Expected value

== History==
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.<ref>{{Cite book|title=History of Probability and Statistics and Their Applications before 1750|language=en|doi=10.1002/0471725161|series = Wiley Series in Probability and Statistics|year = 1990|isbn = 9780471725169}}</ref> This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to [[Blaise Pascal]] by French writer and amateur mathematician [[Antoine Gombaud|Chevalier de Méré]] in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.

He began to discuss the problem in the famous series of letters to [[Pierre de Fermat]]. Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. 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. They only informed a small circle of mutual scientific friends in Paris about it.<ref>{{cite journal |title=Ore, Pascal and the Invention of Probability Theory |journal=The American Mathematical Monthly |volume=67 |issue=5 |year=1960 |pages=409–419 |doi=10.2307/2309286|jstor=2309286 |last1=Ore |first1=Oystein }}</ref>

In Dutch mathematician [[Christiaan Huygens|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. Huygens published his treatise in 1657, (see [[#CITEREFHuygens1657|Huygens (1657)]]) "''De ratiociniis in ludo aleæ''" on probability theory just after visiting Paris. 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]].

In the foreword to his treatise, Huygens wrote:

{{Blockquote|text=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. This does not belong to me. 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. 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 finally I have found that my answers in many cases do not differ from theirs.|sign=|source=Edwards (2002)}}

During his visit to France in 1655, Huygens learned about [[de Méré's Problem]]. From his correspondence with Carcavine a year later (in 1656), he realized his method was essentially the same as Pascal's. Therefore, he knew about Pascal's priority in this subject before his book went to press in 1657.{{cn|date=September 2023}}

In the mid-nineteenth century, [[Pafnuty Chebyshev]] became the first person to think systematically in terms of the expectations of [[random variables]].<ref>{{cite journal|journal=Bulletin of the American Mathematical Society |series=New Series|volume=3|number=1|date=July 1980|title=HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY - A HISTORICAL SURVEY|author=George Mackey|page=549}}</ref>

===Etymology===
Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes:<ref>{{Cite web|url=https://math.dartmouth.edu/~doyle/docs/huygens/huygens.pdf|title=The Value of Chances in Games of Fortune. English Translation|last=Huygens|first=Christian}}</ref>

{{Quote|text=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. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2.|sign=|source=}}

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:<ref>{{Cite book|title=A philosophical essay on probabilities|last=Laplace, Pierre Simon, marquis de, 1749-1827.|date=1952| orig-year=1951|publisher=Dover Publications|oclc=475539}}</ref>

{{quote|… 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. 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. We will call this advantage ''mathematical hope.''}}