One Head Tilts a Flat Prior to 2p, and the Average Bias to Two Thirds
A flat belief about a coin's bias is an input to the calculation, not a conclusion of it, and a single head does not leave it standing. The density tilts to 2p, the cumulative law becomes p squared, the average bias moves to 2/3, and the old answer of one half is demoted to the lower quartile. The general update is the Beta conjugate family, which sends 750 heads in 1000 to Beta(751, 251) with mean 0.749501.