The Boy/Girl Paradox With A Twist

There exists a large population of 2-child families. There are two different observation processes that randomly select mother-daughter pairs from the large population. Each time this occurs, an observation is emitted.

Process A selects a daughter at random from all the daughters of 2-child families, then pairs that daughter with her mother. It can be shown that observations generated by this process describe girl-girl families with a probability of 1/2.

Process B selects a mother at random from all 2-child families with at least one girl and then pairs the mother with one of her daughters. In boy-girl families, this is the only daughter. In girl-girl families a coin-flip is used to select one of the two daughters. It can be shown that observations generated by this process describe girl-girl families with a probability of 1/3.

None of this is controversial - different randomisation processes result in observations with different probability distributions.

Consider now Process C. Process C has two inputs, one from Process A and one from Process B. The function of Process C is to match observations from Process A with observations from Process B and, if a match is detected, to emit a new observation. Otherwise, nothing is emitted.

Q: What is the probability that observations emitted by Process C describe a girl-girl family?