Practical Probability

Probability is an often talked about and often misunderstood concept. There are general mathematical definitions for an event and a probability (an event is a measurable set in a probability space, and its probability is the measure of that set), but they generally have nothing to do with what non-mathematicians talk about. The measure which everyone refers to in all discussions which aren’t specifically about mathematics is measuring by proportion. Everyone is familiar with this in the simple discrete case. For example, what’s the probability of drawing the five of spades from a (shuffled) deck of cards? There’s 1 five of spades and 52 cards, so 1/52.

But what does that actually mean? Once the deck is shuffled, there’s no option. Either you will get it on your first draw or you won’t. The problem is just that you don’t know until you actually draw the card. So, before you draw that card, what do you know? As we said above, you know what fraction of the cards has the property that you want. So what can you say about this draw?

Absolutely nothing.

Probability says nothing about individual outcomes.

What probability does say is that if you sit there drawing a card, putting it back, shuffling the cards and drawing a card again, as the number of times that you do this goes to infinity, the proportion of times that you get the card that you’re looking for divided by the number of times that you’ve tried this will approach 1/52.

The thing is, forever is a long time. If you set about flipping a coin over and over again, if the first 10,000 times you flip the coin you get heads, but ever after you alternate heads and tails, by the time you’ve flipped the coin 1 billion times, you’ll have gotten heads 49.998% of the time. And to make things more fun, the probability that you don’t get runs of all-heads or all-tails is vanishingly small.

That is to say, when you’ve got a large number of events, you expect weird coincidences to happen.

Now, you’ll notice that all what I just said has been talking about the future. That’s because probability’s domain is, properly speaking, the infinite. The future is unlimited, so the future belongs to probability. The past is, by contrast, very limited. There’s only one battle of thermopylae and no matter what we do we’ll never get another one in 480 B.C. It’s pointless to talk about the frequency of some outcome relative to all possibilities as you run the battle over and over again; it ran once, and that’s all you get.

Probability has no meaning when you’re talking about the past. I mean that literally: when someone talks about the past and uses the language of probability, they’re actually just talking about their own emotions. (Certainty is an emotion.) Unless their meaning is “I believe that this happened” or “I disbelieve that this happened”, they cannot define their terms so as to give their words a coherent meaning.

(There are, however, arguments for what to do in the future which sound like arguments for what happened in the past. In particular, saying that something was 95% likely means that if you assume that it happened, and you keep doing this about things forever, eventually you’ll only be wrong in 5% of your assumptions. This isn’t an argument about the truth of any particular historical event; it’s only a guide for what to assume about the past so as to minimize the proportion of times that your assumption is wrong.)