Your first set of calculations solves the problem that you asked - what is the AVERAGE PERCENTAGE of redheads in a room at the stroke of noon on the first day of the month? You take the percentages and you add them up. Each percentage is calculated based on an observed ratio fixed in time each month, and then averaged.
The second set of calculations is different, though. You are conflating all the events into one event, so you are getting the answer to a different question: what is the percentage of redheads to non-redheads based on four separate events in the aggregate?
The first answer to my mind seems to be the only one that addresses your question exactly. The other one is creating some sort of a mean. The point is that your original question posits that you're trying to ascertain what the percentage is likely to be the next first of the month, so you calculate each event separately and then average them. The second one is aggregating the numbers in a different way that won't provide as accurate an answer to the basic question.
Of course, I'm not a mathematician and never really enjoyed math that much, so I can only explain in vague terms and tell you why I think something is right or wrong using my own logic.
ἡ δὲ κἀκ τριῶν τρυπημάτων ἐργαζομένη ἐνεκάλει τῇ φύσει, δυσφορουμένη, ὅτι δὴ μὴ καὶ τοὺς τιτθοὺς αὐτῇ εὐρύτερον ἢ νῦν εἰσι τρυπώη, ὅπως καὶ ἄλλην ἐνταῦθα μίξιν ἐπιτεχνᾶσθαι δυνατὴ εἴη. – Procopius
Ummaka qinnassa nīk!
*MySmiley*
