Monthly Archives: November 2011

Flipping Out

While we’re on the subject of statistics-related classroom activities with a “wow factor,” let me bring up my favorite: guessing whether a sequence of coin flips is real or fake.

BS detector

For me, it really brought home the idea that math is an amazing BS detector. Sure, we tell kids to learn math so you can balance your checkbook, figure out the tip at a restaurant, blah blah blah. But consider these very reasonable counterarguments: (1) Yawn, and (2) Calculators/computers do all that for us anyway.

So you have to fire back: you wanna get screwed over? When you sign up for student loans at a terrible rate because the loan officer was friendly and you couldn’t even guesstimate the math in your head, you’ll be stuck with awful payments for the next 10 years. When your phone company advertises “.002 cents per kilobyte” but charges you .002 dollars per kilobyte instead, a hundred times as much, you should call them out on it.

You may never have the luck to acquire a superhero spider sense, but we mortals can certainly hone our number sense. People will try to con you over the years, but if you keep this tool called “math” in your utility belt I guarantee it’ll save your butt a few times down the line.

Coin trick

Anyway, the coin flip thing itself may be more of a cute demo than directly practical — but it’s really really cute. Watch:
You split the class into two groups. One is going to flip a coin 100 times in a row and write down the resulting sequence of heads and tails. The other is going to pretend they did this and write down a made-up “random” sequence of heads and tails. The teacher leaves the room until both groups are done, then comes back in and has to guess which sequence came from real coin flips and which is the fake. And BAM, like magic, no calculation required, the teacher’s finely-honed number-sense makes it clear which is which.
Can you tell from the pair below?
(example copied from Gelman and Nolan, 2002, Teaching Statistics)

Enterprising statisticians have noticed that, in a sequence of 100 truly random coin flips, there’s a high probability of at least one “long” streak of six or more heads in a row (and same for tails). Meanwhile, people faking the data usually think that long streaks don’t look “random” enough. So the fake sequence will usually switch back and forth from heads to tails and back after only 2 or 3 of each, while the real sequence will have a few long streaks of 5 or 6 or more heads (or tails) in a row.

So is your number sense tingling yet? In the example above, the sequence on the left is real while the right-hand data was faked.
(I’m not sure where this demo originates. I first heard of it in Natalie Angier’s 2007 book The Canon, but it’s also described in Gelman and Nolan’s 2002 book Teaching Statistics mentioned above, and in Ted Hill’s 1999 Chance magazine article “The Difficulty of Faking Data”. Hill’s article is worth a read and goes into more detail on another useful statistical BS detector, Benford’s Law, that can detect patterns of fraudulent tax data!)

So what?

Lesson learned: randomness may look non-random, and vice versa, to the untrained eye. Sure, this is a toy example, but let’s generalize a bit. First, here we have random data generated in one dimension, time. This shows that long winning or losing streaks can happen by pure chance, far more often than most people expect. Say the sports team you manage has been on a winning (or losing) streak — does that mean the new star player is a real catch (or dud)? Maybe not; it might be a coincidence, unless the streak keeps running much longer than you’d expect due to chance… and statisticians can help you calibrate that sense of just how long to expect it.

Or imagine random data generated in two dimensions, spatial data, like mapping disease incidence on a grid of city blocks. Whereas before we had winning/losing streaks over time, now we’ll have clusters in space. We don’t know where they’ll be but we are sure there’s going to be some clustering somewhere. So if neighborhood A seems to have a higher cancer rate than neighborhood B, is there a local environmental factor in ‘hood A that might be causing it? Or is it just a fluke, to be expected, since some part of town will have the highest rates even if everyone is equally at risk? This is a seriously hard problem and can make a big difference in the way you tackle public health issues. If we cordon off area A, will we be saving lives or just wasting time and effort? Statisticians can tell, better than the untrained eye, whether the cluster is too intense to be a fluke.

It’s hard to make good decisions without knowing what’s a meaningful pattern and what’s just a coincidence. Statistics is a crazy powerful tool for figuring this out — almost magical, as the coin flip demo shows.

Spinner Prescription

In the last post I described a problem with Dan Meyer’s otherwise excellent expected-values teaching tool: you’d like to wow the kids by correctly predicting the answer a month in advance, but the given setup is actually too variable to let you make a safe prediction.

Essentially, if you’re saying “Let’s do a magic trick to get kids engaged in this topic,” but the trick takes a month to run AND only has a 30% chance of working… then why not tweak the trick to be more reliable?

spin it many more times?

Part of this unreliability comes from the low number of spins — about 20 spins total, if you do it once every weekday for a month. The “easy” fix is to spin Dan’s spinner many more times … but it turns out you’d have to spin it about 4000 times to be 90% confident your prediction will be right. Even if you have the patience, that’s a lot of individual spins to track in your spreadsheet or wherever.

use a MORE reliable spinner?

