The game is as follows: you have three slots: [1][2][3] with the same 9 fair types for each slot.
When all the slots are the same its a win, doesn't matter the type.
[5][5][5] = win.
What is the probability of winning in such a game?
What i calculated: there are 9 winning variations because any 3 of the same 9 types is win and there are 729 variations (9×9×9) so 9/729 = 1.234%
What ChatGPT said: 1/729 = 0.13%
$p(win\mid x_1 = j) = p(x_1 =\cdots = x_r = j\mid x_1=j) = [idp]= \frac{1}{n^{r-1}}$.
$p(win) = \sum_{j=1}^r p(win\mid x_1 = j)p(x_1=j) = \frac{1}{n^{r-1}}\underbrace{\sum p(x_1=j)}_{1} =\frac{1}{n^{r-1}}$.
In this case, $n,r=9,3$ so $p(win) = \frac{1}{81}$.
Combinatorial, perhaps easier, approach: $p(win) = \frac{\# \text{winning}}{\#\text{total possible}} = \frac{n}{n^r} = \frac{1}{n^{r-1}}$.