Consider a finite set $A$ consisting of $n$ distinct elements. Draw $m$ independent elements $a_1, \dots, a_m$ with replacement from $A$. Define $B=\{a_1, \dots, a_m\}$. What is $P(A\subset B)$? Clearly, this is zero if $m<n$, increasing in $m$, and approaches unity as $m\to \infty$ for every fixed $n$.
The exact expression for $P(A\subset B)$ seems to be involved. Is there a nontrivial lower bound on $P(A\subset B)$ that depends on $m$ and $n$?