I have a small technical question related to the paper 'Agreeing to Disagree' (Aumann, 1976) by the Nobel laureate in Economics Aumann.
What I do not understand at all in what follows is why the posteriors for the event $A$ will be $\frac23$ and $\frac13$ respectively.
The crucial point is the following: "Suppose that the agents $1$ and $2$ have a uniform prior on the parameter of a coin, and let $A$ be the event that the coin will come up $H$ (heads) on the next toss. Suppose that each person is permitted to make one previous toss, and that these tosses come up Hand $T$ (tails) respectively.
If each one's information consists precisely of the outcome of his toss, then the posteriors for $A$ will be $\frac23$ and $\frac13$ respectively. If each one then informs the other one of his posterior, then they will both conclude that the previous tosses came up once $H$ and once $T$, so that both posteriors will be revised to $\frac12$".
Thank you very much in advance!