When In Doubt, Always Switch

The New York Times has a good online game that helps you understand how the probabilities of the famously counterintuitive Monty Hall Problem work: link

Try playing the game 20 times and always switching your choice. Your rate of success should converge to 67%. Now play the game another 20 times, never switching your choice. Your rate of success should approach 33%. Click on "How It Works" to get an explanation of why this happens.

This is currently in the news because a researcher thinks that a good deal of psychological studies are flawed because they do not account for this phenomenon.