How to make a game theoretic version of these chess statements?

Question

Consider this in a 2 player board game (like chess) there are $3$ possible outcomes. We forget about resignations in the game:

1. Player $1$ wins $E_1$.
2. Player $2$ wins $E_2$.
3. There is draw (we include the outcome of the game going on forever in this outcome) $E_3$ - we assume we are in the subset, $E_3$ does not take place:

Now, for simplicity we assume it's a game of you versus yourself and you implement the same winning-strategy (or set of algorithms) $S_1$ to win against yourself.

My question is assuming it is a game which does not discriminate on the availability of resources and has a symmetry of rules applied (for example think of chess where both players have same number of pieces and the rules for the white - *are almost - are the same for black)

In this case I was thinking of using Bayes theorem in the sense of relating the events:

$$ P(E_1\mid S_1)={\frac {P(S_1\mid E_1)P(E_1)}{P(S_1)}} $$

Now to calculate $P(S_1\mid E_1)$ one can start with the win condition and go back all possible allowed moves until the hit the initial board condition (note this does not include $E_3$). The probability of $E_2\mid S_1$ is:

$$ P(E_2\mid S_1)={\frac {P(S_1\mid E_2)P(E_2)}{P(S_1)}} $$

Now, in a game of chess there is asymmetry as white moves first! Therefore, if I count the total number of moves $N$ in the sense white moves first followed by black moving $2$'nd. Hence, for black to win (/checkmate) it will be an even number of moves.

Hence, the probability will be:

$$ P(S_1\mid E_2) = \frac{\sum_{i}^{K_2} 1}{\sum_{i}^{K_2} N_i} = \frac{K_2}{\sum_{i}^{K_2} N_i}$$

where $K_2$ is the total number of games in which satisfy the event: $S_1\mid E_2$ and

$$ N_i \equiv 0 \text{ mod } 2 $$ $$ \implies \sum_{i}^{K_2} N_i \equiv 0 $$

Let, this is be statement $2$.

Again, for white:

$$ P(S_1\mid E_1) = \frac{\sum_{i}^{K_1} 1}{\sum_{i}^{K_1} N'_i} = \frac{K_1}{\sum_{i}^{K_1} N'_i}$$

where $K_2$ is the total number of games in which satisfy the event: $S_1\mid E_1$ and

$$ N'_i \equiv 1 \text{ mod } 2$$ $$ \implies \sum_{i}^{K_1} N'_i \equiv K_1$$

Let, this be statement $1$

Question

What all subtle assumptions have I made for statement $1$ and statement $2$ to be true in the game of chess and can game theoretic version of this statement be made (and apply it to other "games")? For example, it was any other game then if it may not be necessary that

1. The win condition is only obtainable when for player $1$ and player $2$ when the number of moves is odd and the number of moves is even, respectively.
2. The algorithms implementable be such $S_1 = S_2$ can happen where $S_2$ is the algorithms used by player $2$