Another fix might be to change the spinner so that it works reliably given only 20 spins. First, we don’t want any of the sectors too small, else we might not hit them at all during our 20 spins, and then it becomes unpredictable. It turns out the smallest sector has to be at least about 1/9th of the spinner if you want to be 90% confident of hitting that sector at least once in those 20 spins.
(Let y \sim \mathrm{Binomial}(p=1/9, n=20) . Then p(y<0) = 1-p(y=0) = 0.905.)
If we round that up to 1/8th instead, we can easily use a Twister spinner (which has 16 equal sectors).
After playing with some different options, using the same simulation approach as the previous post, I found that the following spinner seems to work decently: 1/2 chance of $100, 3/8 chance of $150, and 1/8 chance of $1500. After 20 spins of this spinner, there’s about a 87% chance that the “$1500″ will have been the winning bet, so you can be pretty confident about making the right bet a month in advance.

Unfortunately, predicting that spinner correctly is kind of unimpressive. The “$1500″ is a fairly big slice, so it doesn’t look too risky.

spin just a few more times and use a safer spinner!

What if we spin it just a few more than 20 times — say 60 times, so two or three times each day? That’s not too much data to keep track of. Will that let us shrink the smallest slice, while keeping predictability high, and thus making this all more impressive?

Turns out that if we know we’ll have about 60 spins, we can make the smallest slice 1/25th of the spinner and still be confident we’ll hit it at least once. Cool. If we want to keep the Twister board, and have the smallest slice be 1/16th of the circle, we actually have a 90% chance of hitting it at least twice. So that makes things even more predictable (for the teacher), while still making it less predictable (to the kids) than the previous spinner.

More messing around led to this suggested spinner: 1/2 chance of $100, 5/16 chance of $200, 1/8 chance of $400, and 1/16 chance of $2500. The chance that “$2500″ is the right bet after 60 spins of this spinner is about 88%, so again you can make your bet confidently — but this time, the “right” answer doesn’t look as obvious.

In short, I’d recommend using this spinner for around 60 spins, rather than Dan’s spinner for 20 spins. It’s not guaranteed to be “optimal” but it’s far more reliable than the original suggestion.
If anyone tries it, I’d be curious to hear how it went!

Spinner Doctor

The setup

Dan Meyer, a (former?) math teacher with some extraordinary ideas, has a nifty concept for teaching expected values:

“So one month before our formal discussion of expected value, I’d print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I’d seal my own bet in an envelope.

I’d ask a new student to spin it every day for a month. We’d tally up the cash at the end of the month as the introduction to our discussion of expected value.
So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?”

I absolutely love the idea of warming up their brains to this idea a month before you actually teach it, and getting them “hooked” by placing a bet and watching it play out over time.

The Challenge

But there’s a problem: at least as presented, the intended lesson isn’t quite true. I’m taking it as a challenge to see if we can fix it without killing the wow-factor. Let’s try.

As I read it, the intended lesson here is: “if you’re playing the same betting game repeatedly, it’s good to bet on the option with the highest expected value.”
And the intended wow-factor comes from: “none of the options looked like an obvious winner to me, but my teacher knew which one would win!”

But the lesson just isn’t true with this spinner and time-frame: here, the highest-expected-value choice is actually NOT the one most likely to have earned the most money after only 20 or 30 spins.
And the wow-factor is not guaranteed: none of the choices is much more likely to win than the others in only 20-30 spins, so the teacher can’t know the winning bet in advance. It’s like you’re a magician doing a card trick that only works a third of the time. You can still have a good discussion about the math, but it’s just not as cool.

I’d like to re-design the spinner so that the lesson is true, and the wow-factor still happens, after only a month of spins.

WAit, is there really a problem?

First, what’s wrong with the spinner? By my eyeball, the expected values per spin are $100/2 = $50; $300/3 = $100; $600/9 = $67ish; $5000/27 = $185ish; and $12000/54 = $222ish. So in the LONG run, if you spin this spinner a million times, the “$12000″ has the highest expected value and is almost surely the best bet. No question.

But in Dan’s suspense-building setup, you only spin once a day for a month, for a total of 20ish spins (since weekends are out). With only 20 spins, the results are too unpredictable with the given spinner — none of the five choices is especially likely to be the winner.

How do we know? Instead of thinking “the action is spinning the spinner once, and we’re going to do this action twenty times,” let’s look at it another way: “the action is spinning the spinner twenty times in a row, and we’re going to do this action once.” That’s what really matters to the classroom teacher running this exercise: you get one shot to confidently place my bet at the start of the month; after a single month of daily spins, will the kids be wowed by seeing that you placed the right bet?

I ran a simulation in R (though sometime I’d like to tackle this analytically too):
Take 20 random draws from a multinomial distribution with the same probabilities as Dan’s spinner.
Multiply the results by the values of each bet.

