Pairs of matching symbols slot machine

Hello everybody I'm happy to be here :)

Me and my friends start to develop a slot game 5x3. Im very new in this field of gambling mathematics and I learn step by step.

After a long discussion , they want to trigger a event (bonus) not with a single simbols ( 5 in a payline) but with combi of 2 Symbols. When the Sym1 has the Sym2 on the right with the distance of a symbol we trigger a event (bonus)

From my understanding based on 5 reels we have 3 possible winning on reels:

SYM1 - Reel1 || SYM2 - Reel3
SYM1 - Reel2 || SYM2 - Reel4
SYM1 - Reel3 || SYM2 - Reel5


How can I calculate the hits and probability in order to understand how often this bonus games is triggered ?? Finally I would like to find the RTP.

If was just 1 symbol, is very easy but here is about combi of 2 symbols.

Thanks you so much in advance !!!

For an industry-standard online slot we may assume that each reel is independent of the others and that the result is determined in advance, no matter what "control" the player may appear to be given. So, for each reel you know the probability of SYM1 or SYM2 turning up (it should be the number of copies of SYMx on the reel divided by the number of symbols on the reel). Let $$p_{i,j}$$ be the probability of SYM$$i$$ appearing on reel $$j$$. The probability of SYM1 and SYM2 appearing correctly to trigger the bonus round is then: $$p_{1,1}p_{2,3}+p_{1,3}p_{2,5} + p_{1,2}p_{2,4}$$ (you can check and see that this corresponds to the three cases you listed in the question).
Note: typically you can see three (or more) symbols per reel on the screen, and since SYM1 and SYM2 don't have to be on a win-line you need to take that into account when calculating the probabilities $$p_{i,j}$$. For example, if there are $$78$$ symbols on reel $$1$$ and three symbols are shown per reel, and SYM1 appears twice on that reel, there is a $$2/78$$ chance of SYM1 appearing in each of three places, so $$p_{1,1}=6/78$$.