how can I explain this in a way that people can understand it as easily as possible
You are correct that the “long-run average” description is slightly wrong. But the weighted average explanation presumes a level of mathematical sophistication that I think almost no one has, who doesn’t already know about expected value. I suspect that at best that explanation will manage to communicate the idea, “expected value is complicated math.”
It’s also possible to shoehorn the intuitive “long run average” explanation into a more mathematical one, if you say that when you repeat an experiment over and over again, the expected value is the limit that the long run average converges toward.
If you have enough time to explain the analogy of probability as a density (or set of discrete masses) defined over the sample space, then you can explain that the expected value is your “center of mass,” or less precisely the balance point, which is also simple and easy to understand.
The question was:
You are correct that the “long-run average” description is slightly wrong. But the weighted average explanation presumes a level of mathematical sophistication that I think almost no one has, who doesn’t already know about expected value. I suspect that at best that explanation will manage to communicate the idea, “expected value is complicated math.”
It’s also possible to shoehorn the intuitive “long run average” explanation into a more mathematical one, if you say that when you repeat an experiment over and over again, the expected value is the limit that the long run average converges toward.
If you have enough time to explain the analogy of probability as a density (or set of discrete masses) defined over the sample space, then you can explain that the expected value is your “center of mass,” or less precisely the balance point, which is also simple and easy to understand.