Why do people find statistical
puzzles such as the Monty Hall problem
so difficult to answer
? Why do they so often supply 'the wrong answer'? Why do they often stick to their solution even when 'the correct solution' is pointed out to them?
I think I know. The reason is that all of these problems have unstated assumptions, assumptions that are required to model the problem with statistics. Often, the provided answer starts and ends with the mathematics, but that is completely beside the point: the question is how you arrive at the model on which the computation is based; often, the difficult part is what the unstated assumptions are.
In the Monty Hall problem, the unstated assumption is: Monty Hall knows which door has the prize, and always opens an empty door. This is what allows us to view the situation as one out of a repeatable series. And that is an absolute requirement for applying statistics in the first place: you have to know the set of potential events the particular event is drawn from in order to tell its probability, because that is what statistical probability is, by definition.
In the case of Monty Hall, the unstated assumption is fairly obvious, but in some related problems (see the rec.puzzles FAQ for a list) there are several reasonable, conflicting assumptions: they lead to different outcomes. A good example, in my opinion, is the puzzle in a family of two children, of which one is a boy, what are the odds the other is a boy as well?
Often, it is simply unclear what can be assumed, in which case statistics cannot be applied, either.
Example: what is the probability that my age is over 20?
You can't apply statistics to this question without assuming a population from which I'm drawn: all humans? all Everything noders at the time you read this? All Everything authors? The question doesn't imply any, so the only correct answer is: sorry, but you'll have to tell me before I can give you a statistical probability.
A stronger example: what are the odds of intelligent life existing elsewhere in the universe?
Here, we cannot even begin to answer without stating some fundamental assumptions on which factors have been critical in the emergence (creation if you wish) of life on earth, and how often they recur elsewhere; and let's be honest, nobody knows.
Therefore, I consider all answers to such problems that consist merely of statistical formulas stupid and more wrong than an 'incorrect' numerical outcome could possibly be.
PS to Geez: in daily life, when we talk about chance, we very rarely have a neatly defined solution space in the way that statistics assumes, maybe it takes a while to get used to the concept.