Consider $n_m$ men indexed with labels in $\mathcal{I}\equiv \{1,...,n_m\}$ and $n_w$ women indexed with labels in $\mathcal{J}\equiv\{1,...,n_w\}$. Each men can be of type $A$ or $B$. Each women can be of type $A$ or $B$.
I run an algorithm that creates $\min\{n_m, n_w\}$ couples (1 man+ 1 woman, people are monogamic). The algorithm allows people to remain singles.
$\forall i \in \mathcal{I}, \forall j \in \mathcal{J}$: let $\mu_{ij}=1$ if $i,j$ form a couple and zero otherwise. Moreover, $\mu_{i0}=1$ denotes $i$ remaining single and $\mu_{0j}=1$ denotes $j$ remaining single.
Just by looking at the output of the algorithm and taking frequencies, I want to compute $$p_{AA}\equiv Pr(\mu_{ij}=1|\text{$i$ of type $A$,$j$ of type $A$ })$$ $$p_{BA}\equiv Pr(\mu_{ij}=1|\text{$i$ of type $B$,$j$ of type $A$ })$$ $$p_{AB}\equiv Pr(\mu_{ij}=1|\text{$i$ of type $A$,$j$ of type $B$ })$$ $$p_{BB}\equiv Pr(\mu_{ij}=1|\text{$i$ of type $B$,$j$ of type $B$ })$$ $$p_{A0}\equiv Pr(\mu_{i0}=1|\text{$i$ of type $A$ })$$ $$p_{B0}\equiv Pr(\mu_{i0}=1|\text{$i$ of type $B$ })$$ $$p_{0A}\equiv Pr(\mu_{0j}=1|\text{$j$ of type $A$ })$$ $$p_{0B}\equiv Pr(\mu_{0j}=1|\text{$j$ of type $B$ })$$ I am really confused on how to compute these conditional probabilities and any help would be appreciated. I think the confusion stems from the fact that people are assumed to be monogamic; for example, if $\mu_{ij}=1$ then $\mu_{ih}=0$ for any $h\neq j$.
Let me provide some simple examples to show you what I think is the correct way to proceed.
Example I: Suppose $n_m=4$ with types $A,A,B,A$ and $n_w=4$ with types $A,A,A,B$. Suppose also that the couple types coming out from the algorithm are $$ (A,A), (A,A), (B,A), (A,B) $$ What I am tempted to do is $$p_{AA}=\frac{\text{Number of couples of type A,A}}{\text{Maximum number of couples of type A,A that is feasible}}=\frac{\text{Number of couples of type A,A}}{\text{Min between number of men of type A and number of women of type A}}=\frac{2}{3}$$ and, by similar arguments, $$p_{BA}=\frac{1}{1}, p_{AB}=\frac{1}{1}, p_{BB}=\frac{0}{1}$$ Also $$p_{A0}=\frac{\text{Number of single men of type A}}{\text{Number of men of type A}}=\frac{0}{3}$$ and, by similar arguments, $$p_{B0}=\frac{0}{1}, p_{0A}=\frac{0}{3}, p_{0B}=\frac{0}{1}$$
Example II: Suppose $n_m=11$ and $n_w=8$, where $7$ men are of type $A$, $4$ men are of type $B$, $2$ women are of type $A$, and $6$ women are of type $B$. Suppose also that the couple types coming out from the algorithm are $$ (A,B), (A,B), (B,A), (B,B), (B,B), (A,A), (A,B), (A,0), (B,0), (A,0), (A,0), (0,B) $$ where $0$ denotes "missing" partner because single. As before, what I am tempted to do is $$p_{AA}=\frac{1}{2}, p_{BA}=\frac{1}{2}, p_{AB}\frac{3}{6}, p_{BB}=\frac{2}{4}, p_{A0}=\frac{3}{7}, p_{B0}=\frac{1}{4}, p_{0A}=\frac{0}{2}, p_{0B}=\frac{1}{6}$$
Question: are the calculations above correct for computing conditional probabilities, and, if they are, is there any way to express $p_{AA}$ as a function of $p_{AB}, p_{BA}, p_{BB}, p_{A0}, p_{B0}, p_{0A}, p_{0B}$?
Regarding the second question, my thought is that $$ p_{AA}=\begin{cases} 1-Pr(\mu_{ij}=1| \text{ $i$ of type A}, \text{ $j$ of type B or missing}) \text{ if number of men of type A $\leq$ number of women of type A}\\ \\ 1-Pr(\mu_{ij}=1| \text{ $i$ of type B or missing}, \text{ $j$ of type A}) \text{ if number of men of type A $>$ number of women of type A}\\ \end{cases} $$ Hence, in Example I $$ p_{AA}=1-\frac{1}{3} $$ and in Example II $$ p_{AA}=1-\frac{1}{2} $$
Apologies, there's a lot to manage here. This is a partial answer that tries to cover several possible assumptions. Let me restate the problem.
You have some process and it outputs a bunch of pairs $(i,j)$, after taking some parameters (in particular, the number of people to pair and the number of each type of person).
So far so good. For any given settings, we can now compute the probabilities of various outcomes. Just run the program a bunch of times. However, I get the impression you want to compute something specific. Unfortunately, your $p_{AA}$ definitions have me confused. Currently, it is written $Pr(\mu_{i,j}| i,j \text{ labeled A})$, but note we haven't defined/fixed i,j! There are a few ways to read this.
$p_{AA}$ is the fraction of (i,j) paired together both with label A, out of all possible pairs both labeled A. This gets arbitrarily small as the number of people gets large, so that's unsatisfying. However it's also the closest to what's written. This would be the probability of picking a paired couple, assuming you picked both with label A.
$p_{AA}$ is the fraction of (i,j) both labeled A paired together, out of the largest achievable number of such pairs. For example, if there are 6 A men, 3 A women, and only 2 AA pairs returned, then we get $\frac 2 3.$ this is what you compute in the question, and it computes "the fraction of potential AA pairings that actually occur." Doesn't seem very conditional.
Perhaps the process is random even if we fix the parameters. Then we could define $p_{AA}(i,j)$ for specific $i,j,$ and ask what the probability is for $i,j$ to be paired, assuming both get labeled A. Then we might find something like ”the first man and woman are paired by the program half the time if they are both labeled A." In this case we would fix (or pick a distribution) for the parameters, and run the program repeatedly, counting the number of times $i,j$ both labeled A and the number of times they were also paired together. The estimate would be the ratio of these.
Math can tell you how to compute something, though it's not terribly good at telling you what you should compute.