a payout of $\sum^{\infty}\$0.50=\$\infty$ every time you play the game! Not such a bad thing, but it leads to my main question (shortly after).

Suppose you hold a party with $30$ people in it, and you want to find the probability that any two of them will have a birthday on the same day. Do you expect that to happen, or not?

Again, common everyday intuition says it seems unlikely that any two people out of thirty will have a birthday on the same day, but, again, mathematically, it is more likely than not. The exact probability is $1-\frac{365!}{365^n(365-n)!}\approx 0.7063$. So is it time to ask the question?

Why do some mathematical ideas seem counter-intuitive? Mathematics isn't based off of physical observations; it's an abstract concept, so shouldn't it explain our world better, not worse?

The above game (which I was told is St. Petersburg paradox) is only an example of what I mean when I say "counter-intuitive". Among others, ones I can name off the top of my head are the Monty Hall problem, Benford's Law, and the Banach-Tarski paradox. Those all have specific aspects to which a normal non-mathematician would turn their heads in confusion.

I really hope my question isn't too philosophical for this site.

This question has been in my head for as long as I can remember, so I decided to post some of my thoughts. Mathematical laws don't just hold for our world or our universe. It holds for all universes. For example, maybe the Banach-Tarski paradox makes perfect sense in $34$ dimensions. Or maybe the second dimension finds the concept of $\pi$ being irrational hard to grasp, whilst we find it easy. The most important thing to note is that mathematics is always right. It doesn't matter what we think. We're stupid. But in the long run, math has and always will get out on top.

Is the reasoning in the previous paragraph correct? The answers so far are good, but they don't really address counter-intuitivity in general, instead specific problems. Several answers below state something along the lines of "some ideas seem counter-intuitive because we've adapted to it; that is to say, it is best for the human race". Can any of you think of a practical application of counter-intuitive ideas in the evolution of humankind? I certainly can't.

So what do you think? I know my question doesn't have a solid answer, and I know it might be put on hold because of it (please don't though!). I just want to put my question out there, and hope it gets answered.

Thanks for reading!

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    $\begingroup$ Wait a minite. Say I flip heads ten times in a row, and on the next flip I get a tail. Then are you saying I lose my buy-in and all the money ($1023 I believe) that I won on the first ten flips? Do I have to keep on flipping till I get a tail, or can I quit any time I want? (If you would only let me keep my winnings from each time I flipped a head, this would be the famous St. Petersburg Paradox with its paradoxical infinite expectation.) $\endgroup$ – bof Oct 10 '14 at 5:18
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    $\begingroup$ Risk evaluation is not logic, you can do it without infinity with this game: You have to bet 10.000 dollars and have a 1 in 100 chance of winning 5.000.000 dollars, The expected outcome is positive, but most people won't do it, because there is a 99% chance to simply lose 10 grand and gain nothing. $\endgroup$ – Falco Oct 10 '14 at 8:41
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    $\begingroup$ Why do you think that an abstract concept should explain our world better, not worse? $\endgroup$ – user 170039 Oct 10 '14 at 8:59
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    $\begingroup$ Your intuition was the result of evolutionary pressures which included guessing accurately about which fruits are edible and whether there's a leopard in that tree; human intuitions about probabilities are notoriously terrible because overestimating risks is far more adaptive than underestimating risks. It would be surprising if a process adapted to think that shadows might be leopards gave correct insights about abstract formal systems even a tiny fraction of the time. $\endgroup$ – Eric Lippert Oct 10 '14 at 16:08
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    $\begingroup$ Note that each round takes at least some non-zero amount of time to play, so while the expected winnings given infinite time may be infinite, the probability of winning anything meaningful within your lifetime is essentially zero. $\endgroup$ – R.. Oct 10 '14 at 20:20
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Counterintuitive results come from our inductive thinking. We tend to think that if all, or even most of the objects we have met so far have a certain property, then all objects with similar characteristics will have that property.

If you look at the 19th century mathematicians, they first thought that a continuous function is differentiable with at most countable exceptions, but then a nowhere differentiable continuous function was defined -- and now we know that amongst the continuous functions, most of the functions are nowhere differentiable.

As you proceed in mathematics you learn that the objects that interest us are often the pathological objects, if you consider the broader picture. Most functions from $\Bbb R$ to itself are not continuous, of the continuous ones, most are not differentiable anywhere, of the differentiable, most are not continuously differentiable, and so on. Similarly of the subsets of $\Bbb R$ most of them are not Lebesgue measurable, of the Lebesgue measurable, most of them are not Borel measurable. And similarly for the real numbers, most of them are not rational, or even algebraic.

