If by shuffle you mean that Monty just gets to keep trying until he picks a goat, you're right. If instead you mean that they randomize all three doors and sta

If by shuffle you mean that Monty just gets to keep trying until he picks a goat, you're right. If instead you mean that they randomize all three doors and start over, the probability is 1/2 because half the time that the player has a goat you start over and get to pick again, which perfectly cancels out the double probability of having a goat. This can be analyzed pretty easily using the three possibilities you outlined earlier. Suppose the player gets door 1 and Monty always picks door 2. Each time they shuffle, you get one of the following with equal probability: A. car-goat-goat B. goat-car-goat C. goat-goat-car A means the player wins, B means we re-shuffle, C means the player loses. It's pretty easy to see that A and B are equally likely outcomes, so they become 50/50 odds after all the re-shuffling is done.