Probability of marking at least one row in given matrix

Question

Let there be a matrix $\alpha=(a_{i,j})_{i\in [m], j\in [n]}$, where $a_{i,j}\in\{0,1\}$ And every row has exactly $r\le n$ ones.

We independently with probability $p$ choose some columns from this matrix and mark everything in these columns.

What is a probability $P$ that we have marked every ones from at least one row?

I would like to see in answer a connection to properties of matrix $\alpha$, such as rank or determinant

For example if we take: $$\alpha=\left[\begin{array}{ccc}1&1&1&0&0&0\\0&0&0&1&1&1\\\end{array}\right]$$

Then probability looks like $P=2p^3-p^6$

So i suspect that in general we have $P=1-(1-p^{r})^{rank(\alpha)}$

But i may be wrong

Regards