Paradoxes, Math And Why You Should Never Sit Ben Roethlisberger
    By Hank Campbell | November 13th 2012 07:09 PM | 17 comments | Print | E-mail | Track Comments
    About Hank

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    We all love paradoxes, those seemingly consistent logical brain-teasers where we sort out what can and should and might and must happen and that invariably lead to self-contradictory arguments.

    If you are like me and my friends, there is nothing you enjoy more than sitting around during half-time of the Steelers game and arguing over Maxwell's Demon - the many ways to violate the Second Law of Thermodynamics, namely that heat transfer happens, from warmer to colder, until equilibrium is reached.  If I put an ice pack next to Ben Roethlisberger's blazing hot 145 third down QBR (Total Quarterback Rating), for example, the ice pack will not cool down, it will warm up.  So it has always been, so shall it must be. It is common sense. 

    But Maxwell's Demon turns us all into circular reasoning dolts. If we allow James Clerk Maxwell's pesky demonic sprite to run free, letting faster molecules in and out of doors and ruining equilbrium, we end up with perpetual motion machines and the Broncos winning in the playoffs and other craziness. Statisticians love the entropy of the Second Law of Thermodynamics so if you release the Demon, they start taunting you with how great a team the Cleveland Browns are. We don't want that to happen.

    Does this look like a football play?  Maxwell came up with this paradox long before the game was invented, so you see where Americans got the idea for X's and O's - they were slow and fast-vibrating molecules, which meant more heat.

    But Maxwell's Demon is not the only paradox worth debating over an Iron City beer, there are lots of others known to lots of people.  The Monty Hall paradox, for example, deals with conditional probability.  It was based on a game show and it has three boxes, two of which are empty (or has a goat behind a door, if that helps - I'd rather get an empty box than a goat) and one of which has the keys to a new car.  You, as the contestant, have to decide the smartest choice; take a small amount of money and give up your box with its 1 in 3 chance of a new car or risk getting nothing at all. The game show host, Monty, then even reveals one of the three boxes to be empty and offers you a larger amount of money for your still unknown box.  Do you take the money, since you seem to have a 50/50 chance of a car or nothing?  No, you offer to switch boxes with him.  Confounding? Aren't there equal probabilities, since there are only two boxes?  Read on...

    Yes, I won the car the first try because the math says so - except it shouldn't, if you ask most people. Link: UCSD 

    Professor Jim Al-Khalili is a Science 2.0 favorite.  He has been mentioned here too many times to count because he is the University of Surrey Professor of Public Engagement in Science and the kind of physicist you want to argue with at halftime during Steelers games. 

    In Paradox: The Nine Greatest Enigmas In Physics, he covers all the good ones, including the difficult Maxwell's Demon along with the more common Grandfather Paradox, were you to go back in time and kill your grandfather, and Schrödinger's Dead And Alive Cat.

    It's a breezy 200 pages, an easily digestible book that will still confound you, no matter how clearly he explains it.   Along the way, he teaches you how to think a little better.

    The Monty Hall Paradox, for example, can be statistically gamed a little - if you have three boxes and Monty Hall knows which one has the car and which two are empty, by opening one to make you feel like better odds, he could be setting you up.  If he offers you $1,000 for your box, he may want you to think it has the car and wants to offer you just enough you get greedy and want to hold onto it, assuming it has a much more valuable car, and getting nothing.

    By offering to switch boxes you statistically increase your chances of getting the car. Here it is visually using doors and goats.

    Of course, this assumes Monty knows which door or box gives you a car.  What if even he does not know which box has the car but is instead offering money blindly?  You can no longer rely on his prior knowledge and neither can he.  What should you do then?  You still switch if you can, you'll be right 66% of the time.

    Are you a modern netizen and can't fathom these problems in your head?   UC San Diego has an online version.  Try it as many times as you want but the math rules, just like Roethlisberger in third down passing situations.  Or second down passing situations.  Basically, if you have three doors in your Monty Hall scenario and all three do not have Big Ben behind them, your chances of Sunday afternoon happiness go way down.

