A fair die averages 3.5—not a face you can roll
Expected value is the probability-weighted average of outcomes, not a result you ever have to see. A fair six-sided die has expectation 3.5; American roulette’s single-number bet returns about minus one nineteenth of a dollar. That long-run average is what casinos and insurers quietly bank on.
Statisticians call the probability-weighted average of outcomes the expected value (or expectation, mean, or first moment). When it is finite, the law of large numbers says independent repeats of the same experiment make the sample average settle on that number. For a variable with finitely many outcomes it is simply each value times its probability, summed; more generally it is a Lebesgue integral against the underlying probability measure.
The idea crystallized in the mid-seventeenth century around the “problem of points”: how to divide unfinished game stakes fairly. In 1654 the Chevalier de Méré pressed Blaise Pascal on the puzzle; Pascal and Pierre de Fermat, corresponding privately, both concluded that a future gain’s worth should scale with the chance of receiving it. Christiaan Huygens published related rules in 1657 in De ratiociniis in ludo aleæ after visiting Paris, extending the principle to more than two players and helping found probability as a theory. Pafnuty Chebyshev, in the mid-nineteenth century, was among the first to treat expectations of random variables systematically. Pierre-Simon Laplace stated “mathematical hope” explicitly in his 1814 Théorie analytique des probabilités; the letter E for expected value is traced to W. A. Whitworth in 1901.
Concrete checks make the abstraction bite. Rolling a fair die yields faces 1 through 6, each with probability one sixth, so the expectation is 3.5—not a face that exists. In American roulette a one-dollar straight-up bet wins thirty-five dollars with probability one thirty-eighth and otherwise loses the stake, giving expected gain of minus one nineteenth of a dollar; over about 190 such bets the net loss is typically near ten dollars. Countably infinite outcome sets use the same weighted-average idea with infinite series, and the construction extends componentwise to random vectors and matrices.
Source: Expected value
