I have a question concerning slot machines. In particular, I'm wondering how to compute the Return to Player (RTP; total amount won divided by the total amount that was wagered) given all the winning configurations of the reels and the respective probabilities. I can compute the RTP via computer simulations, let's say I run a code simulating a large numbers of run of the game, then compute $(\text{money}_{in}/\text{money}_{out})\cdot100$. But here I am interested in doing it straight away using math, if possible.
To be more precise, let us assume that I have a simple Slot machine with 3 reels and 10 symbols on each reel: $s1,s2,s3,\ldots,s8,s8,s8$. Oveserve that the symbol $s8$ appears 3 times, just to get different probabilities of the outcomes.
The player win if one of the three following configurations appears:
$\text{conf}_1: s1 \ s1 \ s1\\ \text{conf}_2: s1 \ s1 \ s8\\ \text{conf}_3: s8 \ s8 \ s8 $
The probability of getting the three configurations is $p_1=1/1000$, $p_2=3/1000$, $p_3=27/1000$, respectively.
Finally, for any amount of the bet B, assume that the player wins 3B for the first configuration, 2B if the second, and B for the third.
Notice that the pairs of configurations 1-2 and 1-3, can occour on the same run (on different lines of the slot machine), while configurations 2-3 are mutuallly exclusive.
Using the information above, i.e. winning configurations, winning probailities, and the amount of money retruned to player for a single bet B, is it possible to compute the theretical RTP?
Thank you very much