Probability of winning tic tac toe game. You and your friend like to play tic tac toe, but are not very good at it. On each turn, the player picks a random empty square to place their mark. The game ends when the first person gets three in a row, as usual. If you make the first move, what is the probability that you win the game?  If you make the first move, what is the probability that you win the game? If the answer can be expressed as $\frac{a}{b}$, a and b are coprime, find a+b.

My attempt:

No of probable ways of arranging tic tac toe symbols so as to win the game

=3horizontal ways+3 vertical ways+2 criss-cross ways=8



The total number of ways in which 3 spaces can be filled out of 9 spaces=9C3.

Therefore, Total number of outcomes=The total number of ways in which 3 spaces can be filled out of 9 spaces (and) One extra space where either of the two tic tac toe symbols can be filled up=9C3*2



Probability=8/(2*(9C3))=1/21
This is Problem E1324 from the American Mathematical Monthly, June-July 1958, p. 447. According to the solution the first player wins with
probability 
$$\frac{1}{126}\left(62 + 36\cdot \frac{13}{40}\right)=\frac{737}{1260}.$$
See the details HERE.
Everybody takes this question wrong. For if you truly play, all games end in a draw. But the question begs, what are the possibilities? You have $9$ places, $X$ moves 1st. there are $3$ moves for $X$ corner, side, middle. $O$ moves 2nd, off an $X$ corner move there are $5$ possibilities for $O$, an $X$ side move there are $5$ possibilities for $O$, and off an $X$ middle move there are $2$ possibilities for $O$. So at the end of $2$ turns there are only $12$ possible games so for. So the question begs, how many possibilities are there at the end of $9$ moves?  How many does $X$ win? How many does $O$ win? How many end in a draw?