# Luck(?) of the Draw

What is luck?  Is luck?  And, if you vote yea, is a belief in luck an obstacle to understanding probability?

This question came up on Twitter a couple of nights ago when Christopher Danielson and Michael Pershan were discussing Daniel Kahneman's recent book, Thinking, Fast and Slow.  Specifically, they were talking about the fact that Kahneman doesn't shy away from using the word luck when discussing probabilistic events.  This, of course, is the kind of thing that makes mathematically fastidious people cringe.  And Danielson and Pershan are nothing if not mathematically fastidious.  Spend like five minutes with their blogs.  So Danielson twittered this string of twitterings:

According to Danielson, luck a "perceived bias in a random event."  And, according to his interpretation of Kahneman, luck is composed of "happy outcomes that can be explained by probability."  Let me see if I can define luck for myself, and then examine its consequences.

# What is luck?

I think, at its heart, luck is about whether we perceive the universe to be treating us fairly.  When someone is kind to us, we feel happy, but we can attribute our happiness to another's kindness.  When someone is mean, we feel sad, but we can attribute our sadness to another's meanness.  When we are made to feel either happy or sad by random events, however, there is no tangible other for us to thank or blame, and so we've developed this idea of being either lucky or unlucky as a substitute emotion.

But happy/sad and lucky/unlucky are relative feelings, and so there must be some sort of zero mark where we just feel...nothing.  Neutral.  With people, this might be tricky.  Certainly it's subjective.  Really, my zero mark with people is based on what I expect of them.  If a stranger walks through a door in front of me without looking back, that's roughly what I expect.  And, when that happens, I do almost no emoting whatsoever.  If, however, he holds the door for me, this stranger has exceeded my expectations, which makes me feel happy at this minor redemptive act.  If he sees me walking in behind him and slams the door in my face, he has fallen short of my expectations, which makes me sad and angry about him being an asshole.

And, in this regard, I think that feeling lucky is actually a much more rational response than being happy/sad at people, because with random events at least I can concretely define my expectation.  I have mathematical tools to tell me, with comforting accuracy, whether I should be disappointed with my lot in life; there is no need to rely on messy inductive inferences about human behavior.  So I feel lucky when I am exceeding mathematical expectations, unlucky when I'm falling short, and neutral when my experience roughly coincides with the expected value.  Furthermore, the degree of luck I feel is a function of how far I am above or below my expectation.  The more anomalous my current situation, the luckier/unluckier I perceive myself to be.

Let's look at a couple examples of my own personal luck.

1. I have been struck by lightning zero times.  Since my expected number of lightning strikes is slightly more than zero, I'm doing better than I ought to be, on average.  I am lucky.  Then again, my expected number of strikes is very, very slightly more than zero, so I'm not doing better by a whole lot.  So yeah, I'm lucky in the lightning department, but I don't get particularly excited about it because my experience and expectation are very closely aligned.
2. I have both my legs.  Since the expected number of legs in America is slightly less than two, I'm crushing it, appendage-wise.  Again, though, I'm extremely close to the expected value, so my luck is modest.  But, I am also a former Marine who spent seven months in Iraq during a period when Iraq was the explosion capital of the world.  My expected number of legs, conditioned on being a war veteran, is farther from two than the average U.S. citizen, so I am mathematically justified in feeling luckier at leg-having than most leg-having people in this country.

# Conclusion

Of course for some people the lottery is terrible.  People have gambling problems.  People spend way too much money on all kinds of things they probably shouldn't.  But that doesn't mean that everyone---or even most people---that play are suckers.  Eating the occasional King Size Snickers probably won't get your foot chopped off; smoking the occasional cigarette probably won't kill you (sorry, kids), and buying the occasional lottery ticket will likely have about zero net impact on your finances.  Besides, isn't it worth it to dream, for even a day, of having indoor hot tubs?  They're so bubbly.