This is why you run into "normal" and "regular" terms in mathematics. We model the basic axioms of an object based on a smidgen of intuition (which may or may not be a well-developed mathematical intuition), but then we learn that there are other objects as well, so the original objects are added an extra hypothesis and we call them "normal" or "regular". And then we develop better mathematical intuition, and the cycle continues to grow.

Finally, since mathematics is not based on physical observation, I don't see why it should describe physical reality "better" or "worse". It shouldn't describe physical reality at all. It can be used to model reality, but since mathematics require infinite precision, and our senses can give us a very limited bound of input, we can never truly model the physical reality via mathematics, since we don't fully grasp it.

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    $\begingroup$ Good answer. "It shouldn't describe physical reality at all" - don't tell Max Tegmark! You may wish to consider updating an incomplete sentence, viz. "...most of them are not Borel measurable, of the Borel measurable. " $\endgroup$ – Epsilon Oct 14 '14 at 1:15
  • $\begingroup$ Nick, thanks for noticing. $\endgroup$ – Asaf Karagila Oct 16 '14 at 18:39
  • $\begingroup$ so the original objects are added an extra hypothesis and we call them "normal" or "regular" I'm not sure what grammatical structure you intend here. $\endgroup$ – user21820 Nov 11 '15 at 16:03
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"Why do some mathematical ideas seem counter-intuitive? Mathematics isn't based off of physical observations;..."

The bold portion above could be paraphrased as asking how come mathematics can do things our intuition cannot.

The bold portion below could be paraphrased as asking shouldn't mathematics be able to do things our intuition cannot.

"... it's an abstract concept, so shouldn't it explain our world better, not worse?"

$$$$

Contrary to what your intuition is telling you, these two ideas are not in contradiction.

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  • $\begingroup$ +1. This answers the real question of OP. (Though I don't deny that resolving the conflict in OP example is a good point, but it's unsatisfying without this part) $\endgroup$ – justhalf Oct 11 '14 at 0:33
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    $\begingroup$ @justhalf Thank you. I think it's a shame we don't do more to teach this aspect of mathematics to students much earlier in their careers. I think it go long way towards making kids less uncomfortable with abstraction. $\endgroup$ – David H Oct 11 '14 at 0:49
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There are several reasons that mathematics seems counter intuitive.

  • For several issues our intuition models the universe badly. For example;
    • We tend to discount very rare events as impossible and we also tend to value a high impact higher than we should compared to high frequency impacts.
    • We fail to model scale correctly. See for example this less wrong article
    • We tend to model infinity as a really big number.
    • We tend to overemphasise effects that are close to us at the expense of things far away. This is particulary noticable in that we tend to discount long term effects.
  • The mathematical model can be of something different than what our intuition is modeling. This is the case in your example.
    • The model assumes that the entity backing the bet can give infinite money. Our intuition will tend to assume that the entity behaves like a concrete legal person with bounded cash.
    • The model operates on cash value, our intuition might just as well operate on expected utility.
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  • $\begingroup$ But this isnt related to intuition... This is related to hypostasis,i.e., confuse model and reality, that is a very different thing. Model is model, not reality. And reality is reality, not model. You are comparing both and saying that because they dont meet our intuition is bad. LOL. By definition model isnt reality... our intuition is perfect because is true that are different things. LOL. $\endgroup$ – Masacroso Oct 10 '14 at 9:04
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    $\begingroup$ @Masacroso I am comparing the model that our intuition provides with the model that we use in formal methods. Both are models in the sense that they make predictions about a system and neither is actually the real thing that we are modeling, if such a thing exists. $\endgroup$ – Taemyr Oct 10 '14 at 10:02
  • $\begingroup$ I cant says that intuition make any model. What make model is reason, see religions. Its not the intuition, it is the reason. If you read definitions of reason and intuition you can see that they are basically opposed. The term "model" is very wrong for me to attach to intuition. As Aristoteles says is the reason, not the feeling, what fail. $\endgroup$ – Masacroso Oct 10 '14 at 10:27
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    $\begingroup$ @Masacroso In my comment I make it explicit what I mean by model. Ie. something that allows us to make predictions about the modeled system. It seems clear to me that under this meaning intuition models a system, because it causes us to make prediction about how a system behaves. $\endgroup$ – Taemyr Oct 10 '14 at 10:41
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    $\begingroup$ @Masacroso A system is not an abstraction - The way we consider a system is. A system is several interactive entities, the entities are real and their interactions are real, so the system is real. Reality is something external to the individual so it's not a feeling. $\endgroup$ – Taemyr Oct 10 '14 at 12:49
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In your first example, the St. Petersburg Paradox, the expected value of the pay-off is infinite. That is a firm mathematical result following from the axioms of probability.

