I think there are two reasons for this: (1) people tend to use the tools that they are most familiar with or that are currently foremost in their minds; and (2) higher mathematical education does not necessarily improve one’s intuition.
To elaborate:
(1) I once went through a period of time (maybe, a semester) when I would immediately think of induction when approaching any problem. I would often see the inductive solution first, before any others. (Perhaps, I still do this to a lesser extent.)
(2) Mathematical education teaches you how to formalize an argument and provides you with the terminology and concepts to understand questions that would otherwise be incomprehensible. However, I don’t think it can teach you how to find that “bright idea”. There is no telling when you will get that bright idea that renders the problem transparent. Even when you get that bright idea, you often cannot explain how you thought of it.
Once I solved this problem in an extremely complicated way. When I presented to my professor, he said that it was correct, but then suggested a much simpler way that I had completely missed.
I also knew someone who had what I call “divine intuition”; it was almost magical. Whenever you gave him a problem, he would usually respond immediately, in an offhand kind of way, with an answer that made you think, “Yes, now that you put it that way, it’s completely obvious; very elegant.” When he couldn’t give a complete answer, he would often say, “I can’t prove it, but I think this is the answer.” Frequently, he would turn out to be correct.
I was hoping it would be last night, too. I very nearly lost a few friends over this one, and I’m hoping someone here can point out the mistake.
One of my BA-in-Math friends ASSERTED that everyone here was wrong, that the answer COULDN’T be any of the choices. The probability, by the something-or-other theorem, tended toward 1-(1/e). I proceeded to list all the possible combinations for n=2, 3, 4, 5, and 6, counted that exactly 1/2 of them were “successful” outcomes, and she STILL told me we were all wrong. Does that 1-(1/e) figure mean anything to anyone?
That’s the limiting probability that at least one person on the airplane sits in their own seat if everybody chooses seats at random. Your friend is thinking of derangements.
** av8rmike**, if your friend can’t be convinced by the multiple sound arguments given here, but you think she might be swayed by Click and Clack’s “official” answer, something is very very wrong. :dubious:
Their answer didn’t convince me. Ray said “Nothing matters until either my seat gets taken or his seat gets taken. If my seat gets taken by some displaced passenger, then I have zero chance of getting my seat.” With just this, it wasn’t obvious to me why this would lead to a 50/50 answer.
Then I had one realization that settled it for me. Assume the first passenger picked a seat other than his own. At some point, a later passenger, whose seat is taken, may pick the first passenger’s seat, and if he does so, the remaining passengers will all find their assigned seats available. This wasn’t obvious to me until I drew a diagram. Now their answer makes sense. So you can fast-forward through the scenario until this specific passenger picks either your seat, or the first guy’s seat, and at that point the outcome is settled. If he picks yours, you won’t get yours, and if he picks the first guy’s, you will get yours. And it’s obvious that this will be 50/50.