Source: (Harvard Statistics 110: see #17, p. 29 of pdf).
A couple decides to keep having children until they have at least one boy and at least one girl, and then stop. Assume they never have twins, that the "trials" are independent with probability 1/2 of a boy, and that they are fertile enough to keep producing children indefinitely. What is the expected number of children?
Solution: Let $X$ be the number of children needed, starting with the 2nd child, to obtain one whose gender is not the same as that of the firstborn. Then $X-1$ is Geom (1/2), so $E(X) = 2.$ This does not include the firstborn, so the expected total number of children is $E(X + 1) = E(X) + 1 = 3.$
My argument: Because of symmetry and for concreteness, we can assume that the firstborn is a girl. Let $X$ be the number of girls before the first boy. Then, $X$ is Geom (1/2). So, the expected total number of chlidren is $E(X)+1=(1-p)/p +1= (1-0.5)/0.5+1=2.$ QED.
Why would mine be incorrect?
Best Answer
Your reasoning is incorrect because even though you're assuming that the firstborn is a girl, asking for "the number of girls before the first boy" includes cases where that number is zero (in which case your assumption from before doesn't hold).
Alternatively, you've forgotten to count the boy that's eventually born (after $1+X$ girls).