A simple, discrete, independent and isolated test has two possible outcomes, A and B. The probability of each outcome when the test is run is 0.5 (one half). I run the test 10,000 times. Every time the result is A.
a) What is the probability of getting outcome A 10,000 times in a row?
b) If I run the test one more time, what is the probability of getting outcome A?
I remind you, the above is a math test question.
I think the answers are:
a) 1/(210,000) (one over (two to the power ten thousand))
b) 0.5 (one half)
Now, suppose I ask you a different question:
This morning, I spent my time rather frivolously tossing an English Fifty Pence Piece. This is a standard coin minted in the UK to acceptable quality levels. It has a clearly distinguishable head and tail, and because of its shape, rarely lands stably on its edge.
I tossed the coin ten thousand times, give or take. Every time it came down heads. I was careful to vary both throw and catch height. Sometimes I allowed it to land on the floor, sometimes I threw it over my shoulder and sometimes I kept my eyes shut (to avoid quantum).
My questions are:
a) what are the chances of that happening?
b) will you take a bet of one dollar if I say that I reckon the next toss will come down heads?
I contend that there is only one approach, and only one answer to the mathematically expressed, theoretical question about outcomes A and B.
However, the real world example may be answered in any of three ways. Two are mutually exclusive, the third might be seen as a modified combination of the other two.
The first approach is the purely mathematical. It gives the same precise answer to a) as I gave above, and the same precise answer to b) as I gave above. Mathematically, it is correct. But I have been telling you an anecdote about a real-world event that took place this morning. Theoretical mathematics assumes that the test is subject only to the stated constraints. (In Rumsfeldian Notation, there are zero known unknowns and zero unknown unknowns.) The result is that we cannot trust the mathematical answer in the real world. Will you take the bet just because the math says that heads and tails are equally likely?
The second approach is based on what is often called The Law of Averages. This law arises from an incomprehension of the math of probability - or an attempt to infer a real world rule from a mathematical statement of probability. The Law of Averages states that if two outcomes are equally likely, then the more times you test, the more the number of times each outcome occurs will tend to equalize. It supposes, therefore, that each time I get heads, tails becomes slightly more likely in the next toss, so after 10,000 heads, tails is extremely likely. If you think like this you'll have taken the bet without hesitation. You will be just as wrong as the mathematician.
The third approach is known as induction or inductive reasoning or proof by induction. It doesn't matter too much why it is called that. In matters of repeated events, induction assumes that whatever happens most often is most likely. In real world situations, this assumption is the only approach which enables you to allow for unknown factors without even trying to know them. If you are naturally suspicious, you will already have assumed that I haven't told you everything about this mornings events, or that I am lying to you (shame on you!). Consequently you will assume you will lose your money and you won't take the bet. This is wise, but the wrong reason to refuse the bet.
If you reason from the evidence available: 10,000 reported heads; you will conclude that there is some unknown factor that is making heads much more likely. This is inductive reasoning. You use this for a whole lot of other things. It frequently gets called common sense, or intuition. In the real world, induction – evidence – is much more reliable than mathematical theory, and both are more reliable than the Law of Averages.
The Law of Averages and induction both arise from our tendency to turn everything into stories. The Law of Averages is a sort of "Story of Probability", while induction is a "story of evidence". Our ability to turn things into stories is essential in real life, where there are always huge numbers of unstated constraints - unknown unknowns. Without stories, we can't predict the future, and we need to predict the future to survive. If you don't believe this, just answer me this: do you wait until you are hungry before you eat, or do you eat at set times to avoid hunger? The former is reacting to the present, the latter is predicting the future by telling yourself a story. Probably one about a little bear who didn't eat his dinner. Which since this is my story, was probably salmon, rather than porridge. Or do bears have that for breakfast? Or was that little girls? One thing is certain. Little girls rarely eat bears. There is probably a story that convinces them not to.
I blogged recently about the chimera of Planning. Planning is also storytelling — people will judge your plan a good one if it makes a convincing story.
Much of people's behaviour is determined by which stories about the future they believe, and which they don't. Soap Opera exploits this by creating characters who constantly believe the wrong stories. The tension between the story that the character believes, and the one the audience expects, is the origin of almost all drama.
Even those of you who won't take my bet because you are suspicious are succumbing to a belief in a story that conflicts with the story I told you. The story you believe is that the coin must sometimes come down tails if both are equally likely. The mathematician will tell you this is not so, and you are a fool to believe the story. He is wrong. Conflicting stories, both of the past (evidence) and the future (expectation), are our best and strongest indicators of error, accident and misdoing.
Indeed, I did not toss a fifty pence coin 10,000 times. I do not own a fifty pence coin, and I don't have time for that sort of thing. I was telling you a story. I'm sure that's obvious by now.