8. $\frac{41}{2145}$.

First, color a $4 \times 4$ chessboard in two colors, black and white, such that there are 8 squares of each color, and each row and each column contains an even number of squares of the same color.
Discuss in three cases.
(1) If the first column is two black and two white, then there are $\mathrm{C}_{4}^{2}=6$ ways to color this column. Considering the number of black squares in each row of the last three columns, there are $1+2 \times 3 \times 2+3 \times 2 \times 2+3 \times 3=34$ ways to color them.
(2) If the first column is four black, then the last three columns have $3+\mathrm{C}_{3}^{2} \mathrm{C}_{4}^{2}=21$ ways to be colored.
(3) If the first column is four white, then the last three columns have 21 ways to be colored.
For each of these coloring methods, fill the black squares with the odd numbers from $1,2, \cdots, 16$, and the white squares with the even numbers, to get a valid filling method.
Therefore, the required probability is
$$
\frac{(6 \times 34+21 \times 2) \times 8!}{16!}=\frac{41}{2145} \text {. }
$$
8. Place the positive integers $1,2, \cdots, 16$ randomly into the 16 squares of a $4 \times 4$ chessboard, with each square filled with exactly one number. The probability that the sum of the numbers in each row and each column is even is $\qquad$.