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All the probability math therein, for a problem whose solution is highly intuitive (if you swap, you'd be just as inclined to swap envelopes indefinitely, is al

All the probability math therein, for a problem whose solution is highly intuitive (if you swap, you'd be just as inclined to swap envelopes indefinitely, is all you need to realize), reminds me of this quote: "The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift." - Albert Einstein Shameless plug for a blog I like that has little math, yet uses much intuition to solve five of the biggest outstanding problems of physics: http://finbot.wordpress.com http://finbot.wordpress.com

justinsb 13y ago
Theoretical resolution of "intuitively obvious" paradoxes such as these are important. If we cannot find a theoretical resolution, it can indicate a flaw in our theories, and that will likely provide a more accurate set of theories. The problems around the speed of light gave rise to Einstein's theories of relativity. Another of those flaws ("the set of all sets that are not members of themselves") gave rise to modern set theory and formal logic. Still a nice quote from Einstein though :-)
uspace 13y ago
It wasn't Einstein http://ianchadwick.com/blog/it-wasnt-einstein-who-said-it/ http://ianchadwick.com/blog/it-wasnt-einstein-who-said-it/
jamii 13y ago
Terence Tao has some interesting writing on the interplay between rigour and intuition: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ http://terrytao.wordpress.com/career-advice/there%E2%80%99s-... Nevertheless, this problem is interesting because people tend to trip up on the maths. The actual behaviour, even in simulation, is very unintuitive (the mean gain from swapping envelopes does not converge to zero over time, and in fact does not converge to anything at all but oscillates wildly). It's a good introduction to the subtleties of probability theory in non-finite spaces.

justinsb 13y ago

Theoretical resolution of "intuitively obvious" paradoxes such as these are important. If we cannot find a theoretical resolution, it can indicate a flaw in our theories, and that will likely provide a more accurate set of theories. The problems around the speed of light gave rise to Einstein's theories of relativity. Another of those flaws ("the set of all sets that are not members of themselves") gave rise to modern set theory and formal logic. Still a nice quote from Einstein though :-)

uspace 13y ago

It wasn't Einstein http://ianchadwick.com/blog/it-wasnt-einstein-who-said-it/ http://ianchadwick.com/blog/it-wasnt-einstein-who-said-it/

jamii 13y ago

Terence Tao has some interesting writing on the interplay between rigour and intuition: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ http://terrytao.wordpress.com/career-advice/there%E2%80%99s-... Nevertheless, this problem is interesting because people tend to trip up on the maths. The actual behaviour, even in simulation, is very unintuitive (the mean gain from swapping envelopes does not converge to zero over time, and in fact does not converge to anything at all but oscillates wildly). It's a good introduction to the subtleties of probability theory in non-finite spaces.