*The last in a series of seven fables/lessons/meditations on probability.*

The teacher kept a garden, a grove of cypress and cedar with a pond at the center.

“You will build me a garden wall, alternating white stones and black,” the teacher told the student. “It must achieve the randomness and beauty of nature. I want to look at my wall and see a reflection of the cosmos.”

“How am I supposed to do that?” the student asked.

“Before you place each rock, flip a coin,” the teacher said. “If it comes up heads, place a white rock; if tails, place a black rock. That way, the sequence will be truly random, like nature.”

The teacher left, and the student began placing stones. But before long, flipping the coin grew tedious. Couldn’t she just pick a random color herself? The teacher would never know the difference. By the end of the day, she’d ringed half the garden with stones—white, black, black, white—and her fingers were cut and blistered.

The returning teacher glanced briefly at the wall. “The stones do not look random. You didn’t flip the coin.”

“How can you tell?” the student asked.

“The mind is full of false patterns,” the teacher said, “and this is what I see when I look at this wall. I do not see the random beauty of nature. I see the strained grunts and groans of a pattern-hunting mind.” The teacher shook her head. “Try the other half of the garden tomorrow. And this time, give me true randomness.”

When the student returned to work the following day, she obeyed the teacher’s instructions, and flipped a coin for each of the 500 stones. The teacher came home that evening and nodded. “Better work today.”

“What’s the difference?” the student asked.

“Look at your first day, the false randomness,” the teacher asked. “What is the longest streak of black stones that you lay, uninterrupted by white?”

“Four in a row.”

“Now look at the wall of true randomness.”

The student looked, and found long stretches of stones all the same color. One point had ten blacks in a row. By comparison, the first day of work looked quite regular, in its alternation between white and black.

“The truly random wall is full of streaks,” the student said. “But why is that?”

“Think,” the teacher said. “What is the probability of a streak of ten in a row?”

“About 1 in 500.” The student’s eyes widened. “And I laid 500 stones each day. So you’d *expect* to have a streak of ten somewhere in that sequence.”

“Yes,” the teacher said. “Your mind reads great meaning into streaks, and little meaning into their absence. But in true randomness, streaks are inevitable.”

“What I thought looked random, was actually full of patterns. The pattern was a lack of streaks,” the student said, pointing to the first day’s wall. “And what is actually random appears full of patterns, if you believe that streaks have meaning.”

The teacher nodded. Then they sat together for a while, admiring the clusters of cedar and cypress, and the gentle ripples in the water.

**Further Thoughts**

This fable echoes that of The Wise Monkey. Random processes (like flipping a coin) inevitably produce streaks and clusters that our minds interpret as meaningful patterns.

We struggle with randomness from both sides. As discussed earlier, we read significance into patterns that lack any interesting cause. Moreover, when we try to fake randomness, we hesitate to include coincidences and long streaks, so we create things that are too conspicuously even and balanced. So, paradoxically, in randomness we see patterns, and in patterns we see randomness.

This idea’s applications extend beyond debunking false prognosticators—fortune telling, superstition, mutual funds, and the like. In its extreme version, this idea suggests that when we pour effort into explaining things, we’re often wasting our time.

“Amazing—that NBA player just hit 8 three-pointers in a row! What’s gotten into him?” Maybe nothing. Based on his career percentage you’d expect him to do that once or twice.

“This company’s profits are soaring! What a genius CEO, right?” Well, with enough companies trying different strategies, some are bound to thrive by luck alone—not necessarily because of brilliant foresight.

“This school’s test scores are phenomenal! What lessons can we draw?” Perhaps none—if talented teachers were distributed randomly across the country, a few lucky schools would receive a share far bigger than average, and we’d expect their students to excel.

We don’t want to overextend this notion. It’s not that causal explanations are never valid or valuable. But when you’re seeking to explain some anomalous success or failure, it’s worth asking—in a completely random world, wouldn’t we expect a few outliers? Isn’t it possible that this remarkable result comes—at least in part—from blind chance?

*Get the pdf, or check out other stories in the series!*

*I’d like to thank my father, James Orlin, for providing some foundational ideas for these stories, as well as helpful feedback and conversations. Also for being one wise soul.*

I love these fables! They make me (an old math-phobe) want to study math this way forever. What’s the best way to contact you off list? I’m frequently in Oakland (like now) from NYC, would like to connect re MotivationEquation.org

Just send me an email! Ben Orlin at gmail.

Another good one.

Nosey Practical Advice: I think that there could be more brick wall drawings that would illustrate the point a little better, without making things any more technical. Let’s say a great big wall, 30 bricks wide by 20 bricks high, that the teacher zooms in on and shows runs of 5 or 10 blocks of the same color, but the teacher cannot find any such runs in the student wall.

I guess that my scientific nature does not allow me too see randomess, or a lack thereof, in a sample of 16. 🙂

That’s good advice. I may go back and reillustrate if I get the chance. As it is, you sort of have to take my word on which is random and which is nonrandom (although I was totally honest, I swear! 😉 ).

Yes we tend to see patterns in true randomness. But the argument that we “expect to have a streak of ten somewhere in that sequence” of 500 is not very clear. What does “expect” mean in this context? Is it the expected number of streaks of length 10 (or more)? Is ten the expected length of the longest streak? Or perhaps we just mean that it is likely (in some sense not precisely given) that we will see a streak of 10. (or of 10 or more.)

