Assume an event \(E\) with an unknown probability \(P(E)\). Given that the event \(E\) has occurred, find the probability distribution of \(P(E)\).
In other words, there is an event whose probability is unknown. We know that the event happens on the very first trial. Based only on this information, how likely is it that the event always happens with a \(50\%\) probability? How about \(0\%\), or \(100\%\)? Find a probability distribution to describe this.
This problem was presented to me by a friend, who had thought of it but didn't know how to solve it. I tried, but I don't have a solution either, and I am interested to see what kind of techniques can be used to solve it.