Looking for an intuitive explanation of expected outcome

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.

So if there's 30% chance to save 10 people's lives, that's the same as saving 3 lives.

No, it is not "the same", i.e., equivalent for all purposes. The two scenarios have one particular statistic, the mean lives saved, equal. That does not make them "the same" for all purposes. (Assuming that there was also a 70% chance of saving 0 lives in the first scenario, which you didn't specify, but seemed implied.)

It's too easy to equivocate here.

One could just as well say that the two scenarios do not have equal expected outcomes. The maximum likelihood outcome for the first is 0 lives saved, while the maximum likelihood outcome for the second is 3 lives saved.

Or, one could say that the value to you of your two scenarios is not the same, if you have a preference between them.

An unqualified "Same" is something to taboo. The "Same" according to what equivalence classes? According to what measure? Many don't even see the issue. Once they've calculated "sameness" according to some verbal or numerical equivalence class, they think that "logic dictates" they must treat the two things "the same" too. Wrong. Choose your equivalence classes according to your purpose, instead of constraining your choices according to your equivalence classes.

Expected value is defined as a weighted average, so this covers your first example. I have trouble understanding your setup in the second example. How is your lacking $10 to survive is related to the 30% chance of saving 10 people?

Maybe this will help, maybe not.

You meet a bored billionaire who offers you the chance to play a game. The outcome of the game is decided by a single coin flip. If the coin comes up heads, you win a million dollars. If it comes up tails, you win nothing.

The bored billionaire enjoys watching people squirm, so he demands that you pay $10,000 for a single chance to play this game.

If you are risk-neutral, you should be willing to pay any amount less than $500,000 to play the game. In fact, you can calibrate your actual risk tolerance by considering a series of similar questions, asking how much you would be willing to pay for a certain chance at a certain sum.

Most people will intuitively grasp that it "makes sense" to pay for the right to play the game, and that it stops seeming obvious at some value.

I think what may be confusing about expected outcome is the name. You don't actually expect to get 5 dollars out of this :) . You don't even expect to get, say, 5 million dollars after 1 million games, such would be rather unlikely. You do expect to get 5$ per game if you played infinitely many games, though, and if you are playing such games on small amounts of money you can choose the game to play based on expected outcome.

I'm glad to see that my idea was understood. I have read all the replies, but unfortunately I've came up with all those ideas when trying to prove the idea to myself.

You meet a bored billionaire who offers you the chance to play a game. The outcome of the game is decided by a single coin flip. If the coin comes up heads, you win a million dollars. If it comes up tails, you win nothing.

The bored billionaire enjoys watching people squirm, so he demands that you pay $10,000 for a single chance to play this game.

I've thought of this, but it's only an intuitive thing and doesn't directly prove my approach. Or if it does, I'm missing something.

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.

I've thought of this too, but all it does is to result in a different percentage, closer to the expected outcome. But it's still a percentage, and it remains such if I don't reach an infinite number of trials.

Forget about the intuitive explanation, is there any evidence at all that 50% chance of winning 10$ is the same as 100% chance of winning 5$, in terms of efficiency? I can hardly imagine the expected value approach to be not valid, but I can't find evidence either. Most of the people I want to explain it to would understand it.

I have trouble understanding your setup in the second example. How is your lacking $10 to survive is related to the 30% chance of saving 10 people?

It's not related. It was a separate example.

You're explaining expected value and it's absolutely true. It's the law that tells you what decision to make.

If there's an intuitive explanation, I haven't found it yet. All I know is that there's a reliable cluster of people who

• prefer a certain $500 to a 15% chance of a $1,000,000.
• will never bet with you on anything, no matter how sure they are.
• call a $5 scratch ticket "paying five dollars for entertainment"
• believe that it's impossible to be a professional gambler / poker player
• say things like, "the reason people lose money in stocks in they get too greedy...you have to put your money in, wait for the price to go up, then sell it!"

Even weirder are the ones who know the math, agree with you that something is a good bet for them to take, and then refuse to bet anyway! Like math and decisions occupy completely different worlds, and the heuristic "if you gamble, you'll lose" takes precedence over EV.