What is the motivation for the definition of the expected value?

Let's go way back to the time of Fermat. You're a gambler. You play a game in which there are a bunch of events $i$ that randomly occur and give you $x_i$ units of money, and the gamemaster charges you $C$ units of money to play. You'd like to know whether he's ripping you off. So you play (or simulate) the game many, many times, and find that if you play the game $N$ times for large $N$, it turns out that approximately $p_i N$ of the time you get event $i$, where $p_i$ is some constant. So your total profit after playing the game $N$ times is approximately

$$\sum_i x_i p_i N - NE = N \left( \sum_i x_i p_i - C \right).$$

Now, if this number were positive, the gamemaster would be losing money in the long run, so would quickly go out of business. If this number were negative, you'd know you were getting ripped off, so you'd probably stop playing. The only way that the game is fair in the long run is if $C$ is exactly equal to the expected output of the game.

If the space of possible events is continuous rather than discrete, then the sum needs to be replaced by an integral. This just expresses the fact that Monte Carlo integration works.

Professor Peter Whittle, Cambridge University, argued that average (expected value) was a more intuitive notion than probability, frequentist or not. For this reason he axiomatized expectation rather than probability. Kolmogorov axiomatized probability in 1933, and this is regarded as a milestone (millstone?) in the history of Probability Theory.

Whittle wrote a book in 1970 called Probability via Expectation. This book has been translated into Russian and other languages, and is now in its 4th Edition, 2000. Read the introductory chapter here: Probabilty via Expectation

Now, if you think that Peter Whittle's view of randomness is eccentric, then read what the great Rudolf Kalman, of Kalman Filter fame, has to say about Kolmogorov probability here:

www.scribd.com/document/62685671/Kalman-Probability-in-the-Real-World

where he answers the question: Why is probability not a satisfactory way of looking at randomness?

Axiomatic Probability or Expectation? Take your pick -- randomly, of course.