> nr.spins <- 20
> spins=rmultinom(1,size=nr.spins,prob=c(1/2,1/3,1/9,1/27,1/54))
> spins
     [,1]
[1,]   11
[2,]    7
[3,]    2
[4,]    0
[5,]    0
> winnings=spins*c(100,300,600,5000,12000)
> winnings
     [,1]
[1,] 1100
[2,] 2100
[3,] 1200
[4,]    0
[5,]    0

For example, in this case we happened not to hit the “$5000″ or the “$12000″ at all. But we hit “$100″ 11 times, “$300″ 7 times, and “$600″ twice, so someone who bet on “$300″ would have won the most money that month.
Now, this was just for one month. Try it again for another month:

> spins
     [,1]
[1,]    8
[2,]    9
[3,]    1
[4,]    2
[5,]    0
> winnings
      [,1]
[1,]   800
[2,]  2700
[3,]   600
[4,] 10000
[5,]     0

This time we got “$5000″ twice and whoever bet on that would have been the winner.
Okay, there’s clearly some variability as to who wins when you draw a new set of 20 spins. We want to know how variable this is.
So let’s do this many times — like a million times — and each time you do it, see which bet won that month. Keep track of how often each bet wins (and ties too, why not).

nr.sims=1000000
bestpick <- rep(0,5)
tiedpick <- rep(0,5)
nr.spins <- 20
for(i in 1:nr.sims){
    spins=rmultinom(1,size=nr.spins,prob=c(1/2,1/3,1/9,1/27,1/54))
    winnings=spins*c(100,300,600,5000,12000)
    best <- which(winnings==max(winnings))
    if(length(best)==1){
        bestpick[best] <- bestpick[best]+1
    } else{
        tiedpick[best] <- tiedpick[best]+1
    }
}

Results are as follows. The first number under bestpick is the rough proportion of times that “$100″ would win; the last number is the rough proportion of times that “$12000″ would win. Similarly for proportion of ties under tiedpick, except that I haven’t corrected for double-counting (since ties are rare enough not to affect our conclusions).

> bestpick/nr.sims
[1] 0.0145 0.2124 0.0712 0.3780 0.3029
> tiedpick/nr.sims
[1] 0.00199 0.02093 0.01893 0.00000 0.0000

(Ties, and the fact it’s just a simulation, mean these probabilities aren’t exactly right… but they’re within a few percentage points of their long-run value.)
It turns out that the fourth choice, “$5000″, wins a little under 40% of the time. The highest-expected-value choice, “$12000″, only wins about 30% of the time. And “$300″ turns out to be the winning bet about 20% of the time.
Unless I’ve made a mistake somewhere, this shows that using Dan’s spinner for one spin a day, 20 days in a row, (1) the most likely winner is not the choice with the highest expected value, and (2) the teacher can’t know which choice will be the winner — it’s too uncertain. So the lesson is wrong, and you can’t guarantee the wow-factor. That’s a shame.

dang. What to do, then?

Well, you can try spinning it more than once a day. What if you spin it 10 times a day, for a total of 200 spins? If we re-run the simulation above using nr.spins <- 200 here’s what we get:

> bestpick/nr.sims
[1] 0.000000 0.012258 0.000287 0.393095 0.589246
> tiedpick/nr.sims
[1] 0.000000 0.000332 0.000037 0.004780 0.005079

So it’s better, in that “$12000″ really is the best choice… but it still has only about a 60% chance of winning. I’d prefer something closer to 90% for the sake of the wow-factor.
What if you have each kid spin it 10 times each day? Say 20 kids in the class, times 10 spins per kid, times 20 days, so 4000 spins by the month’s end:

> bestpick/nr.sims
[1] 0.000 0.000 0.000 0.106 0.892
> tiedpick/nr.sims
[1] 0.00000 0.00000 0.00000 0.00157 0.00157

That’s much better. But that’s a lot of spins to do by hand, and to keep track of…
Of course you could run a simulation on your computer, but I assume that’s nowhere near as convincing to the students.

What I’d really like to see is a spinner that gives more consistent results, so that you can be pretty sure after only 20 or 30 spins it’ll usually give the same winner. A simple example would be a spinner with only these 3 options: 1/2 chance of $100, 1/3 chance of $300, and 1/6 chance of $400.

> bestpick/nr.sims
[1] 0.0574 0.6977 0.2371
> tiedpick/nr.sims
[1] 0.00200 0.00783 0.00596

That’s okay, but there’s still only about a 70% chance of the highest-expected-value (“$300″ here) being the winner after 20 spins… and anyway it’s much easier to guess “correctly” here, no math required, so it’s not as impressive if the teacher does guess right.

Hmmm. Gotta think a bit harder about whether it’s possible to construct a spinner that’s both (1) predictable and (2) non-obvious, given only 20 or so spins. Let me know if you have any thoughts.

Edit: I propose a better solution in the next post.