Considerations for a win-loss statistics

The following considerations are not yet completed, but they can give some hints in which direction one can dig further into the problem.

From a theoretical point of view one can partition the board into subfields of possibilities. The following main cases will be distinguished:

1. Case 1: Beginner 'X' starts with the mmiddle cell (2,2);
2. Case 2: The beginner 'X' starts not in the middle field but in one of the corners (1,1), (1,3), (3,1), (3,3)
3. Case 3: The beginner 'X' starts not in the middle field but in one of the fields between the corners (1,2),(2,1),(3,2),(2,3)

 
Case (1)
 
!_  _  _  !
!         !
!_  X  _  !
!         !
!_  _  _  !

Case (2)
 
!X  _  _  !
!         !
!_  _  _  !
!         !
!_  _  _  !

Case (3)
 
!_  X  _  !
!         !
!_  _  _  !
!         !
!_  _  _  !

These main cases will be subdivided with the aid of the second turn as follows:

1. Case 1.1: Beginner 'X' starts with the mmiddle cell (2,2); then the opponent moves into one of the corners.
2. Case 1.2: Beginner 'X' starts with the mmiddle cell (2,2); then the opponent moves into one of the cells between the corners.
3. Case 2.1: The beginner 'X' starts not in the middle field but in one of the corners (1,1), (1,3), (3,1), (3,3); then the opponent moves into the middle field.
4. Case 2.2: The beginner 'X' starts not in the middle field but in one of the corners (1,1), (1,3), (3,1), (3,3); then the opponent moves into one of the corners.
5. Case 2.3: The beginner 'X' starts not in the middle field but in one of the corners (1,1), (1,3), (3,1), (3,3); then the opponent moves into one of the cells between the corners.
6. Case 3.1: The beginner 'X' starts not in the middle field but in one of the fields between the corners (1,2),(2,1),(3,2),(2,3) ; then the opponent moves into the middle field.
7. Case 3.2: The beginner 'X' starts not in the middle field but in one of the fields between the corners (1,2),(2,1),(3,2),(2,3) ; then the opponent moves into one of the corners.
8. Case 3.3: The beginner 'X' starts not in the middle field but in one of the fields between the corners (1,2),(2,1),(3,2),(2,3) ; then the opponent moves into one of the cells between the corners.

How can we compute the probability of these different cases?

Let us take as example case 1.