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](https://mathoverflow.net/questions/222769/matching-with-probabilistic-edges) (which referenced a [paper by Erdős and Rényi](https://www.renyi.hu/~p_erdos/1964-14.pdf)), 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.



Probability bound for perfect matching
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.


