Sunday, June 28

Let's back up to the coin-flipping...

Thinking about coin-flips has been (I'll admit it) a lifelong obsession of mine. For many reasons, most of which have to do with philosophical questions like: Is reality deterministic or is it all underpinned by chance? And the better question, what is chance?

In my last blog entry I mentioned all of humanity taking a moment out of their day (or night, as it would be to many of them) and flipping a fair coin 20 times. I mentioned that as startled as those people might be to whom it happened, some would inevitably flip 20 heads in a row, with no tails turning up. These people would then probably go run to buy a lotto ticket, head for the nearest church, mosque, or synagogue, or do something else equally silly. Why silly, you say? After all, what is the chance of flipping heads 20 times in a row (in a 'run' of only 20)?

The odds of getting 20 heads in a row would be the chance of getting 'heads' on each individual flip (0.5) to the power of the number of flips (in this case, 20) - this simple calculation being used since all flips are independent events and the 50/50 odds remain the same for each flip. If the flips were somehow linked probablistically to one another, we'd have to use a different methodology. Anyway, we get the simple calculation 0.5^20, which is 0.00000095367431640625, or about one chance in a million (actually 1 in 1,048,576).

Therefore if the 6,750,000,000 people alive right now each flipped 20 fair coins in a row, one time each, then stopped (we'll forget the fact people are being born and dying all the time and view this as a snapshot), you'd expect about 6,750 people to flip ALL HEADS, which would surely amaze them and all their friends. People are not distributed equally across the landmasses of the world, but if they were, what would be the average distance between these "lucky" all-heads-flippers? The land surface area of the earth (which of course is not all neatly contiguous either) is roughly 150,000,000 sq km, or 57,500,000 sq mi, so on average each owner of an all-heads set would occupy around 8500 square miles of land to themselves before bumping into another such person. But since this presumes them to be at the center of their "plot" and the figures are in square miles and not linear miles, we have to use geometry to get the distance between each actual person. If the plots are square (so that they tessellate nicely, though they really "should" be circular, but again never mind that: it scarcely changes the numbers, and we're simplifying a bit at each step as it is), then each side of each "lucky box" would be the square root of 8500, or about 92 miles long. We know by the Pythagorean theorem that the hypotenuse through such a square would be about 130 miles long. Since each person in this thought-experiment sits exactly in the middle of such a hypotenuse, and if the square plots are tessellated as mentioned (continuity of dry land be damned), then it means the average linear distance from one all-heads-flipper to another would be about 130 miles. Too far to shout, and probably outside that person's main circle of (nonvirtual) friends.

That's a bit of a bummer - it would indeed give rise to people thinking they'd been chosen by God or were supernaturally lucky or some such nonsense. On the other hand, since nothing is perfectly evenly distributed in a probabilistic outcome of an event like coin-flipping, we should ideally create a bell curve using the standard deviation for that event, and also (crucially) factor in the reality that people are not even close to being evenly distributed upon the earth's land surface, but are overwhelmingly clumped into urban areas which account for just a tiny fraction of that surface area. I'll leave the math to those interested, but it's safe to say that if this experiment WERE actually run at this moment, the odds of at least two of those people (who flipped twenty heads in a row without a single 'tails' coming up) being within shouting distance of one another, or at least sharing one good friend, are enormously favorable. It sounds crazy, but it would almost surely happen. In fact, if we ended up with no such cases, we should suspect divine tampering!

This kind of "luck," when you deal with sufficiently enormous numbers, leads to those seemingly miraculous things such as the same person getting struck by lightning twice, or winning the lottery twice, or indeed of me watching well over an hour's worth of Michael Jackson videos (which I DID, and which I had never previously done in my life) on YouTube less than 24 hours before he died. Obviously I had no foreknowledge of his impending death. Seems spooky, and it was a bit, but does it make be believe in God or that I was somehow moved by some divine force to queue up all those videos that night? No. It was bound to happen to someone (more than one person, most likely), and the fact that it happened to me is only "spooky" or remarkable to me, and probably not to anyone else (or at least not to rational people). If I'd read a story in a newspaper about the same thing happening to someone else in a far away country, I would have dismissed it as chance right away; therefore I must do the same thing in my case. Every seemingly remarkable event, if statistically probable, has to happen to someone, after all.

