Probability bound for perfect matching Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the remaining edges. From this question (which referenced a paper by Erdős and Rényi), we know that the probability that there exists a matching between $A$ and $B$ approaches $1$ as $n\rightarrow\infty$.
What can we say about the probability itself for any particular value of $n$? What is a lower bound for it, in terms of $n$ and $p$? I tried to deduce it from the above paper, but the way it is written it is quite hard to read off what this probability should be.
 A: Let $q=1-p$ be a probability that there is no edge between two given vertices. For any subsets $A'\subset A$, $B'\subset B$, $|B'|+|A'|=n+1$, denote by $X(A',B')$ the following event: there are no edges between $A'$ and $B'$. The absence of a perfect matching is a union of these events by Hall theorem. 
Thus the following is an upper bound of the probability that there is no perfect matching:
$$
{\rm prob}\, \bigcup X(A',B')\leqslant \sum {\rm prob}\, X(A',B')=\sum_{k=1}^n \binom{n}{k} \binom{n}{k-1} q^{k(n-k+1)}.
$$
As usual, it may be improved by using the fact that these events are positively correlated by Kleitman lemma or how do you prefer to call it: 
$$
{\rm prob}\, \bigcup X(A',B')=1-{\rm prob}\,\bigcap \overline{X(A',B')}\leqslant 1-\prod {\rm prob}\,\overline{X(A',B')}=\\1-
\prod_{k=1}^n (1-q^{k(n-k+1)})^{\binom{n}{k} \binom{n}{k-1}}.
$$
As for the lower bound of the probability that there is no perfect matching, we may use part of inclusion-exclusion: say, if we consider only $2n$ events with $\min(|A'|,|B'|)=1$, denote them $Y_1,\dots,Y_{2n}$, we may use
$$
{\rm prob}\, \bigcup Y_i\geqslant \sum {\rm prob}\, Y_i-\sum_{i<j} {\rm prob}\, Y_i\cap Y_j=2nq^n-n(n-1) q^{2n}-n^2q^{2n-1}.
$$
I guess that in most asymptotic regimes these two bounds are enough, but if you see that no, please specify which dependence of $p$ and $n$ are you interested in.
