How to calculate RTP (Slot Machine Return) if,
- The number of symbols is $3$, i.e. black, white, red
- Each symbol appears equally, there may be e.g. $9$ red ones
- symbols are independent, on each of the $9$ fields it is possible to draw $1$/$3$ of a given symbol
- Winning means drawing $3$ symbols of the same level
- There are $3$ paylines simultaneously, that is Top, Middle and Bottom.
- Lines won can add up
- Of course, there can also be $1$ or $2$ win lines at once
- The winnings table is not important, it is important that the RTP is around $95 \%$
Best Answer
In each line, there are $3^3=27$ possibilities and $3$ of them are winners, so the probability of winning is $\frac3{27}=\frac19$, and the probability of losing is $\frac89$.
The probability of winning on all $3$ lines is $\left(\frac19\right)^3=\frac1{729}$.
The probability of winning on exactly $2$ lines is $3\left(\frac19\right)^2\frac89=\frac8{243}$, because there are $3$ ways to choose the losing line.
The probability of wining on exactly one line is $3\left(\frac89\right)^2\frac19=\frac{64}{243}$, because there are $3$ ways to choose the winning line.
The probability of losing on all lines is $\left(\frac89\right)^3=\frac{512}{729}$
As a sanity check, $$\frac1{729}+\frac8{243}+\frac{64}{243}+\frac{512}{729}=\\ \frac1{729}+\frac{24}{729}+\frac{192}{729}+\frac{512}{729}=\\ \frac{729}{729}=1$$