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 $10 to survive is related to the 30% chance of saving 10 people?
It's not related. It was a separate example.
I'll first explain how I see expected outcome, because I'm not sure my definition is the same as the widely accepted definition.
If I have 50% chance to win 10$, I take it as there are two alternative universes, the only difference being that in one of them, I win 10$ and in the other one, I win nothing. Then I treat the 50% chance as 100% chance to be in both of them, divided by two. If winning 10$ means I'll save myself from 1 hour of work, when divided by two it would be 30 minutes of work. In virtually all cases, when it's about winning small sums of money, you can simply multiply the percentage by the money (in this case, we'll get 5$). Exceptions would be the cases analogous to the one where I'm dying of an illness, I can't afford treatment, but I have all the money I need except for the last 10$ and there isn't any other way to obtain them. So if there's 30% chance to save 10 people's lives, that's the same as saving 3 lives.
If you have no idea what you're talking about, then at least you can see a proof of my problem: I find it hard to explain this idea to people, and impossible for some.
I'm not even sure if the idea is correct. I once posted it on a math forum, asking for evidence, but I didn't find any. So, can someone confirm whether is true, also giving any evidence?
And my main question is, how can I explain this in a way that people can understand it as easily as possible.
(it is possible that it's not clear what I meant - I'll check this thread later for that, and if it turns out to be the case, I'll edit it and add more examples and try to clarify and simplify)