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I am a french student in mathematical engineering. I had to go trough an intensive 3 year "preparation" to pass a "concours" to go to High School. In mathematics, I have been taught a lot of algebra, analysis and geometry.

The most interesting results I have encoutered were the results which involve two or three of the fields mentionned above. I remember some exercises wich seemed to be difficult multivariate analysis problems but were in fact simple algebra ones.

One of my favorites is:

Find $$ \inf_{a,b \in \Re}\left(\int_{0}^{1}t^2(\ln(t)-a-bt)dt\right)$$

Such link is very interesting because it gives new ways to solve problems by building links between theories, but it also helps to understand both theories involved. This is with this sort of results that I find mathematics very beautiful, simple and complex at the same time.

I've "recently" been introduced to probability theory, being taught the basics and more. But I've failed to encouter such interesting results. Maybe I got used to see beautiful results or I don't grasp them entirely. I saw the link between analysis and probability through pdf, stochastic calculus, between algebra and probability trough conditional expectation... but I wasn't struck. Probability seems to be a tool, a sophisticated tool but a tool.

More recently I've seen some results, for example: A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$ which makes me think I've missed something. Both in understanding all the aspects of probability theory and in building links between probability theory and other fields.

Could you bring me some other striking examples of the power of probability theory? My criteria is either a use wich simplifies a complex problem from another field or a result which puts light on an uncommon aspect of probability theory.

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Lucas Morin
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    I love the probabilistic method which proves existence of structures that are hard to create, esp those which do no depend on local properties. – Calvin Lin Jun 27 '13 at 15:35
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    Does the existence of normal numbers count as a striking example? – Did Jun 27 '13 at 16:32
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    Exploring this site will give you some examples. For instance,

    http://math.stackexchange.com/questions/429007/proof-of-inequality-sum-limits-k-0n-binom-n-k-frac5k5k1-ge-frac2/430221#430221 http://math.stackexchange.com/questions/181871/computing-lim-limits-n-to-inftyn-int-0-pi-2xfx-cos-n-xdx/181888#181888 http://math.stackexchange.com/questions/51926/a-stronger-version-of-discrete-liouvilles-theorem/51935#51935

     –  Jun 27 '13 at 17:25
  • @CalvinLin Thanks this is interesting. – Lucas Morin Jun 28 '13 at 09:36
  • @Did This is interesting too, specially the part about information theory. you should devellop your comment and post an answer ! – Lucas Morin Jun 28 '13 at 09:39

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I'm not sure if that's quite what you're looking for but I believe it is a striking result - at least a very unexpected one. It's a puzzle so it's stated in a very simple way and you don't need very sophisticated tools to solve it (or, more likely, to understand the solution, cause in my opinion it's extremely hard to come up with it on your own). Here it goes:

There's a prison with 100 inmates in it and a prison governor. Inmates are numbered from 1 to 100. The governor wants to play a game. He writes numbers $1 - 100$ on pieces of paper and he puts them into 100 boxes. The boxes are also labeled with numbers, but they don't necesserily correspond to the numbers on sheets in them. The governor calls each prisoner into his office (one at a time, everyone will be called exactly once, they can't communicate with each other after being called to the office). A prisoner chooses 50 boxes he wants to open. If he finds his number (opens a box with his number inside) he wins, otherwise all the prisoners die (even those who won before). Number stays in the same box throughout the whole game. If all the prisoners win they are set free, else they all die.

Prisoners can only communicate with each other before the game starts. Find a strategy for them to win with probability bigger than 30%.

When I heard the puzzle I was convinced it wasn't possible, since if each prisoner chose the boxes randomly then each of them wins with probability $1/2$, so the probability of all of them winning would be ${2^{-100}}$ in this case and it seems quite unlikely that you can make it much better - but it turns out you can. You can actually make it much much better - bigger than $\frac{3}{10}$, quite surprising, isn't it?

here's a hint, a very big one, but it's rather hard to give a hint that wouldn't almost completely spoil the puzzle

more than 30% of permutations of 100 elements don't have a cycle longer than 50

I've heard this puzzle in my probability class, hope you enjoy it as much as I did

mm-aops
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