Tic Tac Toe – Probability – Part 1

Just this Tic-Tac-Toe thing came up the mind. Was wondering what could be the probability of winning a 3x3 Tic-Tac-Toe game.

Calculation:

1. The probability of player A playing first is 1/2

2. Say A places his coin on center (1/9) and if player B does not place the coin in corner (4/8), A wins. So chance of A winning = 1/9x4/8

3. Say A places in a corner (4/9) and if player B does not take the center (7/8) then A wins

4. Say A places in the side centers (4/9) and if B does not play in the adjacent 2 squares or opposite 2 squares (4/8) then A wins

Combining the 4 steps: 1/2x( (1/9x4/8) + (4/9x7/8) + (4/9x4/8) ) = 1/3

So what could be the probability of winning in a NxN board? 1/N? Given that N>=3?

Shhhh!!!! I am working!!!