    Paradoxes don't make sense?  We can argue the specifics all week but if you want to tickle your brain with Fermi and The Twins and the other paradoxes I mentioned here, pick up a copy of "Paradox" and jump into arguments that are sometimes thousands of years old.

    Science 2.0 rating: 5 Bloggys!



    Dear Hank,

    while Maxwell's daemon can be called a paradox, I would not say the same of the monty hall problem, which has nothing paradoxical (it is a very simple calculation involving the addition of fractions that gives the correct answer).

    para doxa = against (common) opinion
    monty hall = paradox (monty hall is even a classical example for such!)
    Maxwell Deamon = not really a paradox
    I am curious, why isn't Maxwell's demon a paradox?  Is the demon part of the system?  

    I had kind of thought that that prior knowledge in each case - a game show host who knows where the car is, a demon controlling a trap door - made these rather similar.
    para doxa = against (common) opinion
    So modal reality is a paradox?  I knew it! :)

    Seriously, regardless of the etymology, there's a bit more to paradox than merely being against common *opinion* - there's the appearence of a contradiction. I think that's why Maxwell's demon is paradoxical - at first sight it ought to "work". If anything, Monty Hall is just sloppy thinking.

    Yes, of course, the vital is that it seems that there is a contradiction where there is none. That is why EPR and SR-twin are solid paradoxa, so is Monty ("nothing changed", so how can the probabilities change), however, Maxwell does not really fit here in my opinion. Where is the paradox? The twin paradox is almost immediately there once the lay person has the basics about LT contraction, but assuming a demon, well, it is certainly not anywhere in the theory as far as I know.
    This is a good point and Al-Khalili talks about Monty Hall but calls it a veridical (common sense) paradox, not the physics kind, because it goes against common sense, then he writes at the end of chapter one 'Enough of this frivolity - a bunch of proper physics problems awaits us' so the verbage on Monty Hall is mine, not his.  Getting the others (like Maxwell) into shape would have meant me writing a chapter in a book, and since he just wrote that book I did not want to redo it, but Monty Hall is manageable in a short article.

    The Monty Hall paradox is not really about "the math rules".

    There's a one in three chance you choose the car initially and it would be wrong to swap. In the other two cases you choose a goat and, because Monty shows you the other goat, it would be right to swap.
    I'd call that logic rather than statistics.

    I wonder if Al-Khalili has solved the Schrodinger Cat paradox as well? :)

    In the article I list that he goes into the cat and the grandfather paradox, the twins, fermi and more.
    The point I was making is that most paradoxes just require clear thought.

    The twins paradox is just the result of looking at the time dilation term in the Lorentz transform, which is independent of "direction of movement", while ignoring the "time offset" term, which most definitely is not. There is no paradox if you apply relativity correctly.
    The grandfather paradox (assuming time travel is possible at all) is only a paradox if you assume there are no fixed points in the equation of state. The words used in the paradox do not match the reality of physical systems.
    The Fermi paradox assumes that intelligent life is common in the universe. It goes away if life is rather rare.

    However, the Schrodinger cat "paradox" is not an error in reasoning, it requires a major shift of ontological paradigm to "solve" it.  Pace Sascha - I know you could say that being stuck in direct realism is a logical error, but most people don't make the mistake of carelessly assuming something about reality, they fall down by clinging stubbornly to a common-sense view of reality endorsed by the 19th and 20th century paradigm of "objective science"... Because it hits at our very concept of reality, I'd say that Schrodinger's cat goes way beyond mere paradox. 

