In German, E stands for "Erwartungswert", in Spanish for "esperanza matemática", and in French for "espérance mathématique". The symbol has become popular since then for English writers. The use of the letter E to denote "expected value" goes back to W. We will call this advantage mathematical hope. This division is the only equitable one when all strange circumstances are eliminated because an equal degree of probability gives an equal right for the sum hoped for. … this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract " Théorie analytique des probabilités", where the concept of expected value was defined explicitly: If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2. That any one Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. Neither Pascal nor Huygens used the term "expectation" in its modern sense. In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables. Therefore, he knew about Pascal's priority in this subject before his book went to press in 1657. From his correspondence with Carcavine a year later (in 1656), he realized his method was essentially the same as Pascal's. But finally I have found that my answers in many cases do not differ from theirs.ĭuring his visit to France in 1655, Huygens learned about de Méré's Problem. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from the same principle. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention. In the foreword to his treatise, Huygens wrote:
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The book extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability. Huygens published his treatise in 1657, (see Huygens (1657)) " De ratiociniis in ludo aleæ" on probability theory just after visiting Paris. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. They only informed a small circle of mutual scientific friends in Paris about it. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively however, they did not publish their findings.
This principle seemed to have come naturally to both of them. The principle is that the value of a future gain should be directly proportional to the chance of getting it.
They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. Soon enough, they both independently came up with a solution.
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He began to discuss the problem in the famous series of letters to Pierre de Fermat. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. This problem had been debated for centuries. The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished.
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