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!
Sunday, June 28
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