However, the buy-in amount (or the decision to play given a specified buy-in) depends upon other considerations -- for example, your risk preferences. This requires a behavioral model and is not a purely mathematical inference. Risk preferences could be modeled mathematically using a utility function and depend on other mathematical results such as probability of ruin, variance of the pay-off, etc.

If you continue to play and bet all of your accumulated gains on subsequent turns, the probability of ruin approaches 1. Would you still play for the infinite expected gain?

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    $\begingroup$ There's also the point that the payout is only infinite because of the slim chance you get an exceedingly vast amount of money, but for most people the difference between winning 10 dollars in the lottery and 10 thousand dollars is greater than the difference between winning 10 million and 10 billion. After a point, acquiring more money has diminishing psychological returns. $\endgroup$ – Jack Oct 10 '14 at 15:49
  • $\begingroup$ @Jack: Precisely -- that's where the utility function comes in. As I recall, this paradox was resolved by introducing log utility. That leads to a buy-in of a few dollars which conforms with most peoples preferences. $\endgroup$ – RRL Oct 10 '14 at 16:03
  • $\begingroup$ @RLL Log utility may solve the specific St. Petersburg paradox, but it is easy to modify it so that log utility no longer resolves it: just exponentiate the gains in all cases (to cancel the log). In fact for any unbounded utility function the paradox can be resuscitated, so what is really unrealistic is unbounded utility functions. Life is finite, and for any reasonable measure of happiness there is certainly a bound on how happy any gains can make us. $\endgroup$ – Marc van Leeuwen Oct 11 '14 at 10:01
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Short answer: Because we're not good at intuiting results in infinite situations (whereas practical, finite versions tend to make more intuitive sense), and we can be deceived by probability too. I harbour a suspicion that this latter one happens when we're working without much information or context; the kind of thing that our evolutionary psychology hasn't prepared us for, I suppose.

Two of your examples revolve around infinite quantities: your lottery game (the St Petersburg paradox) and the Banach-Tarski paradox. We're not good at intuitions around infinity, which makes a certain amount of sense. And if you reduce the St Petersburg lottery to finite possible winnings (or finite time to play), it actually becomes quite in line with intuition.

I like to think of the Monty Hall paradox as a bit of sleight-of-hand played with information. You may intuitively realise that your first guess is probably wrong (2/3 chance), but you think that, since picking any door would have led to the same conclusion, the "switch" door is no better. But really, since your first guess is probably wrong, that means the right door is probably not yours--and the host has conveniently eliminated the only other wrong door! (To put it another way, you're being offered a chance to play a new guessing game with a 50-50 chance of winning, instead of the original game, which you probably lost.)

As for Benford's law, I'm not really seeing what's unintuitive, but I only just read about it for the first time, thanks to your question.

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Other people seem to agree with your initial claim that the payoff is infinite. Maybe it's because I have a cold, but I'm not seeing it. First off, I'm assuming that the money won stacks (in other words, if you get $10$ heads in a row and quit, you get $1+2+4+8+\cdots+512=1023$ dollars, versus just getting $512$). In that case, your expected payout is $2^{-10}\cdot1023$ which is just slightly less than $1$ ($1023/1024$)... and for the case of an infinite games, your expected payout would be $1$. (Note that as written right now, your problem doesn't appear to be identical to the St. Petersburg Paradox; not sure if this is an edit issue or something else)

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    $\begingroup$ I'm sorry, but I'm not asking about the game. I just gave it as an example of something that's counter-intuitive. $\endgroup$ – Edward Jiang Oct 11 '14 at 1:09
  • $\begingroup$ You are quite right, the OP botched the definition of the St. Petersburg Paradox. I also pointed this out in my comment on the question. I assume the "other people" didn't bother to read the OP's description of the game, because they are already familiar with it and assume OP described it correctly. $\endgroup$ – bof Oct 11 '14 at 11:11
  • $\begingroup$ @bof: Or because that wasn't the point of the question, and in a surprising case mathematicians were able to transcend the mistake and talk about the actual question, rather than ignoring the actual question and nitpick the small mistakes? :-) $\endgroup$ – Asaf Karagila Oct 11 '14 at 11:23
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    $\begingroup$ @AsafKaragila I is an engineer; pretty sure ignoring the actual question and nitpicking the small mistakes is right there in my job description $\endgroup$ – Foon Oct 11 '14 at 13:20
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    $\begingroup$ @Foon: Not really. Your job is to abuse mathematics and take a 40% safety factor. Mathematicians are the ones whose job description requires them to be super nitpicky and ignore the actual question. $\endgroup$ – Asaf Karagila Oct 11 '14 at 13:34

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