The tricky part to explain about the Monty Hall problem is that there is extra information after the host opens the door. Because you picked a door, the choice

The tricky part to explain about the Monty Hall problem is that there is extra information after the host opens the door. Because you picked a door, the choice the host had of which door to open is sometimes constrained and that gives you extra information as well. Everybody tries to do it solely with probabilities, which is fine, but it becomes far more intuitive when you point out the extra information.