I have a probability question that seems easy, but I somehow can't wrap my head around it.
Suppose we have a coin. Probability that coin toss will come out heads is some unknown value X. First toss came out as heads. What would you be your best guess about the value X (so, if you guess is y, your task is to minimize $ |X - y| $)?
For me it seems like given the result of the first experiment, coin is just a little bit more likely to be loaded in a way that heads come out more often, so optimal guess about the likelihood of heads is 1. But I can't formulate it in a proper way or prove it mathematically. Besides, there is an opinion in other (non-math) online community that probability 0.5 would be more likely. I think there is a flow somewhere in my logic.
Can you help me to understand this concept? Thanks.
Update: for anyone interested, the question originally emerged during the discussion of Hindsight bias phenomenon. More precisely, the result of Fischhoff and Beyth experiment seems to be logically correct since differences in the results of predictions were caused by the differences in the information given to the groups. Even if the students were explicitly asked not to consider the result of conflicts as the probability factor, the only thing that experiment states is that we can't throw things out from our subconscious perception of the world at will (and that is obvious from the definition of the subconsciousness itself). So the phenomenon of hindsight bias can not be tested through such experiment or any one alike. The experiment should show difference between mathematical probability and empirical probability given the same initial data.