Fix integers $k,m,t > 0$. Say that I sample $w_1, \ldots, w_t$ uniformly and independently from $\{1,2,\ldots,m\}^k$. For each $i \in \{1,2,\ldots,t\}$, let $S_i = \{w_1,\ldots,w_t\}\setminus\{w_i\}$, i.e., the collection of vectors without $w_i$. Say that $w_i = (n^i_1, n^i_2, \dots, n^i_k)$ is "covered" by $S_i$ if $(\forall j \in \{1,2,\ldots,k\})(\exists \ell \neq i)$ s.t. $n^i_j = n^\ell_j$. Said less formally, $w_i$ is covered if each of its component values appear in at least one other vector (in the same position). In this sense, the covered vector $w_i$ is "redundant". As an example, if my collection is $w_1=(1,2,3,4), w_2=(1,2,5,6), w_3=(1,7,3,4)$, the vector $w_1$ is covered by $\{w_2,w_3\}$, but neither of $w_2$ or $w_3$ is covered.
My question: Let $\mathsf{Win}$ be the event that among the collection of vectors $w_1,w_2,\ldots,w_t$ there is at least one $w_i$ that is covered by its corresponding $S_i$. Is there a closed form expression for $\mathrm{Pr}[\mathsf{Win}]$? If not, a tight lower bound will suffice.
I've looked at this from various angles, without success. I can sort it out if there is a single target vector, say $w_1$, and I want to know the probability that $w_1$ is covered by the corresponding $S_1 = \{w_2,\ldots,w_t\}$. But if we let $\mathsf{Win}_i$ be the event that $w_i$ is covered, note that the events $\mathsf{Win}_i$ and $\mathsf{Win}_j$ are not necessarily independent; thus, one cannot (easily) compute $\mathrm{Pr}[\mathsf{Win}]=1-\mathrm{Pr}[\bigwedge_{i=1}^t \neg\mathsf{Win}_i]$. Thanks in advance for your ideas.