The king of Lydia is planning to attack Persia, and wants to know if the war will be successful. So he goes to ask the nearest oracle. The problem with this oracle is that she only answers in probabilities, but on the other hand, she is always correct (and doesn't give stupid ambiguous answers). That is, out of all the times the oracle says "50%", the thing really happens 50% of the times, and so on. He asks if the war will be successful, and she says that the probability of winning the war is 75%.
Not quite content with the risk, the king decides to get a second opinion. He goes to ask another oracle. She works the same way - always giving correct probabilities. This oracle says that the probability is 90%. The king thinks that sounds a lot better, but now he's confused - what probability should he expect after hearing both oracles?
We assume that the second oracle has not heard about the first prophecy (so the order of the oracles is irrelevant), and that the war is somehow already determined (the king can't, for example, lose on purpose just to spite the oracles).
The king goes to ask his advisors. The first one says, "surely now the probability must be somewhere between 75% and 90%".
The second one says: "By Bayesian logic, no answer at all is equivalent to 50%. If we get a 90% answer and a 50% answer, that means the 50% oracle just didn't know, so the resulting probability is still 90%. If we get 90% and 75%, the resulting probability must be higher than 90%."
The third one says: "We must use the naive Bayes assumption, and postulate that the probabilities are independent. Then we can calculate a resulting probability, using Bayes' theorem."
The fourth one says: "We will also need to know the a priori probability, but by the nature of war, that must be 50%."
The fifth says: "I think the oracles both know the outcome of the war, they just speak in probabilities to annoy us. They pick a probability at random, equally distributed, and then randomly chooses, with the appropriate probability, whether to tell the truth or lie."
- Let's start with advisor four. Will his addition of an a priori probability make any difference?
- Using the idea of advisor three, what probability do we get?
- If advisor five is correct, what probability do we get?
- Without making any such assumptions, what can we say about the probability?