I don't think it's the wording - for me the problem is that I get the intuition that, at the time just after the game show host opened a door with a goat behind

I don't think it's the wording - for me the problem is that I get the intuition that, at the time just after the game show host opened a door with a goat behind it, the system can be in one of four states of equal probability: * Closed door 1 chosen (incorrect) * Closed door 2 chosen (incorrect) * Closed door 1 chosen (correct) * Closed door 2 chosen (correct) ... each with a probability of 1/4. This is probably simply because that's the kind of situation we're familiar with. And it's completely wrong in this case. The actual state space at that point in time is (you can easily verify this by drawing the state transitions down on a piece of paper) * Incorrect door 1 chosen, incorrect door 2 open (1/3, losing) * Incorrect door 2 chosen, incorrect door 1 open (1/3, losing) * Correct door chosen, incorrect door 1 open (1/6, winning) * Correct door chosen, incorrect door 2 open (1/6, winning) ... and I think the reason that's not intuitive at first is that we're unfamiliar with thinking about a system transitioning through states with different probabilities: If you chose the incorrect door at first, the system state after the game show host intervenes is 100% determined (he will open the only one other incorrect door, and these states have 1/3 probability), but if you chose the correct door at first, he has a 50% choice and you will end up in one of two different states (1/3 * 1/2 = 1/6 probability). So, in effect, you have four states of unequal probability after the game show host opens the door, and by switching doors you reverse the probabilities to 1/3 losing and 2/3 winning.