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. Is it true that as $n\rightarrow\infty$, the probability that there exists a matching between $A$ and $B$ approaches $1$?

I posted this on Math Stackexchange, but after a week of bounty there is still no valid answer. I'm also quite certain that this setting has been considered by Erdős or the likes, but I don't know what it is called. Could someone point me to a reference of this?