1/3.
The mechanical conditioning from lesson 2, applied exactly: “at least one girl” rules out only BB. Three equally likely worlds survive — GG, GB, BG — and both-girls is one of them:
P(GG | at least one girl) = (1/4) / (3/4) = 1/3
The seductive-but-wrong shortcut (“one’s a girl, so the other is 50/50”) fails because “the other one” isn’t a well-defined child. Your question didn’t point at a specific kid; GB and BG are two distinct ways to have one girl, and together they outweigh GG two-to-one.
But the park scenario is 1/2 — and that’s the real lesson. At the park you didn’t learn the proposition “at least one is a girl” through a yes/no question about the family. You sampled a child and observed her to be a girl. Enumerate the equally likely (family, child-you-met) outcomes: GG×2, GB, BG produce a daughter-sighting in 4 equally likely ways, and 2 of those 4 come from GG families. P(GG | met a daughter) = 1/2. The GG family is twice as likely to show you a daughter, and that difference in the generating procedure carries real information.
Compare the two scenarios: the surviving fact is identical — this family has at least one girl. The procedure that delivered the fact differs, so the probabilities differ. The general principle:
You don’t condition on the fact you learned. You condition on the event that you learned it, the way you learned it.
This is Monty Hall in disguise (the host opening a door he knows is empty is a different procedure from a door falling open at random, and that’s the entire puzzle). It’s also a working data scientist’s daily hazard, wearing different clothes: the customers who answered your survey, the errors that got reported, the incidents that made it into the postmortem doc — each arrived through a selection procedure, and ignoring that procedure (conditioning on the fact instead of on the telling) is how selection bias sneaks past people who know how to compute conditionals.
The pitfall, distilled: when someone hands you information, ask “out of all the worlds, in which ones would I have received exactly this message?” — not “in which ones is this message true?”
Where this goes: you’ve now conditioned your way through everything stage 0 owes you. Next: the two factorizations from lesson 3 snap together into Bayes’ theorem, and the updating loop this track is named for begins.