Consider this variation of a well-known problem. As a contestant on a game show, you choose one of three doors. One door has a prize behind it, the other two have nothing. The host knows which one has the prize. The host opens a door and reveals what is behind it. If it has nothing, should you change your answer to the remaining door? To make it more concrete, let's say you choose door #1, the host opens door #2 and it is empty. Should you switch to door #3? Here, the advantage of switching depends on (your a-priori knowledge of) how helpful or malicious the host is. For example: 1. Completely helpful: for example, you know that the host will reveal the door with the prize if you didn't choose it. Since he did not, your door has the prize and you should not switch. (If he had, obviously you should choose to the open door.) 2. Completely malicious: you know that the host will always deliberately choose a door that does NOT have a prize. Since your door (#1) initially had a probability of 1/3 and #2 and #3 together had 2/3; and now you know that #2 does not have the prize, switch to door #3 which has double the probability of having the prize. 3. Random: the host picks the door randomly and #2 happens to not have the prize. The probability is 1/2 each that your door (#1) and the remaining door (#3) have the prize. Switch or not, makes no difference. What is fascinating to me is that these probabilities depend on your knowledge of the host's intentions. Two people with different understanding or ideas about what the host is thinking, will compute different probabilities. Assume that the host is god-like and cannot be questioned or approached. Who is right? Is this a legitimate line of thinking that can be or has been developed into a branch in mathematics? Perhaps already a part of game theory?