# Building a Probability Cannon

For just a moment, let's consider a staple of the second year algebra curriculum: the one-dimensional projectile motion problem.  (I used to do an awful lot of this sort of thing.)  It's not a fantastic problem---it's overdone, and often under-well---but it's representative of many of our standard modeling problems in some important ways:

1. Every one of my students has participated in the activity we're modeling.  They've thrown, dropped, and shot things.  They've jumped and fallen and dove from various heights.  In other words, they have a passing acquaintance with gravity.
2. Data points are relatively easy to come by.  All we need is a stopwatch and a projectile-worthy object.  If that's impractical, then there are also some great and simple---and free---simulations out there (PhET, Angry Birds), and some great and simple---and free---data collection software as well (Tracker).
3. We only need a few data points to fix the parameters.  For a general quadratic model, we only need three data points to determine the particular solution.  Really we only need two, if we assume constant acceleration.
4. Experiments are easy to repeat.  Drop/throw/shoot the ball again.  Run the applet again.
5. The model conforms to a fairly nice and well-behaved family of functions.  Quadratics are continuous and differentiable and smooth, and they're generally willing to submit to whatever mathematical poking we're wont to visit upon them without getting gnarly.
6. Theoretical predictions are readily checked.  Want to know, for instance, when our projectile will hit the ground?  Find the sensible zero of the function (it's pretty easy to sanity check its reasonableness---see #1 above).  Look at a table of values and step through the motion second-by-second (use a smaller delta t for an even better sense of what's going on).  Click RUN on your simulation, and wait until it stops (self-explanatory).  And, if you're completely dedicated, build yourself a cannon and put your money where your mouth is.

Of course I've chosen to introduce this discussion with the example of projectile motion, but there are plenty of other candidates: length/area/volume, exponential growth and decay, linear speed and distance.  Almost without exception (in the algebra classroom), we model phenomena that satisfy the six conditions listed above.

Almost.  Because then we run into probability, and probability isn't so tame.  I'll grant that #1 still holds (though I'm not entirely convinced it holds in the same sense), but the other five conditions go out the window.

# Data points are NOT easy to come by.

I can already hear you protesting.  "Flip a coin...that's a data point!"  Well, yes.  Sort of.  But in the realm of probability, individual data points are ambiguous.  The ordered pair (3rd flip, heads) is very different from (3 seconds, 12 meters).  They're both measurements, but the first one has much, much higher entropy.  Interpretation becomes problematic.  Here's another example: My meteorologist's incredibly sophisticated model (dart board?) made the following prediction yesterday: P(rain) = 0.6.  In other words, the event "rain" was more likely than the event "not rain."  It did not rain yesterday.  How am I to understand this un-rain?  Was the model right?  If so, then I'm not terribly surprised it didn't rain.  Was the model wrong?  If so, then I'm not terribly surprised it didn't rain.  In what sense have I collected "data?"

And what if I'm interested in a compound event?  What if I want to know not just the result of a lone flip, but P(exactly 352 heads in 1000 flips)?  Now a single data point suddenly consists of 1000 trials.  So it turns out data points have the potential to be rather difficult to come by, which brings us to...

# We need an awful lot of data points.

I'm not talking about our 1000-flip trials here, which was just a result of my arbitrary choice of one particular problem.  I mean that, no matter what our trials consist of, we need to do a whole bunch of them in order to build a reliable model.  Two measurements in my projectile problem determine a unique curve and, in effect, answer any question I might want to ask.  Two measurements in a probabilistic setting tell me just about nothing.

Consider this historical problem born, like many probability problems, from gambling.  On each turn, a player rolls three dice and wins or loses money based on the sum (fill in your own details if you want; they're not so important for our purposes here).  As savvy and degenerate gamblers, we'd like to know which sums are more or less likely.  We have some nascent theoretical ideas, but we'd like to test one in particular.  Is the probability of rolling a sum of 9 equal to the probability of rolling a sum of 10?  It seems it should be: after all, there are six ways to roll a 9 ({6,2,1},{5,3,1},{5,2,2},{4,4,1},{4,3,2},{3,3,3}), and six ways to roll a 10 ({6,3,1},{6,2,2},{5,4,1},{5,3,2},{4,4,2},{4,3,3})*.  Done, right?

It turns out this isn't quite accurate.  For instance, the combination {6,2,1} treats all of the 3! = 6 permutations of those numbers as one event, which is bad mojo.  If you go through all 216 possibilities, you'll find that there are actually 27 ways to roll a 10, and only 25 ways to roll a 9, so the probabilities are in fact unequal.  Okay, no biggie, our experiment will certainly show this bias, right?  Well, it will, but if we want to be 95% experimentally certain that 10 is more likely, then we'll have to run through about 7,600 trials!  (For a derivation of this number---and a generally more expansive account---see Michael Lugo's blog post.)  In other words, the Law of Large Numbers is certainly our friend in determining probabilities experimentally, but it requires, you know, large numbers.

