“Nothing’s Impossible”   5 comments

Mark Chu-Carroll over at Good Math, Bad Math gives a good simple description of how the financial markets got themselves into this mess.

He also got me thinking about a particular problem that I’ve found jiggering around in the back of my head during my last few forays into the Interwebs: people really don’t understand the Strong Law of Large Numbers, and the immediate consequence of this is that they don’t understand that really improbable events are usually impossible.  Not unlikely.  IMPOSSIBLE.

Let me illustrate.  Go find a pair of six sided dice.  Give ’em a good shake, like you’re in Vegas, and give ’em a toss (you’re encouraged to call out, “Daddy needs a new pair of shoes!”).  Now, without me knowing what the result is, I’ll go ahead and guess “7”, as it’s the most likely result… but of course there’s a substantial probability that I’m wrong.  The actual probability that you rolled a seven is 0.166666666(…)

Now grab a pencil and paper, and write that number down.  Roll the dice 10 more times, and record all of your results.  Add them together and divide by 10, scribble that in another column.  Now roll the dice 10 more times, and add the 20 results together, and divide by 20… scribble that in that other column.  Repeat until you’ve rolled the dice 100 times.

Now you can look over that sheet of 100 dice rolls, and you may very well find some “crazy improbable miracle result!”, like rolling 10 twos in a row… but you’ll find that that second column of numbers averages very close to 7, don’t it?

What the Strong Law of Large Numbers says is that if you roll those dice over and over and over again, all the way out to infinity… the probability that the average of those dice equals seven is 1.

It’s not close to 1, it is 1.  Certainty.  There is no other possibility.

“But Pat,” you may think, “I could keep rolling two the hard way!  It’s possible!  Each die roll is an independent event!  ANYTHING is possible!”

You’re right, each die roll is an independent event… but your conclusion, that therefore “ANYTHING is possible”… no, it’s not.  Taking individual events, anything is possible (I could always roll a “2” on my next die roll)… but while each die roll is an independent event, our average is NOT.  It depends upon all of the other results that we’ve rolled so far… so while each die roll is independent, the average is conditional (note – misunderstanding the converse of this and all of the implications of it is where the Gambler’s Fallacy comes from… some people understand intuitively that the average is conditional, but then assume that this has some “effect” on the individual events).

Now, there’s a lot of convergence theorems in mathematics as well, which predict how quickly your average is going to get arbitrarily close to the expectation, but we don’t want to get toooo complicated here.

Basically, what the math shows is that an outcome requiring an infinitely large collection of improbable events to occur is not improbable… it’s impossible.  Taken with convergence theorems, it’s actually provable that given a sufficiently large collection of improbable events, we can conclude that the outcome is impossible.  Surprisingly to many people… “sufficiently large” isn’t that big at all, particularly when the events are really improbable… particularly when you take into account boundaries.

This is why followers of Popper and Hume are wrong.  This is why, even though science is not an axiomatic system, scientists have a tendency use the terms “proof”, “law”, and “truth”… in a purist sense, they shouldn’t, but in an empirical sense, it’s certainly reasonable for them to do so.

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Posted September 25, 2008 by padraic2112 in math, science, Uncategorized

5 responses to ““Nothing’s Impossible”

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  1. Sorry you must mean “fair dice” which are some sort of mathematical made up dice. I’d say that the average would more likely go to 0.69999875, because dice are injection modeled plastic, and no way they have even edges, 100% uniformity. Also I could wear them down, or slightly weight them with the hot breath and skin cells! But I’m not sticking around to infinity working on that project.

    Now as to how close to 7.0, I’d like to know what the research on drilled (decidedly less fair) versus drilled and filled (casino dice) showed. Its out there somewhere? Maybe the bias in the later was too small to measure. Of course a casino only cares that it is background noise compared to their advantage.

    Still I can’t believe you said “equals seven”. I’m having to take off a point for that. Just one, because of course it would converge to a number approaching seven. Or I should assume the dice that mathmaticians have. Sorry, but engineers won’t allow unstated assumptions. (So yeah, I’m not the target audience)

    I suppose you could also create enough dice from separate molds, rotate the position of the spots, and roll a greater number of dice, maybe one big infinite roll. Then it comes down to how perfectly the spot filling density matches the body material.

    Anyway, I thought the nit intersting enough to pick.

  2. Well, Benji Boy, you’ve forgotten your limit theory. Of course the limit equals seven. But you’re right, I didn’t use strict terminology there.

  3. Actually, you weren’t following me, maybe because as a commenter I can’t show you…

    Pointy E as n from 1 to sideways eight… I am doing the same thing… and the fair dice do go to seven. I agree.

    But since you didn’t call out “fair dice”, which are useless as far as engineers are concerned, I don’t put in 1/6 for the chance of each number on each die. Why not? Because a fair die would be a materials engineering accomplishment equal to the tower to the sky. So the result would skew away from 7. Which way depends on the physical structure of the dice. Drilled dice always skew to higher numbers, painted slightly toward lower numbers, drilled and filled are the best we can do. I think there is a study on just how good they are… you can be sure casinos have a proof from their suppliers.

    So the mold used is imperfect, but if you milled it to the highest precision, it still requires a coating of a release agent. Therefore, foiled again! Now you need the most perfect release agent squeegy and process ever designed! So the individual roll probabilities on each face are close to 1/6, but I’d place the biggest bet that they would not EQUAL 1/6. That’s why I got a different answer.

    Yes, I know… you should be reading this on my blog! So maybe enough applications of six sigma manufacturing analysis can result in dice manufacturing with random imperfections. Then you can roll a very large number of dice a very large number of times and get as close to seven as the process would ever allow.

  4. AH, I got you. You’re pulling that whole “reality” trick that engineers like to pull on mathematicians.

    Okay, smart guy. Here’s your solution. Take a normal six sided die, but instead of using the numerical values on each side as the numerical value of the die roll, re-assign the numerical value on each face randomly before each trial. So on one throw, the pips are:

    1 pip = 1
    2 pips = 2
    3 pips = 3
    4 pips = 4
    5 pips = 5
    6 pips = 6

    and the next throw

    1 pip = 5
    2 pips = 6
    3 pips = 2
    4 pips = 1
    5 pips = 4
    6 pips = 3

    And so on. Assuming your random number generator is sufficiently random, you can correct for physical deformities or dice bias (now you’re going to argue that it’s impossible to create a truly random random number generator…)

  5. That’s a clever solution. I’ll give you the random number generator, since you clearly call one out. I was only imposing reality, since you didn’t state your assumptions. Welcome to the world of cost-plus engineering! This is how unspecified items drive the cost of a six sided piece of plastic to $20k each (that’s the quantity discount).

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