The Monty Hall Problem states that during a gameshow, a contestant can choose one of three doors. One of these three doors contains a car, whereas the other two doors contain a gag prize. After selecting your door of choice, the host will open one of the two gag prize doors. At this point, is it better to switch to the other door, or to stay. The problem I'm looking at right now, and having a little bit of difficulty is the following. Suppose the contestant can only switch once, yet there are 6 different doors to choose from. After selecting the door, the host will randomly choose 3 of the gag prize doors to open and show you. My question is this: What are the respective probabilities at each part of the tree?