*If you've ever taught probability, you know that this type of dice-sense is rampant.  Students consistently collapse distinct events based on superficial equivalence rather than true frequency.  Ask a room of high school students this question: "You flip a coin twice.  What's the probability of getting exactly one head?"  A significant number will say 1/3.  After all, there are three possibilities: no heads, one head, two heads.  Relatively few will immediately notice, without guidance, that "one head" is twice as likely as the other two outcomes.

# Experiments are NOT easy to repeat.

I've already covered some of the practical issues here in terms of needing a lot of data points.  But beyond all that, there are also philosophical difficulties.  Normally, in science, when we talk about repeating experiments, we tend to use the word "reproduce."  Because that's exactly what we expect/are hoping for, right?  I conduct an experiment.  I get a result.  I (or someone else) conduct the experiment again.  I (they) get roughly the same result.  Depending on how we define our probability experiment, that might not be the case.  I flip a coin 10 times and count 3 heads.  You flip a coin 10 times and count 6 heads.  Experimental results that differ by 100% are not generally awesome in science.  In probability, they are the norm.

As an interesting, though somewhat tangential observation, note that there is another strange philosophical issue at play here.  Not only can events be difficult to repeat, but sometimes they are fundamentally unrepeatable.  Go back to my meteorologist's prediction for a moment.  How do I repeat the experiment of "live through yesterday and see whether it rains?"  And what does a 60% chance of rain even mean?  To a high school student (teacher) who deals almost exclusively in frequentist interpretations of probability, it means something like, "If we could experience yesterday one million times, about 600,000 of those experiences would include rain."  Which sounds borderline crazy.  And the Bayesian degree-of-belief interpretation isn't much more comforting: "I believe, with 60% intensity, that it will rain today."  How can we justify that level of belief without being able to test its reliability by being repeatedly correct?  Discuss.

# Probability distributions can be unwieldy.

Discrete distributions are conceptually easy, but cumbersome.  Continuous distributions are beautiful for modeling, but practically impossible for prior-to-calculus students (not just pre-calculus ones).  Even with the ubiquitous normal distribution, there is an awful lot of hand-waving going on in my classroom.  Distributions can make polynomials look like first-grade stuff.

# Theoretical predictions aren't so easily checked.

My theoretical calculations for the cereal box problem tell me that, on average, I expect to buy between 5 and 6 boxes to collect all the prizes.  But sometimes when I actually run through the experiment, it takes me northward of 20 boxes!  This is a teacher's nightmare.  We've done everything right, and then suddenly our results are off by a factor of 4.  Have we confirmed our theory?  Have we busted it?  Neither?  Blurg.  So what are we to do?

# We are to build a probability cannon!

With projectile motion problems, building a cannon is nice.  It's cool.  We get to launch things, which is awesome.  With probability, I submit that it's a necessity.  We need to generate data: it's the raw material from which conjecture is built, and the touchstone by which theory is tested.  We need to (metaphorically) shoot some stuff and see where it lands.  We need...simulations!

If your model converges quickly, then hand out some dice/coins/spinners.  If it doesn't, teach your students how to use their calculators for something besides screwing up order of operations.  Better yet, teach them how to tell a computer to do something instead of just watching/listening to it.  (Python is free.  If you own a Mac, you already have it.)  Impress them with your wizardry by programming, right in front of their eyes, and with only a few lines of code, dice/coins/spinners that can be rolled/flipped/spun millions of times with the push of a button.  Create your own freaking distributions with lovely, computer-generated histograms from your millions of trials.  Make theories.  Test theories.  Experience anomalous results.  See that they are anomalous.  Bend the LLN to your will.

Exempli Gratia

NCTM was kind enough to tweet the following problem today, as I was in the middle of writing this post:

Okay, maybe the probability is just 1/2.  I mean, any argument I make for Kim must be symmetrically true for Kyle, right?  But wait, it says "greater than" and not "greater than or equal to," so maybe that changes things.  Kim's number will be different from Kyle's most of the time, and it will be greater half of the times it's different, so...slightly less than 1/2?  Or maybe I should break it down into mutually exclusive cases of {Kim rolls 1, Kim rolls 2, ... , Kim rolls 6}.  You know what, let's build a cannon.  Here it is, in Mathematica:

Okay, so it looks like my second conjecture is right; the probability is a little less than 1/2.  Blammo!  And it only took (after a few seconds of typing the code) 1.87 seconds to do a million trials.  Double blammo!  But how much less than 1/2?  Emboldened by my cannon results, I can turn back to the theory.  Now, if Kyle rolls a one, Kim will roll a not-one with probability 5/6.  Ditto two, three, four, five, and six.  So Kim's number is different from Kyle's 5/6 of the time.  And---back to my symmetry argument---there should be no reason for us to believe one or the other person will roll a bigger number, so Kim's number is larger 1/2 of 5/6 of the time, which is 5/12 of the time.  Does that work?  Well, since 5/12 ≈ 0.4167, which is convincingly close to 0.416159, I should say that it does.  Triple blammo and checkmate!

