Say we have a "3 faces coin" that will be flipped infinitely many times.
Face one is Heads (H) that comes up with probability $(1-m) p$,
Face two is Tails (T) that comes up with probability $(1-m)(1-p)$, and
Face three is Empty (E) that comes up with probability $m$.
Both $m$ and $p$ are between 0 and 1.
Say the coin flipper never misreads H and T. However, whenever they face E, they will interpret it, with probability $g \geq 0.5$, as evidence in favor of the face that came up more often until that point. (i.e. it will randomly transform the observed E into H or T, depending on the observed relative frequency of T and H).
Particular example: if, at any point in time, the agent has observed 5 realizations in favor of H and 3 in favor of T, then if they face E, they will interpret as H with probability $g$ or as T with probability $1-g$. Once the interpretation is stored, it will also influence the way the future E realizations will be interpreted.
QUESTION IS: what is the probability that the number of H realizations/interpretations is greater than (less than) the number of number of T realizations/interpretations after infinitely many flips? (i.e., I am looking for an analytical/closed-form solution for such probability and that does depend on the parameters $m,p,g$.
Obs. 1: For simplicity, if the number of H's and T's are the same, one could use a tie break that favors H. I conjecture this is not wlog, but it makes things easier.
Obs. 2: This problem seems to be similar to the Gambler's ruin, Polya's urn or some other branching process. But I haven't been able to solve it properly (not a mathematician here). thx!