Bonus fun fact. It's estimated that 6.75% of all the people who have ever lived on Earth since humans became a distinct species (Homo sapiens), are alive right now. That means that in the last 2 million years, roughly 100 billion people have been born. 6.75 billion are living right now. The average human lifespan across those 2 million years, factoring in death by predation, war, and most significantly childbirth, has been something less than 18 years. Of course now it is much greater than that. Let's be generous and use today's global life expectancy, which is about 66 years of age (remember, this includes the developing world, and non-natural death, whatever that means). That means that in 2 million years, there have been about 30,300 lifespans (not to be confused with generations), strung end-to-end, as it were. We are living in only ONE (the latest) of these 30,300 lifespans. For almost 7% of the total population of a species (so far) to be living in 1/30,300 of the duration of that species (so far), is truly mind-blowing. If human population had held steady throughout the 2 million years, we would expect each lifespan to include only 50,000 people. In reality, for all but the last tiny fraction of human history, human population at any given time was probably less than 10,000 worldwide give or take a few. Certain events like the eruption of Toba may have cut that number to even less. Certainly until agricultural times, which only dawned about 10,000 years ago, humans, living as hunter-gatherers, would have subsisted at numbers which today would put us on the "Critically Endangered" list!

One last fun fact. How much fatter has the earth gotten as a result of this recent population explosion? Let's say the average person today weighs 115 pounds (accounting for children, the malnourished, etc, as well as all the fatties)... The addition of 6.75 billion people at that weight to the Earth would mean that the Earth has gotten about 775 billion pounds heavier, all things else being equal. Of course all other things aren't equal, as forests are cut down, animals are driven to extinction, buildings and freeways are erected, and so on. Besides, at a mass of around 10,000,000,000,000,000,000,000,000 (ten septillion) lbs, the earth could probably care less. But this all misses one major point: excluding incoming space debris, all of earth is a closed system, meaning that no mass is gained or lost, only transformed from one type of thing into another. If human beings have increased in number and in size, and if we have built tremendous cities, it has, of course, come at the expense of foodstuffs and building materials and that were already on the earth in the first place. So the planet has even less to worry about: It has not gained ANY weight (mass) from the explosion of humanity at all!


Anonymous said...

It's fascinating to think about our planet as a closed system, but what about the solar energy that continuously flows in, doesn't that give some mass influx in the lines of e=mc^2?

Metamatician said...

Yes, it does, as do cosmic rays and the odd neutrino. Very astute.

At some point I have to simplify though to keep my readership :)

Besides, I think when you consider how little remains is actually gained when EMR is converted to mass, you'll agree that infalling moon dust, mars dust, stardust, stray molecular gas particles, and of course meterors more than dwarf all the EMR, from the sun or otherwise.

Thanks for the contribution. Why anonymous btw?

Metamatician said...

take out the "remains" in the third paragraph above; it's late....

also, I do actually know how to spell meteors.


Metamatician said...

Hi again Anonymous. You're question has been bothering me for several days, so finally I decided to look up the influx of solar energy upon the earth. I found out how many watts of energy the sun put out per second, and how far the earth is on average from the sun. Then I knew how to calculate how much of that sunlight would hit the earth, but it turns out there are many complicating factors that make it extremely difficult to get an accurate figure. First, the sunlight is scattered by the atmosphere more and more as you increase the obliqueness of the angle - in other words, towards the poles. Secondly, there are a tremendous variety of surfaces on earth which the light can hit - this of the different in absorbing vs reflecting properties of asphalt, ice, trees, oceans, snow, and so on. You'd have to make a really good model of the surface of the earth and its albedo (reflectiveness), and it would depend on which exact side of the earth was facing the sun, though of course you could average it all out over time.

But then, if you talk about measuring over a long period of time, you get continents drifting around, glaciers growing and shrinking, volcanism and cyanobacteria literally changing the chemical composition of the atmosphere, the earth's magnetic field changing, and a dozen other factors, not to mention that in the distant past the sun was fainter than it is today, the earth retain some of its own primordial energy (and still generates it through radioactive decay internally), plus our day was once as short as 5-hours (the tidal friction with our huge moon as slowed that to 24 hours over the lifetime of the earth).

Then finally there's the problem that nearly all this solar influx is re-radiated back out into space, either by outright reflection, or by absorption and then re-emission (say, during the night). Otherwise, our planet would just constantly heat up and we'd end up a bit like Venus.

But you're right in that some radiation is taken in by plants and photosynthesized, which seems as though it must call for a conversion to new plant mass (combining sunlight, water, and CO2), though mostly it just acts as a catalyst and powers the operation. I could find any good figures on this. But when you think of E=MC^2, just a tiny amount of mass can be converted to a huge amount of radiant energy, which is why nuclear weapons work so well. So going in reverse, it would take an equally huge amount of converted sunlight to make any appreciable amount of matter...

