[FabHK]

Good intuition. The answer is, indeed, (2-p)/(4-p), where p is the probability that the kid in question is born in the time period in question, ie 1/7 or 1/365, as the case may be.

So, you’re correct.

One can pump the intuition further: If you only know that there are no girls, the result is 1/3. If a boy opens the door, or you know that the older one is a boy, the result is 1/2.

So identifying one of them as the boy increases the result from 1/3 to 1/2.

By saying “one is a boy born on Tuesday”, you give some information, thus nudge the result a bit toward 1/2. Saying “one is born on 12/30”, say, you identify the kid more and nudge the result nearly all the way towards 1/2.

And indeed, if (in the formula above) we set p=1 (no identification), 1/3 results, and if we set p=0 (full identification), 1/2 results.

Quite neat.