The math that is relevant for determining what to “expect” is not immediately obvious, at least to me. The chances of getting a streak of 10 is about 1 in 500 (or more precisely 2 out of 1024) only if we only lay exactly 10 stones. If we are laying 500 then there are (almost) 500 places where such a streak could start, assuming we haven’t seen one already. But it isn’t obvious that you can just multiple 491 times 2/1024 to approximate the probability of observing a streak of 10 or more, because this approach does not adequately partition the space of possible outcomes into disjoint events. For example, a sequence that starts BBWWWBBBB may or may not include the start of the first string of 10 or more black stones.

The distribution of the length of the longest streak is analyzed in an interesting paper by Mark Schilling that you can find here: http://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020742.02p0021g.pdf

It derives the exact distribution using an easy to understand recursive expression that is fairly easy to implement in a spreadsheet for modest numbers of stones. But his method rapidly runs into computational overflow problems otherwise, because 2^n grows so rapidly. However using asymptotic arguments he shows that the expected length of the longest black streak is approximately log_2(n)-2/3, which for n=500 is about 8.3. Using symmetry, the length of the longest streak of either color must be greater by one, or 9.3. The asymptotic standard deviation of the distribution is also given in the paper. It is surprisingly small, and turns out to be independent of n. It is a bit smaller than 2. So 10 is clearly in the range that we would “expect” in that it is likely to be included in any reasonable confidence interval we might choose.

Wow, that is cool analysis, but I think if he tried to explain that in his drawings it might confuse things. The streak of 10 is more of an approximation I’d say, and a reasonable one at that.

But, I really do like your addition to it!

My thoughts exactly!

Ben

I recently blogged about this idea, and an experiment in class with pennies, at http://mrdardy.wordpress.com/2013/11/05/magic-in-stats-today/

It was less fun and inspiring than your zen master, I must admit.

Ben

Is there a one-stop shop for the PDFs, or should I grab them individually?

Thanks SO much for making them available.

Hey, I enjoyed your post! And yeah, the pdf (and solutions) are available at the bottom of this page:

https://mathwithbaddrawings.com/the-bear-in-the-moonlight/

Radiolab had a very interesting and similar broadcast about this topic: http://www.radiolab.org/story/91684-stochasticity/

That is one of my all time have episodes. The follow up – a short called ‘Are We Coins’ is one I assigned to my AP Stats class to discuss randomness.

Thanks, guys – I’m going to check this out today. I really need to start listening to Radiolab. The number of recommendations I’m getting for it is climbing exponentially.

My only quibble is a simple one: Why do teachers insist upon using primitive tools such as coins to generate random numbers? When the students are in landscaping architecture, do we demand they use stone tools instead of iron ones. The concept could just as easily be taught, and with much less resistance, with any number of software programs such as Excel, Numbers for Mac, and so forth. But no, there’s some sort of sadistic desire that students need to bloody their fingers and undergo excruciating boredom to learn something. Have you considered what they learn is that you are not a very good teacher as you lack the skills to inspire or develop creative solutions?

This is a fictional story!

Dice and coins are pedagogically useful random number generators because they’re easy to visualize and have simple sample spaces. But I’ve never heard of a real teacher pushing students to the point of “bloody fingers” or “excruciating boredom”!

Less fictional than you imagine. I read numerous lesson plans that use similar methodology. One of the reasons I dislike astronomy so much was this methodology was part of my secondary school education. And while the fingers were not bloody, (which was a exaggeration referring to stone tools) the boredom was excruciating.

Matlab and other high level computer tools were developed as a response to repetitive and low yield measurement (and educational) projects such as this one. (Performing flow cytometry in the 1990’s was an exercise in wakeful dreaming.) In Excel (or Google’s free open office) the student could be exposed to numerous concepts that are far easier to visualize than these primitive methods. RNG’s invariably create numbers 0…1, so multiplying the result by 10, 100, 1000 creates large heterogeneous data sets. Students could look for values other than odd/even such as greater/less than, between, or exactly equal to, and color code the results accordingly. I understand that some school districts may lack the resources so this would be difficult in-class assignment; however as homework done in the library or using a home computer is a trivial matter. I agree that foundations must be laid and that basic principles need to be understood; nonetheless, failing to use all the tools available is a major impediment to the learning environment.

BTW – blame Cocktail Party Physics for my intrusion.

No intrusion at all! You’re raising worthwhile points. I certainly agree that random number generators are a valuable tool for teaching students about randomness.

Just to be clear, I’d be very worried if anyone interpreted my story as a “lesson plan” or “educational activity”! It’s absolutey not. The whole point of writing it as a fictional story is that, as readers, we get to spend four minutes imagining this repetitive activity and its results, rather than four hours slogging through it. It’d be a horrible waste of time to have a student flip a coin 500 times!

For what it’s worth, I do think there’s pedagogic value in watching random processes (like coin flips and dice rolls) unfold. Not enough to waste 30 minutes on it, but 3 minutes once in a while? Sure. Probability and statistics involve a pretty high level of abstraction (the distribution of sample means, for example) and I’ve found that building on concrete, tactile experiences can make later abstractions more comfortable for students.

Sounds like your astro class really didn’t serve you well, in part because it used outdated tools and tedious activities. I hear that. A good teacher ought to be a master of classroom tools/resources. But I’d point out that it’s easy to teach a good probability lesson without Matlab, and also easy to teach a bad one with it.