The trouble with probability is that no one agrees what it means.
Most people understand probability to be about predicting the future and statistics to be about the frequency of past events. While everyone agrees that probability and statistics should have something to do with each other, no one agrees on what that something is.
Probability got a rough start in the world of math. There was no concept of probability as a discipline until about 1650 – odd, given that gambling had been around for eons. Some of the first serious work on probability was done by Blaise Pascal, who was assigned by a nobleman to divide up the winnings when a dice game ended unexpectedly. Before that, people just figured chance wasn’t receptive to analysis. Aristotle’s idea of knowledge required that it be universal and certain. Probability didn’t fit.
To see how fast the concept of probability can go haywire, consider your chance of getting lung cancer. Most agree that probability is determined by your membership in a reference class for which a historical frequency is known. Exactly which reference class you belong to is always a matter of dispute. How similar to them do you need to be? The more accurately you set the attributes of the reference population, the more you narrow it down. Eventually, you get down to people of your age, weight, gender, ethnicity, location, habits, and genetically determined preference for ice cream flavor. Your reference class then has a size of one – you. At this point your probability is either zero or one, and nothing in between. The historical frequency of cancer within this population (you) cannot predict your future likelihood of cancer. That doesn’t seem like what we wanted to get from probability.
Similarly, in the real world, the probabilities of uncommon events and of events with no historical frequency at all are the subject of keen interest. For some predictions of previously unexperienced events, like and airplane crashing due to simultaneous failure of a certain combination of parts, even though that combination may have never occurred in the past, we can assemble a probability from combining historical frequencies of the relevant parts using Boolean logic. My hero Richard Feynman seemed not to grasp this, oddly.
For worries like a large city being wiped out by an asteroid, our reasoning becomes more conjectural. But even for asteroids we can learn quite a bit about asteroid impact rates based on the details of craters on the moon, where the craters don’t weather away so fast as they do on earth. You can see that we’re moving progressively away from historical frequencies and becoming more reliant on inductive reasoning, the sort of thing that gave Aristotle the hives.
Finally, there are some events for which historical frequencies provide no useful information. The probability that nanobots will wipe out the human race, for example. In these cases we take a guess, maybe even a completely wild guess. and then, on incrementally getting tiny bits of supporting or nonsupporting evidence, we modify our beliefs. This is the realm of Bayesianism. In these cases when we talk about probability we are really only talking about the degree to which we believe a proposition, conjecture or assertion.
Breaking it down a bit more formally, a handful of related but distinct interpretations of probability emerge. Those include, for example:
Objective chances: The physics of flipping a fair coin tend to result in heads half the time.
Frequentism: Relative frequency across time: of all the coins ever flipped, one half have been heads, so expect more of the same.
Hypothetical frequentism: If you flipped coins forever, the heads/tails ratio would approach 50%.
Bayesian belief: Prior distributions equal belief: before flipping a coin, my personal expectation that it will be heads is equal to that of it being tails.
Objective Bayes: Prior distributions represent neutral knowledge: given only that a fair coin has been flipped, the plausibility of it’s having fallen heads equals that of it having been tails.
While those all might boil down to the same thing in the trivial case of a coin toss, they can differ mightily for difficult questions.
People’s ideas of probability differ more than one might think, especially when it becomes personal. To illustrate, I’ll use a problem derived from one that originated either with Nick Bostrom, Stuart Armstrong or Tomas Kopf, and was later popularized by Isaac Arthur. Suppose you wake up in a room after suffering amnesia or a particularly bad night of drinking. You find that you’re part of a strange experiment. You’re told that you’re in one of 100 rooms and that the door of your room is either red or blue. You’re instructed to guess which color it is. Finding a coin in your pocket you figure flipping it is as good a predictor of door color as anything else, regardless of the ratio of red to blue doors, which is unknown to you. Heads red, tails blue.
The experimenter then gives you new info. 90 doors are red and 10 doors are blue. Guess your door color, says the experimenter. Most people think, absent any other data, picking red is a 4 1/2 times better choice than letting a coin flip decide.
Now you learn that the evil experimenter had designed two different branches of experimentation. In Experiment A, ten people would be selected and placed, one each, into rooms 1 through 10. For Experiment B, 100 other people would be placed, one each, in all 100 rooms. You don’t know which room you’re in or which experiment, A or B, was conducted. The experimenter tells you he flipped a coin to choose between Experiment A, heads, and Experiment B, tails. He wants you to guess which experiment, A or B, won his coin toss. Again, you flip your coin to decide, as you have nothing to inform a better guess. You’re flipping a coin to guess the result of his coin flip. Your odds are 50-50. Nothing controversial so far.
Now you receive new information. You are in Room 5. What judgment do you now make about the result of his flip? Some will say that the odds of experiment A versus B were set by the experimenter’s coin flip, and are therefore 50-50. Call this Theory 1.
Others figure that your chance of being in Room 5 under Experiment A is 1 in 10 and under Experiment B is 1 in 100. Therefore it’s ten times more likely that Experiment A was the outcome of the experimenter’s flip. Call this Theory 2.
Still others (Theory 3) note that having been selected into a group of 100 was ten times more likely than having been selected into a group of 10, and on that basis it is ten times more likely that Experiment B was the result of the experimenter’s flip than Experiment A.
My experience with inflicting this problem on victims is that most people schooled in science – though certainly not all – prefer Theories 2 or 3 to Theory 1, suggesting they hold different forms of Bayesian reasoning. But between Theories 2 and 3, war breaks out.
Those preferring Theory 2 think the chance of having been selected into Experiment A (once it became the outcome of the experimenter’s coin flip) is 10 in 110 and the chance of being in Room 5 is 1 in 10, given that Experiment A occurred. Those who hold Theory 3 perceive a 100 in 110 chance of having been selected into Experiment B, once it was selected by the experimenter’s flip, and then a 1 in 100 chance of being in Room 5, given Experiment B. The final probabilities of being in room 5 under Theories 2 and 3 are equal (10/110 x 1/10 equals 1 in 110, vs. 100/110 x 1/100 also equals 1 in 110), but the answer to the question about the outcome of the experimenter’s coin flip having been heads (Experiment A) and tails (Experiment B) remains in dispute. To my knowledge, there is no basis for settling that dispute. Unlike Martin Gardner’s boy-girl paradox, this dispute does not result from ambiguous phrasing; it seems a true paradox.
The trouble with probability makes it all the more interesting. Is it math, philosophy, or psychology?
How dare we speak of the laws of chance. Is not chance the antithesis of all law? – Joseph Bertrand, Calcul des probabilités, 1889
Though there be no such thing as Chance in the world; our ignorance of the real cause of any event has the same influence on the understanding, and begets a like species of belief or opinion. – David Hume, An Enquiry Concerning Human Understanding, 1748
It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge. – Blaise Pascal, Théorie Analytique des Probabilitiés, 1812