Below is an example of a board state showing the probabilities of x winning in each cell. Is there a mathematical way to calculating the probabilities? I understand that the bottom left corner for x is 1 because the game will end but for the other cells i tried combinatorics/permutations but am getting no where unless I draw every possibility.
|X|0.45|0.4|
|:-|:-|:-|
|**O**|**X**|**0.1**|
|**O**|**0.25**|**1**|
In: Mathematics
Probability = Winning Outcomes/Total Outcomes
So for each board state, there are N number of sequences that can happen from there, and X number of ones that result in a win.
So if a cell has 0.45 it means that “45% of all possible sequences where this is the next move result in a win”
Assuming best play, there’s no such thing as a probability of winning in tic-tac-toe, since it’s a solved game. If X goes first, every first move and continuing combination of moves has been determined. The game is always a forced draw if the second player knows the optimal sequence.
Now, it’s different if the second player plays randomly and if the first player plays optimally, or if both players use suboptimal strategies to some degree. To find the probability of winning in those cases will require the problem to be better defined.
In the specific example given, I’m not sure how the probabilities are calculated. While X can instantly win by going bottom right, they can also win if they go bottom middle or top middle, since that would force a win on the subsequent move. That would suggest the probability of winning bottom middle should be higher than top right. There must be some randomness of moves, since given optimal play there are three possibilities for X which all force a win.
You can’t calculate probabilities here because the choices are neither random nor equally likely. I think you’re using the wrong term.
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