nauka: Buffon's needle problem

Evidence: 100 • authority: 72

In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon: Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips? Buffon's needle was one of the earliest problems in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is p = 2 π ⋅ l t . {\displaystyle p={\frac {2}{\pi }}\cdot {\frac {l}{t}}.} This can be used to design a Mo…

Evidence: 38 • authority: 80

On the Uniqueness of the Distribution Function for the Buffon Needle Problem

Evidence: — • authority: 72

In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share the same birthday. The birthday paradox is the counterintuitive fact that only 23 people are needed for that probability to exceed 50%. The birthday paradox is a veridical paradox: it seems wrong at first glance but is, in fact, true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday (the number being less than 1/15th of the number of days in a year), this result is made more intuitive by considering that the birthday comparisons will be made between every possible pair of individuals. With 2…

Evidence: — • authority: 72

Sylvester's four point problem in geometric probability asks for the probability that four randomly chosen points in the Euclidean plane form a convex quadrilateral. Together with Buffon's needle problem, it has been called "one of the prime paradigms in geometric probability theory". The answer depends on the probability distribution from which the points are drawn, and finding a distribution for which this probability is small is closely connected to the crossing number of complete graphs. The problem was posed in 1864 by J. J. Sylvester, who asserted (with fallacious reasoning) that if the points are drawn from the entire plane then the probability is 3 4 {\displaystyle {\tfrac {3}{4}}}…

Evidence: — • authority: 72

In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century. This approach to the problem was published by Joseph-Émile Barbier in 1860. This variation of the Buffon's needle problem can be stated as follows: Suppose there exist infinitely many parallel horizontal lines, spaced a distance D {\displaystyle D} apart, and we were to randomly toss a wet noodle (i.e. an arbitrary plane curve) of length ℓ {\displaystyle \ell } onto them. What is the expected number of times that the noodle will cross a line upon landing? Remarkably, this expected number turns…