Mathematical expectation & long-term profit

Intuitively, the expected value is referred to as the "long-term" average or mean. By "long term" is meant that over the "long term of doing the experiment over and over", you would expect this average.
The mathematical formula is the probability-weighted average of the data.

Example: stakes and profits in a random die game

Consider the following game rule: we wager a euro and we throw a balanced six-sided die and

when a 6 occurs, I earn 5 times my stake, that is 5 euros
when a 5 occurs, I double my wager, that is 2 euros
otherwise, I lose my euro wagered

Would you play that game ? Joueriez-vous à ce jeu ? est-il rentable pour le joueur ? Combien peut-on espérer gagner (ou perdre) ?
En effet il y a beaucoup de cas où on perd notre mise, mais on gagne "beaucoup" lorsqu'on gagne. Comment cela se compense ou s'équilibre-t-il ?

The mathematical expectation is computed as the average weighted by the probabilities (1/6 for a balanced die).

Mathematical formula for calculating expectation

In the previous game, for example, we bet 1 euro, and then:

Profit (euros)	5	2	0
Probability	1/6	1/6	4/6

and the expectation is then the probability-weighted average:

E = 1/6×5 + 1/6×2 + 0×4/6/1/6 + 1/6 + 4/6

and, taking into account that the sum of the probabilities is always equal to 1, we always have the simplified formula and the result:

E = 1/6×5 + 1/6×2 + 0×4/6 ≃ 1,16

and we therefore hope to earn on average around 1.16 cents per game over a large number of games, or a 0.16 cents profit, that is to say 0.16 cents in addition to our invested stake.
This game is therefore not balanced, and it is favorable to the player who bets. Over 100 games, the wagerer can hope to earn around 16 euros.
To answer the question asked: yes, I would play this game, and preferably a lot of times!
The following calulator simulates this random game.

Simulation of random play and payouts

The following calculator simulates, randomly, a game with n possible results (3 in the previous example of the dice), and for which we can freely indicate the payout associated with each result as well as the probability of the result.
Results for a certain number of randomly simulated games are then displayed, with the corresponding profit.
The more the number of games increases, the more the total profit stabilizes around the mathematical expectation: for a large number of games, we should not expect a very significant profit (nor a loss) compared to the calculated expectation. This is a direct illustration to the the Law of Large Numbers.