Flipping our Coin
There’s a little bit of “math” in this post, but please bare with me…I really do have a point.
The probability that a fair coin, when flipped, will turn up heads is 50% (0.5, ½, ‘1 in 2’, however you’d like to phrase it). Even a child can tell you that.
Now, what is the chance that the same coin will come up heads twice in a row (by that I mean, what are the chances before the first throw)? Any high school math student can tell you that it’s ½ times ½, or ¼. The basic rule is simple to remember: the probability of any two independent sequential events both happening is the product of the probability of both; so it’s (1/n)x, where n is the number of possible outcomes (2, in the case of the coin) and x is the number of independent events (how many times we flipped the coin). So, for two coin tosses, it’s ½ x ½ = ¼, for three tosses it’s ½ x ½ x ½ = 1/8, and so forth. Of course, this only applies before you toss the first coin, since any given throw has a ½ chance of being heads; even if you’ve tossed 10 heads in a row, the chance of the next throw being heads is still ½. I’m quite sure that most people have no problem conceptualizing this, so let’s make things a little more complex.
If I flipped ten straight heads, you might think something was a tad askew. What are the chances of getting ten straight heads, you might say? Well, just apply the same logic as before: (1/2)10 = 1/1024. Put into words, if we flip a coin 10 times (let’s call this one ‘set’), we would only expect to have a ‘set’ end up with 10 heads in a row once for every 1024 sets. Make sure you grasp this before moving on.
Now, a 1 in 1024 chance of getting 10 heads in a row seems pretty unlikely. It appears far more likely that I’ve biased the coin in some way. But let’s say that you picked the coin and did the flipping, and you got this result. You’d be shocked, wouldn’t you? In fact, let’s assume you flipped the coin 100 times and got 100 straight heads. You’d be blown away! The chances of that happening are: (1/2)100 = 1/7.9 x 1031. In words, this means one chance in 79 million million million million million (that’s 79 with 30 zeros after it). By comparison, the universe is 13.9 billion years old (more on that later), in which time only about 1017 seconds (1 with 17 zeros after it) have elapsed. Suffice to say, 100 straight heads is pretty unlikely. If you got 100 heads, you would be absolutely astonished, and convinced that the coin was “fixed”.
Now, flip the coin 100 times. The odds are very good, of course, that you’ll get something other than 100 heads. But wait, what are the chances of getting any particular result (that is, any given pattern of 100 heads or tails)? Well, apply the basic probability theory: the chance is ½ for each flip, meaning that the probability of having any particular pattern is (1/2)100, which is the same as the probability of getting all heads.
We’ve already agreed that if you had seen all heads, you would have been astounded by the result. And yet, the pattern you did get inspired nothing but the boredom that comes from being forced to flip a coin 100 times. Why should this be?
The reason you are amazed by 100 heads (or tails) and not any other, equally likely pattern is that you see a recognizable pattern in the 100 heads; everything else just looks random to you, so you ignore it. Human beings are spectacularly good at seeing patterns. Unfortunately, we are also spectacularly bad at interpreting them, so we don’t think twice about the seemingly random patterns.
There are numerous examples of this that occur throughout your life. If you won the lottery, you’d be amazed at your dumb luck (your odds, in a typical “6 from 49” game are 1/13,983,816; the math for this can be found at http://en.wikipedia.org/wiki/Lottery_mathematics). But if someone else wins, you’re not surprised at all (they will be, though, and this will become important later). You inherently accept that as long as enough people are playing the lottery, someone will win it, but that the likelihood that it will be you is very small. This is precisely analogous to the 100 heads example: the chances of getting 100 heads (analogous to you winning the lottery) is very small, but the chances of getting some combination of heads and tails is 100% (analogous to anyone but you winning the lottery).
You can carry the analogy even further, if you’d like. Let’s say we bet on the outcome of the coin flips. For the sake of practicality, let’s lower the number of flips to something more manageable; 24. (1/2)24 is equal to 1/16777216. Now, let’s gather 16777215 of our closest friends, and have each one “bet” on one unique combination of heads and tails; I’ll take 24 heads. What are the odds that I’ll “win” the bet (i.e. we get 24 heads in a row)? Simple: 1/16777216. What are the odds that any other given person will win? 1/16777216. But someone will win, and that person will be shocked to have won. They might even think the game was rigged for them to win. But why should anyone think this? Someone has to win the game.
I share this little example not to highlight my love of coin flipping, or lotteries, or even of probability theory, but rather to highlight how bad our conception of “chance” really is. I cannot tell you how many times I’ve heard something along these lines:
“Life is too complex to have arisen by chance. It must have been created by an ‘intelligent’ designer.”
What are the conceptual “odds” of producing, de novo, life as complex as ours (notwithstanding the fact that “complexity” is a relative, rather than absolute, quantity)? (EDIT: I want to clarify that ‘life’ as we know it is, of course, the product of random mutation and the non-random process of natural selection; in this post, I’m referring more to the conditions that allow for life, rather than the life itself. A small, but relevant clarification, I think) Let’s say, for the sake of argument, that the odds are 1 chance in 10200. 10200 is an enormous number; far greater than the number of milliseconds since the dawn of the universe, far greater than the total number of grains of sand that could fit into a sphere the size of our Sun. Huge.
Now, imagine yourself sitting on the sidelines just before the instant the first life came into being, and assume whatever happens next is totally random. What are the chances that life will develop into what we know today? Well, we’ve already arbitrarily defined it at 1 chance in 10200, so let’s stick with that. That implies that there is a far greater likelihood of some other outcome.
But…remember that there is a 100% likelihood of some outcome. Something has to happen. But each one of those “somethings” is just as likely as any other. Have you figured out where I’m going with this yet?
The person who wins our “coin toss” lottery is always shocked that they won, but the 16777215 other people who played are not shocked in the least. If the flips had come up slightly differently, someone else would have won, and they would have been shocked by it.
Similarly, the outside observer is never shocked by the outcome. To them, every outcome looks the same.
Why, oh why, should we be shocked that life is the way it is, and not some other way? The answer comes from our own arrogance (meant in the least pejorative way possible); we look for patterns. We are the winners of the universe’s “coin flip” lottery, and we are shocked by it. But if things had happened just slightly differently, we would have lost, and something else would have taken our place and had to deal with the same shock.
We can all intrinsically accept this when we’re talking about a lottery or a coin toss, but for some reason we have to ascribe some special meaning to our own existence, as though it were outside of what we intrinsically accept about probability; somehow, we think of ourselves as special.
This is not to say that we are not special; perhaps we are. But reliance upon chance to “prove” it is very poor form indeed.