I still think the total radiation from the sun over a given period of time which is not reflection or re-emitted into space, but which actually gets converted to matter, cannot possibly be very significant when it comes to adding mass to the earth, especially when you compare it to infalling space debris, which is actually pretty considerable.

I'll keep my eye out for more information on this though. Just don't ask me what effect starlight has!

billybytedoc said...

What is the chance of any one of 10 people who started reading through this mathematical marvel making it all the way through? I did.

billybytedoc said...

It is very rare that any energy is converted to mass on this earth. It takes a very large burst of very concentrated energy to create mass.

The energy reaching the earth is converted to heat and then radiated back into space OR reflected back into space initally.

The earth is in a steady state temperature wise. Global warming is an arguable theory, not a proven fact, in spite of what Gore says!

These days it is politically correct to parrot the theory, and we all want to be politically correct don't we???

Metamatician said...

Thanks for being one of the few (the only one?) who made it through!

I agree that the climate is an order of magnitude or two more complex than our models of it, meaning that, basically, we don't understand it at all. We know the rudiments, but cannot come close to modeling the extreme complexity, and unless quantum computing comes to the rescue, maybe never be able to make useful predictions beyond a very short time into the future.

Predicting earthquakes and volcanoes was one thought to be "just a matter of time" but has proved so hard that many seismologists and vulcanologists have just given up on the idea of prediction and more on quick detection and a much better alert system/network, to save lives.

And the earth's temperature and atmospheric chemical makeup have changed rather dramatically in the past and continent have drifted around, glaciers had extended and receding (at one point creating a planet nearly encased completely in ice), and major chain-volcanic eruptions (many supervolcanoes erupting simultaneously, like in Russia or the Deccan Flats in India) have pumped way more CO2 into the atmosphere in a decade than human beings have since the beginning of the Industrial Age.

And the climate always "recovered," not that many species didn't become extinct during these events.

So I think it's pretty arrogant to assume our science at this moment in time can make meaningful statements about any kind of hard fast "climate laws" which would be the minimum (plus lots of data) needed to create a good model. All the computing power in the world is useless if the model is flawed because of a lack of fundamental understanding. Garbage in, garbage out.

I keep an open mind. I'm ready to accept that human beings are playing a role in heating up the planet. I also think the Sun's irregular sunspot activity in the 20th century probably has even more to do with it.

But you're right; this has become a political argument and no longer a scientific one. Every scientist is in someone's pay and petroleum lobbyists and left-wing environmental extremists and everyone in between has a vested in interested in the policies which emerge and the money that follows them.

Metamatician said...

And yes, converting sunlight to matter would be a fusion reaction, so I suppose it's even more rare than I initially thought off the top of my head. Photosynthesis, for example, is probably entirely chemical rather than nuclear.

I'm afraid I now think the sun and other forms of radiation hitting the earth from space are contributing essentially zero mass to the earth, even over it's 4.5 billion lifetime. Answers that question!

billybytedoc said...

Agree Agree

Metamatician said...

Thinking on it a bit more...

If anything the earth has surely loss mass over the eons in the form of gasses in its upper atmospheric layers being lost to space, naturally as a byproduct of the ideal gas laws and also helped along by that solar wind which escaped deflection by the magnetic field. I'm sure it doesn't amount to a lot in the grand scheme of things, though.

Heat has also been lost as the primordial earth cooled, but again the amount of mass that all that heat is equivalent too according to E=MC^2 is probably very little, proportionate to all the mass left behind.

I'd have to sit and actually get some data on all this and do the math, but I have a hunch both of these factors can probably basically be ignored in terms of any big picture you'd like to form in your head of what's incoming and what's outgoing. I still maintain the earth is 99%+ a closed system.

Of that less that 1% of mass change, I'd wager pretty confidently that most of it is in the form of additional mass due to meteorite impacts and slow but steady accretion of 'space dust' that burns up in the atmosphere but rains gently and invisibly down as elementary particles or simple compounds, landing fairly evenly all over the globe.

An Gabhar Ban said...

Wow. I wonder things like that and imagine what I might need to figure out in order to come up with the answer, but you DID it. Cool. So much for my pride in picking the KY Derby winners several years in a row when I was younger though...

Metamatician said...

Thanks Mrs. Goat! I rather liked this post as well. It's fun to flex the old neurons once in awhile. :)

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