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DMS Graduation 1996:
An Experiment with Probablility

This is my talk for the Dunn Middle School graduation ceremony on 31 May 1996, delivered outdoors on the steps of the Middle School. My son Ryan was one of the graduating 8th graders, and my Mom and Dad were in the audience! I explain how to do it at the end.

It's been my custom to do a science experiment/demonstration for my graduation talk. I've been doing this for 13 years, but it's a challenge each time. The experiment has to work outdoors and be visible to a large group. It can't be too long or complicated. And it must generate a metaphor for the occasion. Most years I begin with a cool experiment and reach for a metaphor. But this year I had an idea to talk about, and I wasn't sure what to do with it. Katie settled it a few days ago. "Just make something explode!"

So here goes. This is a can full of natural gas (methane, CH4) that Bruce "Doc" Macomber at Dunn High School filled for me early this morning, since I don't have a gas line here at the Middle School. I'm going to blow it up. Note there are two holes in the can, one in the lid and one near the bottom. This was necessary to fill the can from a gas jet. I'm a little worried because the can has been sitting for a few hours, and I hope the gas hasn't all leaked out.

Doc hooked up a gas line to the bottom hole and let it go until gas came out the top, and then taped it shut. One nice thing about having Doc help with this is that I can blame him if it doesn't work and take credit if it does work! (I'll come back to this idea later.) Doc is a generation past me, and I notice that he used masking tape to cover the holes. Doc is part of the masking tape generation, while I am a duct tape man. So I can blame Doc, and Doc can blame the masking tape. We science teachers can always blame something!

I do have a backup experiment in case this doesn't work.

{Light the can. It should flicker and go out and nothing seems to happen. Walk away disappointed. It SHOULD go off unexpectedly after a few minutes while I'm talking about something else, and pop the lid into the air with a bang. Then talk about the bang and return to the talk.}

I apologize. The can must have lost gas while sitting. Katie, we'll have to blow something up later.

For my backup, I will use one of the most important pieces of scientific apparatus, a roll of duct tape.. Here's my experiment, not very dramatic. I'm going to toss the roll of duct tape and see where it lands.

{Toss it so it rolls and bounces a bit.}

It's actually pretty complicated. It falls, hits, rolls, flops a few times this way and that, and stops. Now here's my question: what is the probability that the duct tape will end THERE?

Probability is a simple mathematical idea that we teach at the Middle School. We usually learn it with things like dice, which can only fall in a few different ways. Probability is defined as the ratio of favorable outcomes to total outcomes. It's the number of ways we want something to happen divided by the number of ways it can happen.

{Hold up a large kids' plastic block: mine has two ruffled sides (like a potato chip) and four other sides with pictures and letters.}

So what's the probability of throwing this block and getting a ruffled side up? That's two favorable outcomes out of six total, so the probability is two out of six, or one-third. Simple.

Now what about the roll of duct tape? What's the probability of it landing there? Offhand, there's only one way to land exactly there. But there's infinitely many other places it can land, since there's always another place between any two points in space, at least if we avoid quantum ideas of space. So the probability of it landing exactly there is roughly one divided by infinity. And what the heck is that?

I don't have an infinity button on my calculator, so we have to get to the answer in a roundabout way. One divided by one is one. One divided by ten is only one-tenth. One divided by one thousand is just one-thousandth. The bigger the number we divide by, the smaller the result. So what is one divided by infinity, the biggest number of all? It must be the smallest number of all, which is zero. In math we can say that the limit of 1/n, as n approaches infinity, is 0. The term vanishes.

So what is the probability of throwing the tape and having it end exactly there? The probability is 1 over infinity, which is zero. It's impossible. It can't happen. And yet, there it is.

This is a paradox. The word "paradox" comes from two Greek words, "para" which means next to or besides, and "dox" which means belief or thought. So a proper paradox is when we have two beliefs at the same time. It's not possible and There it is, both in the same breath. We have to feel some tension.

A lot of paradoxes just seem silly, but this one has at least one foot in reality. My wife Julie and I were involved in a minor chain-reaction fender-bender accident on the freeway a few years ago. No one was hurt, but our car got dinged pretty bad. Afterwards we thought: if we hadn't just made that last yellow light; if we hadn't passed that slow truck; if I hadn't adjusted my seatbelt; if we hadn't looked for that Grateful Dead tape under the seat...A million ifs!...then we wouldn't have been right there at that exact moment to be caught in that accident. Seen as the final step in this long series of trivial events, it seems impossible that all those events could cooperate to have us there at that particular moment.

