Expected value

Expected value, also called the mathematical expectation, is the average value of a random experiment considered over the very long term. It sums up, in a single number, the whole set of possible results and their respective chances, by answering the question: if this experiment were repeated countless times, what would be the average value obtained per trial?

A useful image is that of a balance point. If each possible result were placed on a ruler, at its numerical position, and given a weight equal to its probability, the expected value would be the point where the ruler would balance: the center of gravity of the distribution.

The calculation consists of multiplying each possible result by its probability, then adding up all these products. With a fair six-sided die, you add 1, 2, 3, 4, 5, and 6, each weighted by 1 in 6; the sum is 21 over 6, that is 3.5. The value 3.5 never appears on a die, which is a reminder that the expected value is an average, not a prediction of any specific roll.

This is the essential distinction to keep in mind: the expected value says nothing about a single draw, which remains unpredictable. It describes only the average tendency when the experiment is repeated a large number of times, in direct connection with the law of large numbers.

In gambling, the player's expected gain is almost always negative: on average, the stake exceeds the gain weighted by its probability. It is precisely this structural gap that constitutes the house edge and guarantees its profitability over time.

The notion helps judge the fairness of a game offered on the site. A draw is said to be fair when the expected gain exactly offsets the stake; comparing the expected value with the stakes makes it possible to see, beyond a one-off result, whether a rule favors or disadvantages the participants over the long term.