Interpretation of Expected value of Random Variable

Question

When the sample space in question, is inherently quantifiable and the outcomes have quantifiable relations to each other the expected value is easy to understand, such as the expected value of 3.5 for a die. But what if I have a bag of marbles, 3 red 2 green and 1 blue and I assign the values 1 to red, 2 to green and 3 to blue. The expected value of this would be

$$\sum_{i=1}^3 i\frac{(4-i)}{6}=\frac{5}{3}$$

But what does this mean? And also because the numbering was arbitrary the answer could change depending on the number assignment chosen to each outcome so why is one answer better than another?

Does this have to do with interpretation separately from the pure math?

Answer 1

You're right the answer depends on the values you assign, so if those were just arbitrary values then there's probably not much meaning to it. Maybe those values are prices at which you can sell the marbles. Then your answer gives you the expected amount you earn from picking a marble and selling it.

Answer 2

You are assigning values. You could really assign any values you wanted. The expected value is a weighted mean, so you multiply the value of an outcome by its probability and then sum over all outcomes as well. Depending on your assignment of value, your expected value will be different...