But we don't have to stop there.  What if I remove the condition that Kim's number is strictly greater?  What's the probability her number is greater than or equal to Kyle's?  Now my original appeal to symmetry doesn't require any qualification.  The probability ought simply be 1/2.  So...

What what?  Why is the probability greater than 1/2 now?  Oh, right.  Kim's roll will be equal to Kyle's 1/6 of the time, and we already know it's strictly greater than Kyle's 5/12 of the time.  Since those two outcomes are mutually exclusive, we can just add the probabilities, and 1/6 + 5/12 = 7/12, which is about (yup yup) 0.583.  Not too shabby.

What if we add another person into the mix?  We'll let Kevin join in the fun, too.  What's the probability that Kim's number will be greater than both Kyle's and Kevin's?

It looks like the probability of Kim's number being greater than both of her friends' might just be about 1/4.  Why?  I leave it as an exercise to the reader.

That tweet-sized problem easily becomes an entire lesson with the help of a relatively simple probability cannon.  If that's not an argument for introducing them into your classroom, I don't know what is.

Thanks to Christopher Danielson for sparking this whole discussion.

# A Tale of Two Numbers

A few months ago, we had just finished talking about polynomials and were moving into matrices.  Because a lot of matrix concepts have analogs in the real numbers, we kicked things off with a review of some real number topics.  Specifically, I wanted to talk about solving linear equations using multiplicative inverses as a preview of determinants and using inverse matrices for solving linear systems.  For instance:

$latex begin{array}{ll} 2x=8 & AX=B \ 2^{-1}2x = 2^{-1}8 & A^{-1}AX = A^{-1}B \ 1x = frac{1}{2}8 & IX = A^{-1}B \ x=4 & X = A^{-1}B end{array}&s=2$

As an aside, I threw out this series of equations in the hopes of (a) foreshadowing singular matrices, and (b) offering a justification for the lifelong prohibition against dividing by zero:

$latex begin{array}{l} 0x=1 \ 0^{-1}0x = 0^{-1}1 \ 1x = frac{1}{0}1 \ x = frac{1}{0} end{array}&s=2$

I thought this was just so beautiful.  Why can't we divide by zero?  Because zero doesn't have a multiplicative inverse.  There is no solution to 0x = 1, so 0-1 must not exist!  Q.E.D.

As it turns out, Q.E.NOT.  One of my students said, "Why can't we just invent the inverse of zero?  Like we did with i?"

Again, we had just finished our discussion of polynomials, during which we had conjured the square root of -1 seemingly out of the clear blue sky.  They wanted to do the same thing with 1/0.  What an insightful and beautiful idea!  Consider the following stories, from my students' perspectives:

1. When we're trying to solve quadratic equations, we might happen to run into something like x2 = -1.  Now of course there is no real number whose square is -1, so for convenience let's just name this creature i (the square root of -1), and put it to good use immediately.
2. When we're trying to solve linear equations, we might happen to run into something like 0x = 1.  Now of course there is no real number that, when multiplied by 0, yields 1, so for convenience let's just name this creature j (the multiplicative inverse of 0), and put it to good use immediately.

Why are we allowed to do the first thing, but not the second?  Why do we spend a whole chapter talking about the first thing, and an entire lifetime in contortions to avoid the second?  Both creatures were created, more or less on the spot, to patch up shortcomings in the real numbers.  What's the difference?

And this is the tricky part: how do I explain it within the confines of a high school algebra class?  Well, I can tell you what I tried to do...

