If a family has two children, what is the probability that both children are girls given that at least one of them is a girl?

Instruction: In a simplified model, each child independently has outcome G or B with probability 1/2. Select a two-child family uniformly and condition only on the event that at least one child is G. Find the probability both are G; contrast with knowing the older child is G.

Context: Tests conditional probability and how an observation protocol changes the sample space. The binary equal-probability model is an exercise assumption, not a demographic claim.

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Official answer

Under the stated simplified model, label the two children by birth order. The equally likely outcomes are GG, GB, BG and BB. Conditioning on the event that at least one is a girl removes only BB, leaving three equally likely outcomes.

Only GG has two girls, so the conditional probability is 1/3.

The observation protocol matters. If instead a particular child, such as the older child, is known to be a girl, the remaining equally likely outcomes are GG and GB, giving 1/2. Being introduced to a randomly selected girl can also involve a different selection mechanism; it is not automatically the stated conditioning event.

Assessment: A strong answer identifies the sample space and the event used for conditioning before dividing counts.

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