What is the probability to guess the solution for Sudoku, if all the cells were filled randomly?
We have a 9 by 9 square. Every cell contains an integer from 1 inclusive to 9 inclusive. And every integer was taken 9 times.
There are no filled cells from the beginning.
If we were to assume that each cell was filled with a number from 1-9 uniformly and independently at random (i.e. it is possible, though unlikely, that we accidentally fill every square with $1$'s), we first recognize that there are $9^{81}\approx 2\times 10^{77}$ equally likely ways for us to have filled the grid (seen from rule of product).
Correction: As pointed out in the comments, the OP stated that each integer was taken exactly nine times. Rather than $9^{81}$ equally likely ways that we fill in the sudoku grid where each cell was filled independently of the others, we are interested instead in the number of ways that we can fill the grid with exactly nine of each digit. Using multinomial coefficients we calculate this to instead be $\binom{81}{9,9,9,9,9,9,9,9,9}=\frac{81!}{(9!)^9}\approx 5.3\times 10^{70}$
As per the results obtained by Felgenhauer and Jarvis among others there are approximately $6.7\times 10^{21}$ different sudoku grids (this is ignoring symmetries. Further studies found the number of "essentially different" grids where rotations and reflections and the like were taken into account).
Taking the ratio then, your probability would be
$$\approx \frac{6.7\times 10^{21}}{5.3\times 10^{70}}\approx 1.2\times 10^{-49}$$