    Amir D. Aczel
    Here's one I have not seen discussed in a long time, and is one of my favorites: The St. Petersburg Paradox.Peter and Paul play a game. If a coin comes up "heads" Peter will pay Paul $2; if it's tails but the NEXT toss is "heads," Peter pays Paul $4; if it's TTH, he pays him $8, and so on. What is the mathematical expectation of this game for Paul? Answer: E(X) = Sum (xP(x)) = 2(1/2) + 4(1/4) + 8(1/8) +...= 1 + 1 + 1 + 1...= infinity. But simulations always stop before "infinity" (in practice the process converges; the reason is that the probability is exceedingly small that you go beyond any very large number).
    Amir D. Aczel
    Most of these were named after the creator but this one is named after the magazine that published it.  I wonder why Bernoulli or his biographers were so modest?  Maybe they realized he was never going to be the most famous Bernoulli anyway?

    Can an infinite value paradox be a paradox at all? How theoretical do we have to go before we reach something so theoretical it is objectionable in a thought experiment?  I guess that is part of the problem. :)
    Johannes Koelman
    The paradox is that nobody will pay, say, $ 100,000 to enter this game. Yet, the expected return would certainly warrant doing so.
    By the way, as everybody knows, 1 + 1 + 1 + 1 + ... = -1/2. You should be paid half a dollar to participate!

    Amir D. Aczel
    Ah, the Riemann zeta function!! :)
    Amir D. Aczel
    Amir D. Aczel
    I think that most "real" paradoxes are not in physics--where the paradoxality is just our lack of understanding or bad intuition--but in the realm of infinity and the foundations of mathematics. Hilbert's Hotel is a great paradox--and a real one! They have infinitely many rooms, but they're all taken. So do the 1-1 mapping from Z to Z - {1} and you've got a free room. Now, that's a paradox! Russell's Paradox seems like pure semantics: What does it mean "he shaves every man in Seville who doesn't shave himself" as the simple example goes--but when you apply it to actual sets you get the impossibility of a universal set, and a huge logical hole in ZFC. Godel's proofs open up other "paradoxes' about what we can and can't ever know...
    Amir D. Aczel
    Whoa. I thought if Monty does not know where the car is, then switching is not an advantage.
    Please explain.

    Prior knowledge confusion was what made Marilyn vos Savant's (referenced in another comment) so controversial.  For the probabilities to make sense, you can't do one experiment.  So if Monty does not know what is where, and the boxes are shuffled at random, either of the two remaining doors he picks will have car keys 50 times.  Game over. But 100 times the box will be empty or, in the door example in the article, have a goat behind it.

    So if Monty is blind and you pick A every time, you have a 1 in 3 chance of being right if you keep your box.  In the other 100 cases, 50% of the time it is behind B and he picks C or it is behind C and he picks B but 100% of the time he opens an empty box.

    Basically, by staying you are correct 1/3 of the time but lose 2/3 of the time. But you win if you switch when you originally picked an incorrect door, which you did 2/3 of the time.
    No, no, no, no, that's completely wrong!

    If Monty is blind, you get 50% chance if he opens a goat door.

    If Monty tells you «I will open a door with a goat.», the information you get is «if the car is behind one of the two other doors (event A), then it is behind the non-open door.» But you get no information on event A, hence its former probability (2/3) stays the same (I assume here everything is exchangeable, otherwise, strictly speaking, you have to use a little game theory to get back to this).

    If Monty does NOT know what is behind the door he opens, and opens a door with a goat, you ALSO get information on event A.
    Indeed, if A were true, Monty would have probability one half to open on the car. If A is false, he has probability 1. Bayes' rule yields half a chance for each remaining door. Alternatively, what is behind the two doors is independent on Monty's choice.

    To be more specific, there is:
    * one chance in three the car is behind the door you choose.
    * one chance in three it is behind the door no one has chosen.
    * one chance in three it is behind the door Monty has chosen.

    If the door Monty opens has no goat, you merely conditionate on that event.

    And it makes absolutely no sense to say «Monty does not know whether there is a goat behind the door he opens, but always opens a door with a goat behind it.» In this problem, that's the definition of knowing. If someone else who knows decide for Monty, then he is «Monty». Monty is the one who chooses the door.