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Benquo comments on Looking for an intuitive explanation of expected outcome - Less Wrong
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Your understanding of mathematical expectation seems accurate, though the wording could be simplified a bit. I don't think that you need the "many worlds" style exposition to explain it.
One common way of thinking of expected values is as a long-run average. So If I keep playing a game with an expected loss of $10, that means that in the long run it becomes more and more probable that I'll lose an average of about $10 per game.
You could write a whole book about what's wrong with this "long-run average" idea, but E. T. Jaynes already did: Probability Theory: The Logic of Science. The most obvious problem is that it means you can't talk about the expected value of a one-off event. I.e., if Dick is pondering the expected value of (time until he completes his doctorate) given his specific abilities and circumstances... well, he's not allowed to if he's a frequentist who treats probabilities and expected values as long-run averages; there is no ensemble here to take the average of.
Expected values are weighted averages, so I would recommend explaining expected values in two parts:
Explain the idea of probabilities as degree of confidence in an outcome (the Bayesian view);
Explain the idea of a weighted average, and note that the expected value is a weighted average with outcome probabilities as the weights.
You could explain the idea of a weighted average using the standard analogy of balancing a rod with weights of varying masses attached at various points, and note that larger masses "pull the balance point" towards themselves more strongly than do smaller masses.
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.