Relationship between an expected value and a probability Since my question has received no answers, I will rewrite it in a generalised form:
I am playing a game in which I have a ball with a value of 500. There is a black bag with 6 other balls of values 430, 470, 510, 550, 590, 710. I am allowed to, just once, swap my ball with a randomly chosen ball from the bag, in the hope of getting the highest valued ball.
Clearly, the probability of improving from my current value if 2/3, but I don't want to "just improve", I want the highest value possible.
Is there any way that I can use the expected value from swapping my ball for another, or maybe the average gain vs the average loss, or something along those lines, to produce a probability on me improving my current value that also factors in the magnitude of how much it could potentially improved by?
From my understanding of expected values, if I could play the game an infinite number of times, then I would always swap my ball, provided that the expected value is greater than hat I have, as the more times I did it, the closer I would converge to the expected value. However, since I am only allowed to play the game once, there is still a 1/6 chance that my value could be reduced by 30, and the same chance for a reduction of 70, but there is also the same chance that it could improve by 210 and the same chance for all the other increases. Is it possible to somehow encapsulate all of these gains/losses in a single probability?
 A: You can represent this problem as the decision-maker's choice between:


*

*scenario 1 in which he receives 500 for sure

*scenario 2 in which he receives 430 with probability 1/6, 470 with probability 1/6, etc.
As you suggested, if you keep track only of the expected value of the swap you lose a lot of information about the distribution of gains associated with this decision. In general there is no way to represent the problem other than specifying the full distribution over final outcomes, as above.
To understand the decision-maker's choice you also need to specify his objective. In the expected-utility theory, we would represent such a decision as
\begin{equation*}
\max \mathbb{E} U(x)
\end{equation*}
where $x$ is the final wealth of the decision-maker (in dollars) and $U$ is his utility function over monetary outcomes. For instance, if $U$ is the identity function, the individual simply cares about the expected gain but not about the other parameters of the distribution. If $U$ is a step function at 500 (i.e. $U(x)=0$ if $x<500$, $U(x)=1$ otherwise), the individual simply wants to maximize his probability of earning more than 500 (irrespective of how much he earns in that case), etc.
