I've taught Monty hall for ages. It is hard to get it across to everyone. But one trick I have found useful for advanced students is to increase the number of doors. here goes.
Suppose there are 1000000 doors altogether. One has a prize, the rest goats. You pick a door--say number 25. Monty hall then opens door, 1,2,3,4,5,6,...,24,26,27,28,....
up to 324786 which he skips, then he continues opening doors up to one million. So, the choice is now between:
door 25 which you picked
door 324786 which Monty picked.
Most peoples view of symetry breaks down here. Some how, the door Monty picked is much more special than the door we picked.
As an aside, I have sometimes taught that 1/2 is the right answer. THis is easy to motivate if you make it into a two player game. Monty's utility is to confuse the contestent. In other words, he wants the croud to be screaming 1/2 for switch and 1/2 for stay. So what strategy should he use? Basically one that only optionally opens doors so that the final conditional probability is .5.
Finally, a fun simulation that I've done in class is on finance. It is about the only example I know of where variance is the one true answer (as apposed to standard deviation). I've written it up for your sister in name journal, "american statistician". I've a on line link of:
posted by Dean Foster
August 19, 2008