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The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory.One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.
Pascal's use of humor, mockery, and vicious satire in his arguments made the letters ripe for public consumption, and influenced the prose of later French writers like Voltaire and Jean-Jacques Rousseau. It is in the Provincial Letters that Pascal made his oft-quoted apology for writing a long letter, as he had not had time to write a shorter ...
In the letters, Pascal's tone combines the fervor of a convert with the wit and polish of a man of the world. Their style meant that, quite apart from their religious influence, the Provincial Letters were popular as a literary work. Adding to that popularity was Pascal's use of humor, mockery, and satire in his arguments.
Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n =6, r =2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then
In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a ...
Pierre de Fermat. Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following: [3] If p is a prime and a is any integer not divisible by p, then a p − 1 − 1 is divisible by p. Fermat's original statement was
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Fermat sent the letters in which he mentioned the case in which n = 3 in 1636, 1640 and 1657. [31] Euler sent a letter to Goldbach on 4 August 1753 in which claimed to have a proof of the case in which n = 3. [32] Euler had a complete and pure elementary proof in 1760, but the result was not published. [33] Later, Euler's proof for n = 3 was ...