This is something I find funny that had been told to me during a dinner. There are different versions of this story:
1st - Binary matrix
Suppose you have a binary matrix with arbitrary values.
You know that for some arbitrary column $i$ (for example columns $3$), the probability that any element $j$ of that column is $1$ equals to $0.8$ and that for some arbitrary row $j$ (say $65$), the probability that any element $i$ of the row is $1$ is equal to $0.2$. what is the probability that the specific element at the intersection of this column and row is $1$?
2nd - Binary vector
You have a vector $x$ with $n$ components. You know that among components whose index is even there is $80\%$ of ones. And for components whose index is odd, there is $20\%$ of ones.
Now, you choose a random index, it is even. The probability that $x_i=1$ is $p(x_i=1|i=2k:k\in \mathbb N)=0.8$. Then, you observe the value of the index: $i=144$. The probability is $p(x_i=1|i=2k, i = 144:k\in \mathbb N) = p(x_i=1|i=144)$. And what is the value of $p(x_i=1|i=144)$ ? You can't tell.
So, here is the "paradox", maybe not exactly in the mathematical sense of paradox: there, you have a problem for which you know the answer only if you have partial information.
For the questions.
Where do the "probabilities" come from, since the matrix is arbitrary ? It depends on the meaning you give to probability. If you take it as a frequency, it makes sense. If you take it from some generative process, there are no generative process here.
The two conditions (on the matrix) are contradictory, both cannot hold. I disagree, take any binary $100\times 100$ matrix. Put 80 ones in column $i$ and $20$ ones in row $j$, and voila.