Combinatorics: Tic Tac Toe

Three marbles are tossed onto a tic-tac-toe board at the same time. Each marble has an equal chance of landing in any of the nine squares. What is the probability that the marbles will end up forming a winning line, where all three marbles align either horizontally, vertically, or diagonally?

Answer


\[ \frac{8 \times 3!}{9^3} = \frac{48}{729} = \frac{16}{243}. \]

Pretty self-explanatory.

Notes and comments

Comment 1: Note that order matters in the denominator, hence it also does so in the numerator.

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