Answer
Draw game: 16/126 = 0.126875
Explanation:
Assuming X as first player, O as second player, there are C(9,5) = 126
different ways to arrange five X's (or four O's) within nine positions.
Among these, 16 are draw games: (XXO,OXX,XOO), (XOX,XXO,OXO), (XXO,OOX,XOX), (XOX,XOX,OXO) together with their horizontal and vertical reflections.
There are 12 configurations with a sure win for O: three O's on one diagonal,
for a total of 2 diagonals times 6 choices for the fourth O.
There are 36 "undecided" configurations, i.e. configurations with both three
X's and three O's on a line, according to the following count:
three O's on one side line, that is 4 sides times 6 choices for the
fourth O = 24. three O's on one middle line, that is 2 middle lines times 6 choices for the fourth O = 12.
This leaves 126 − 16 − 12 − 36 = 62 configurations with a sure X win.
The win in an undecided configuration will depend on which side
completes first a triad on a line. Assume a player completes a triad at
his 3rd draw: then the two remaining X's or O's must have appeared in
the first two draws, for a total of C(2,2) =1 possibility each. If the
triad is completed at the 4th draw, then the two remaining X's or O's
had to be distributed within the first three draws, giving C(3,2) = 3
possibilities each. Finally, if the first players completes his triad
at the 5th draw, the two remaining X?s must have appeared within four
draws, giving C(4,2) = 6 possibilities. In accordance, there are four
different possibilities:
X completes at the 3rd draw, O at the 3rd or 4th: X wins, with 1x(1+3) =
4 configurations
O completes at the 3rd draw, X at the 4th or 5th: O wins, with 1x(3+6) =
9 configurations
X completes at the 4th draw, O at the 4th: X wins, with 3x3 = 9 configurations
O completes at the 4th draw, X at the 5th: O wins, with 3x6 = 18 configurations
Thus, the probability that an undecided configuration results in an X win
is (4+9)/40 = 13/40, whilst the probability of an O win is (9+18)/40 = 27/40.
The final probabilities can now be computed:
First player: (62 + 36*13/40)/126 = 0.584850
Second player: (12 + 36*27/40)/126 = 0.288275
Draw game: 16/126 = 0.126875