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.
Requiring the fifty-fifty at every amount you could open forces the weights to satisfy f(x) = f(x/2)/2, whose only solutions are proportional to 1/x, and that integrates to infinity at both ends. Conditional on the pair, the swap gains the smaller amount or loses it with equal chance, which is zero and needs no assumption at all. The article carries a proper spread where the conditional answer is genuinely x/2, and the infinite-mean spread where swapping really is right at every observable amount.
Coins have no memory, which is true, and nobody said this coin is fair, which is the whole problem. A fair coin explains the run with probability two to the minus one hundred while a two-headed coin explains it every time. The article locates the threshold exactly and reconciles the answer with the companion piece on ten heads, which asks a different question about a different setup.
Draw one coin from a thousand, flip ten heads, and the chance it is the two-headed one is 0.5062. Both reflex answers miss, in opposite directions: ninety-nine percent ignores the bag, one in a thousand ignores the flips. Counting patterns gets the exact figure with no Bayes notation at all, and the reason it lands on a coin flip is that 2^10 happens to sit next to the size of the bag.
Una enfermedad que tiene una persona de cada doscientas, y una prueba sin ningún falso negativo. La respuesta refleja ante un positivo supera el noventa por ciento, y se equivoca por más de un factor de diez. Una multitud de mil personas lo enseña antes que el álgebra, y Bayes fija la cifra exacta en 100/1493.