A shuffled deck has $N = n + z$ cards. Out of those $N$, $n$ of them have a distinct positive integer value and $z$ of them have a zero.
The single-player game goes as follows. You start with zero points. You draw cards one after another until you draw one with a zero. At that point, the game stops and you earn points equal to the sum of the values of the previously drawn cards. E.g. if you draw cards $2$, $3$, $8$, $0$ you get $13$ points. If you just draw the card $0$ then you get $0$ points.
If the $n$ cards have values $e_1, e_2,\ldots, e_n$, what is the expected number of points you get in a game?
I have brute-forced some specific examples and in every one of them the answer is $$\dfrac{e_1 + e_2 + \ldots + e_n}{z+1}$$ but I cannot figure out why. All attempts of computing this have resulted in immense formulae.
The $z$ cards acts as dividers in the deck which creates $z+1$ regions.
Example: 5 $z$ cards and 6 regions. Each box represents any number of cards including 0.
$\square z\square z\square z\square z\square z\square$
Each card $e_k$ card has a $\frac{1}{z+1}$ chance of being in the first region (on top of the deck above all $z$ cards), giving that card an expected value of $\frac{e_k}{z+1}$. Summing the expected value of all of the cards gives you the total you already discovered.