I think we often have these thoughts when bad things happen. One of Treebeard's Rules that I often tell the kids is: Take credit when it works, blame something (anything!) when it fails. (This is especially handy with computers!)

But I think the principle holds for good things as well. The point is that complex events in the real world can and do seem improbable to the point of being impossible. And yet, there it is. That is the paradox.

As a philosopher and math/science person, there's all kinds of issues here about freedom and determinism, cause and effect, infinity and chaos. But we're here for a graduation, to celebrate these kids sitting here today about to leave the Middle School for good. Not to argue philosophy.

What's the probability of all of us being here today to celebrate this occasion? How many separate events and decisions over the years does this moment depend on? That's easy. Probability zero. Absolutely impossible. And yet here we are. This is worth celebrating!

You graduates, here's some advice from three disparate sources, all quotes.

On Tuesday we had a full morning of "standardized testing," and we watched the Hitchhiker's Guide to the Galaxy video in the afternoon to relax. I was thinking about this speech, not paying much attention, when I heard the words "Don't Panic." Good advice!

Second, from Alice in Wonderland (actually the second book, Through the Looking Glass), Alice says to the White Queen, "One can't believe impossible things." And the Queen answers, "I dare say you haven't had much practice. When I was your age I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." Do it!

Last of all, the school sign I see every day as I pull into school, "Carpe Diem," seize the day, take the opportunity when it comes, go for it. It can make a huge difference in where you go from here, possible or not!

{End of speech.}

{Jump here when the can goes off, hopefully not too soon. Say what's appropriate. Then try to recover gracefully back to the main text.}

The question isn't why did the can go bang. After all, it's full of gas! But why did it wait? Three years ago we studied fire in Science class and learned about the fire triangle. To have combustion, you must have fuel, air, and heat all at the same time. At first the can is just fuel, no air. It burns at the very top because there is air there. But as it burns, air is pulled in at the bottom and mixes with the gas. At some point there's enough air to make the mixture explosive, and bang! I'm impressed that it happens so suddenly. I figure that when it starts to burn fast, it also sucks in air fast. This part is self-regulating.

It's interesting that air leaks as well as gas leaks can be dangerous. (This is what the movie Backdraft was about.)

I tried this same experiment a few years ago for a graduation experiment, and it didn't work. But I live up in the mountains and I have propane not natural gas. Propane is more dense, and it just runs out the bottom hole, so there's no mixing.

Now I can ask my question again. What is the probability of the lid landing exactly where it does? Of course the answer is zero. But here the chain of events is even more complicated than with the duct tape since it depends on such things as random air currents and diffusion rates.

Researchers in the fields of chaos theory and complex dynamics talk half-seriously about the Butterfly Effect. That a butterfly fluttering by in Mexico City last winter could set up tiny air currents that could lead to a thunderstorm here this summer. In our case at least, a butterfly, or any one of us moving about, could effect air currents that could effect the exact moment when the gas-air mix becomes explosive, and so could effect where the lid finally lands.

That's a lot of "could"s, but that's the point. With infinite possibilities, the probability of any one of them must be zero!

{Return to main text.}

How to do it:

What I used for the speech:

To make the exploding can, you need a strong metal can with a friction lid that's airtight but not too tight. A gallon paint can will do. (I'd like to try a BIG can!)

In the center of the lid, drill or punch a hole about 1/4 inch in diameter. Near the bottom of the can on the side, make another hole that fits a rubber hose from your gas jet. This hole should probably be as small as possible to delay the gas mixing. Press the friction lid in firmly. Place the hose from the gas jet in the lower hole, and let it run until the can is full and you can smell gas coming out the top. Then disconnect the hose and cover the holes with duct tape (NOT masking tape, sorry Doc!) to hold the gas. Also tape around the lid.

When it's time, remove all the tape and light the gas at the top hole. Step back and wait for results. Do not go near the can even though the flame seems to have gone out! But keep up the talk and build suspense or surprise. When I did it, the lid flew about 4 feet in the air with a good flame, so warn anyone close by.

When I did this for graduation, it actually didn't work as planned, since it went off almost immediately. My speech was towards the end of the ceremony, and I figure that the can lost gas and gained air through the masking tape while sitting. Really seal that can if it has to sit!

You must use natural gas for this, not propane. Propane is dense enough to leak out the bottom. It doesn't work.

This experiment is described in the classic UNESCO book 700 Science Experiments For Everyone, Doubleday & Company, 1958/1962. It's in Chapter XIII, Experiments with Heat, near the end.

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