Let's suppose that j is a legitimate mathematical entity in good standing with its peers, just like i.  Since we've defined j as the number that makes 0j = 1 true, it follows that 0 = 1/j.  Consider the following facts:

$latex begin{array}{l} 2 cdot 0 = 0 \ 2frac{1}{j} = frac{1}{j} \ frac{2}{j} = frac{1}{j} \ 2 = 1 end{array}&s=2$

In other words, I can pretty quickly show why j allows us to prove nonsensical results that lead to the dissolution of mathematics and perhaps the universe in general.  After all, if I'm allowed to prove that 2 = 1, then we can pretty much call the whole thing off.  What I can't show, at least with my current pedagogical knowledge, is why i doesn't lead to similar contradictions.

Therein lies the broad problem with proof.  It's difficult.  If there are low-hanging fruit on the counterexample tree, then I can falsify bad ideas right before my students' very eyes.  But if there are no counterexamples, then it becomes incredibly tough.  It's easy to show a contradiction, much harder to show an absence of contradiction.  I can certainly take my kids through confirming examples of why i is helpful and useful.  But in my 50 min/day with them, there's just no way I can organize a tour through the whole scope and beauty of complex numbers.  Let's be serious, there's no way that I can even individually appreciate their scope and beauty.

The complex numbers aren't just a set, or a group.  They're not even just a field.  They form an algebra (so do matrices, which brings a nice symmetry to this discussion), and algebras are strange and mysterious beings indeed.  I could spend the rest of my life learning why i leads to a rich and self-consistent system, so how am I supposed to give a satisfactory explanation?

Take it on faith, kids.  Good enough?

Update 3/20/12: My friend, Frank Romascavage, who is currently a graduate student in math at Bryn Mawr College (right down the road from my alma mater Villanova), pointed out the following on Facebook:

"We need to escape integral domains first so that we can have zero divisors!  Zero divisors give a quasi-invertibility condition (with respect to multiplication) on 0.  They aren't really true inverses, but they are somewhat close!  In $latex Z_{6}$ we have two zero divisors, 3 and 2, because 3 times 2 (as well as 2 times 3) in $latex Z_{6}$ is 0."

In many important ways, an integral domain is a generalization of the integers, which is why they behave very much the same.  An integral domain is just a commutative ring (usually assumed to have a unity), with no zero divisors.  If there are two members of a ring, say a and b, then they are said to be zero divisors if ab = 0.  In other words, to "escape integral domains," is to move into a ring where the Zero Product Property no longer holds.  This means that, in non-integral domains, we can almost, sort of, a little bit, divide by zero.  Zero doesn't really have a true inverse, but it's close.  Frank's example is the numbers 2 and 3 in the ring of integers modulo 6, since 3 x 2 = 0 (mod 6).  In fact, the ring of integers modulo n fails to be an integral domain in general, unless n is prime.  CTL

# 0!rganized Emptiness

On the back of the fundamental counting principle, my class has just established the fact that we can use n! to count the number of possible arrangements of n unique objects.  This is fantastic, but we don't always want to arrange all of the n things available to us, which is okay.  We've also been introduced to the permutation function, which has the very nice property of counting ordered arrangements of r-sized subsets of our n objects.  Handy indeed.

Today we made an interesting observation: we now have not one, but two ways to count arrangements of, let's say, 7 objects.

1. We can fall back on our old friend, the factorial, and compute 7!
2. We can use our new friend, the permutation function, and compute $latex bf{_7P_7}$

Since both expressions count the same thing, they ought to be equal, but then we run into this interesting tidbit when we evaluate (2):

$latex _7P_7 = frac{7!}{(7-7)!} = frac{7!}{0!}&s=2$,

which seems to imply that 0! = 1.  To say this is counterintuitive for my kids would be a severe understatement.  And in this moment of philosophical crisis, when the book might present itself as a palliative ally, students are instead met with this:

To prevent inconsistency?  How in the world are kids supposed to trust a mathematical resource that paints itself into a corner, only tacitly admits such, and then drops a bomb of a deus ex machina in order to save face?  I haven't been so angry since the ending of Lord of the Flies.  Especially when this problem appears two pages later:

Okay, 8!.  So how many ways can I arrange my bookshelf with a zero-volume reference set?  One: I can arrange an empty shelf in exactly one way.  And, since we already know that n! counts the ways I can arrange n objects, it follows naturally that this 1 way of arranging 0 things must also be represented by 0!.

There are a lot of good proofs/justifications available for the willing Googler, but this one, to me, seems like the most natural and straightforward for a high school classroom.  At a bare minimum, it's much, much better than, "Because I need it to be true for my own convenience."

Only a math textbook could take something so lovely and make it seem dirty.