Should you take this discount on marbles?

Solution 1:

You can build a dynamic program to solve the problem, then look at the solutions to see when the discount is used. For instance, let $Q(r,g,c)$ be the expected points following an optimal strategy starting with $c$ coins, $r$ red balls and $g$ green balls in the bag. You first note that whenever you pull a red ball from the bag you keep is aside; whenever you pull a green ball from the bag, you put it back in the bag.

The general equation is this: $$Q(r,g,c) = \max \Big[ {r\over{r+g}}Q(r-1,g,c-1)+{g\over{r+g}}(1+Q(r,g,c-1)), \\ \sum_{i=0}^{10} P(i,r,g) * (10-i+Q(r-i,g,c-9))\Big]$$ where $P(i,r,g)$ is the probability of picking $i$ red balls out of a bag of $r$ red and $g$ green. Initial conditions: $Q(0,g,c)=c$ if $g>0$, and $0$ if $g=0$. $Q(r,g,0)=0$ for all $r$ and $g$.

A cursory glance at some runs (for large $r$ and $g$ and $c$), it seems the shortcut is used whenever the proportion of green balls in the bag is above some threshold so it might be possible to show that directly.