Re: An interesting proposition
See the envelope paradox: it is quite remarkable that the
probability is close to 1/e of being successful (actually,
the real probability differs from 1/e by less than 1/1000!).
Here, ! denotes factorial.
The standard answer is this: if there are N pieces of paper,
go through N/e of them and remember the maximum number seen.
Then pick the next piece of paper that has a higher number
(obviously, you may lose if the first 1/e fraction of them
includes the highest). The remarkable result is that even
as N approaches infinity, the probability of success
approaches 1/e or about 0.36788. This is not intuitively
obvious!
A much more interesting question is this: under what
conditions would you believe that the above strategy will
not do nearly as well as an alternative strategy? Also,
suppose that the numbers need not be unique. There are some
very clear examples especially when N is very large!
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