Since my last post, I’ve had a couple of other thoughts on the deceiving nature of statistics. In the era of such scientific achievements, people are understandably inclined to view scientific study and statistical analysis as the most proper means of determining truth. However, properly understood, statistics are merely a means of making a “best guess” based on limited information, and not a magical method for determining actual patterns. So let’s examine the coin toss…

I’ve spoken with a couple people lately for whom it’s a dull surprise that, when flipping a coin, we shouldn’t expect the coin tosses to result in a heads, tails, heads, tails pattern. The reason they seemed to believe this is that, assuming a properly balanced coin, the chances of any given coin toss landing on heads is 50/50. So that means, in theory, if I toss a coin twice, chances are I’ll get 1 heads and 1 tails. Four times? 2 of each. Six times? 3 of each, and so on. These people therefore expect that, when you toss the coin twice, if it landed on heads the first time, it will likely land on tails the following toss, and so on.

As another example, a lot of people have this same sort of misunderstanding of statistics so as to think that a random mix of black and white pebbles should be likely to generate a roughly checkerboard pattern– black, white, black, white– when, in fact, a checkerboard pattern is very un-random (or to use a real word, ordered).

However, lets look at the instance of 6 a coin being tossed 6 times. What are the chances that this pattern will be maintained throughout the entire span of the six tosses? In other words, what are the chances of getting either a heads-tails-heads-tails-heads-tails or tails-heads-tails-heads-tails-heads? In fact, the chances are no better than getting 6 heads or six tails. For any number of tosses, the chances of getting an alternating pattern is equal to getting a straight/constant result.

The reason for this is that the probability of any exact pattern is exactly equal. Getting the pattern heads-heads-tails-heads is no more or less likely than the pattern heads-tails-heads-heads or heads-heads-heads-heads. The reason why 3 heads and 3 tails is the most likely result of 6 coin tosses is because of the 64 possible patterns, 3 heads and 3 tails is the most common sum, accounting for 20 of the 64 possible patterns.

Notice that, although 3 heads and 3 tails is the most likely result, it’s not an overwhelmingly likely result. if you had to bet on whether the result will be exactly 3 heads/3 tails, the smart money would be on “no”. As a noted, a 3 to 3 split only accounts for 20 of the 64 patterns, meaning there’s roughly a 30% chance of reaching that result. However, the combinations reaching a result of 2/4 either way (either 2 heads and 4 tails or 4 heads and 2 tails) is more likely than that. In other words, the most probable result is probably not going to happen.

Now, all of this (along with my other post) is to say that, though statistical analysis is an extremely useful tool, statistics is only a predictive tool in an essentially unpredictable world, and it’s easy to put too much stock in the power of numbers to reveal truth.

There’s this funny thing I like to point out to those who are too quick to dismiss things as impossible: According to statistics, the most statistically unlikely things are likely to happen. In fact, we assume the impossible will happen, we rely on it, and if statistically improbable events never occurred, we would live in a very strange world.

Before you jump all over me for spouting nonsense, think about it for a second. Let’s take, for instance, event A, and the chances of event A happening to any given person at any given time is about 1 in one-billion. So when you say, “What are the chances of event A occurring to me right now?” The answer is, “That’s almost impossible. It’s a one-in-a-billion shot.”

But that applies to any given time, so look at any particular person, over the scope of their entire lives. How many moments are there in a person’s life? So instead of, “What are the chances that event A will happen to me right now,” let’s ask “What are the chances event A will happen to me ever?”

Now look at the billions of people in the world, each of them with so many moments in their lives. So suddenly the chances of event A happening to some person at some time become quite good. This much should be pretty obvious. Think of the lottery. What are the chances of me winning the lottery today? Astonomically small. What are the chances that someone will win, sometime? It’s very unlikely that no one will every win the lottery again.

Now, (and here’s where I think it gets interesting) consider all the other 1 in a billion possibilities. There’s event A, (let’s say winning the lottery) but there’s also event B (meeting your long-lost twin), C (getting struck by lightning), D (travelling backwards through time), each unlikely. What are the chances that one of them, A, B, C, or D, will happen to someone at some time? It’s pretty well guaranteed. But that’s only four unlikely possible events. How many possible unlikely events are there? Uncountable.

When you figure this in, that something statistically unlikely will happen to someone at some time is pretty much guaranteed. In fact, it becomes likely that very unlikely things are happening all the time. It becomes statistically impossible for statistically-impossible things to not happen. Of course, this relies on a certain indeterminacy about which unlikely things will happen when and to whom, but it’s still worthwhile to remember that impossible things happen every day.