ABSTRACT
The fallacy, as Bertrand himself pointed out, is to assume that if the first of the coins in the chosen box is gold the chance that the other is gold is the same as the chance that the other is silver. It isn’t, the silver is less likely. You are twice as likely to see a gold coin first if your box is GG than if it is GS; so seeing that one of your coins is gold tells you it is twice as likely you have GG as GS. Similarly, seeing that one of your coins is silver tells you it is twice as likely you have SS as GS.
