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How to solve puzzles by graphing the rebounds of a bouncing ball: 1963 Oct: About two new and two old mathematical board games 1963 Nov: A mixed bag of problems 1963 Dec: How to use the odd-even check for tricks and problem-solving 1964 Jan: Presenting the one and only Dr. Matrix, numerologist, in his annual performance 1964 Feb
Buffon's needle was the earliest problem in geometric probability to be solved; [2] 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 =.
In the initial problem, the 100 prisoners are successful if the longest cycle of the permutation has a length of at most 50. Their survival probability is therefore equal to the probability that a random permutation of the numbers 1 to 100 contains no cycle of length greater than 50. This probability is determined in the following.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers (see also Random number generation) and observing that fraction of the numbers that obeys some property or properties. The method is useful for obtaining numerical solutions to problems too complicated to solve analytically.
The solution to this problem can be used to estimate the probability that two large random numbers are coprime. Two random integers in the range from 1 to n {\displaystyle n} , in the limit as n {\displaystyle n} goes to infinity, are relatively prime with a probability that approaches 6 / π 2 {\displaystyle 6/\pi ^{2}} , the reciprocal of the ...
there is a notion of conjugation of random variables, satisfying (XY) * = Y * X * and X ** = X for all random variables X,Y and coinciding with complex conjugation if X is a constant. This means that random variables form complex commutative *-algebras. If X = X * then the random variable X is called "real".
In Ars Conjectandi (1713), Jacob Bernoulli considered the problem of determining, given a number of pebbles drawn from an urn, the proportions of different colored pebbles within the urn. This problem was known as the inverse probability problem, and was a topic of research in the eighteenth century, attracting the attention of Abraham de ...