Values attempts of intuitive definition of the concept of probability

The expected value of a random variable is the long-term average of its possible values when values have been realized a large number of times. It is equal to the sum of the products of the values and their probabilities.

let’s say that any random variable can have N number of possible values: v1, v2, v3, …, vN (N can be any number greater than 1). Let’s also denote the probabilities of the values as P(v1), P(v2), P(v3), …, P(vN).

With these notations in mind, the expected value (EV) of a random variable for a single observation is given by the following expression:

For example, the expected value of rolling a six-sided die is 3.5, because the average of all the numbers that come up converges to 3.5

Like the law of large numbers, the expected value of a random variable is a bridge between theoretical expectations and empirical observations. The two are so closely connected that the main formulations and proofs of the law of large numbers are centered around the expected value of an arbitrary random variable.