Suppose we have a floor made of parallel strips of wood, each the same width 2b, and we drop a needle of length 2a onto the floor. The probability P n of them being coprime tends towards when n tends towards infinity.Ī proof is available on the Cesàro's page. If we want to use this result, we have to reword the following sentence as the set of natural integers couples is infinite: if we choose two natural integers less than n, The probability of two randomly selected integers being coprime is. The most famous theorems probably are CesàroĪnd Buffon ones, that we remind here (as the regular visitors already know where to find them on my website -)). However, some results are somewhat facinating and show Pi in domains where we wouldn't expect it (him !) to appear. ![]() ![]() I would call these theorems some "lotto probabilities" (!) since it deals more with proportions, counts and areas estimations most of the time, than real probability theormes stemming from inherent measure theory ! Pi appears in several isolated theorems often considered as belonging to the probability field. You are allowed to have a summary -)Īnd the theorems related to probabilitiesġ - Some notions about the Brownian MotionĪsymptotic probabilities for small Brownian ballsĪ - Pi and the theorems related to probabilities This page is quite recent so it's less messy now than before. And I did not find all the proofs of what I present in the following paragraphs, so if you know one of them, please contact me ! I'm open minded to any suggestion, of course. This is indeed not easy to explain everything in one page ! I hope this page will grow as I will collect some new stuff. This page is part of three pages dealing with the complex relations between Pi and the randomness field. Home Version history Guestbook Who I am Some pictures (fr) Acknowledgements Last modifications Contact
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