An actor is presented with three choices (doors), A, B, and C. There are two outcomes randomly distributed behind each door: a reward and two failures (a car and two goats). Each door has an equal chance of containing any of the prizes (without replacement) and the actor chooses a door. The game then reveals one of the non-chosen outcomes (if the player chose B, then the game will reveal either A or C) to have a failure (goat) behind it. The player is then given an option to switch their door to the un-selected, non-exposed door or to keep their choice. On it’s face, it may appear that there is no difference between keeping your door or switching to the final alternative (.5 probability of having a goat), but this is wrong. Contrary to conventional guessing, switching doors results in the prize two-thirds of the time while keeping your door results in a prize one-third of the time. A few different mathematical and logical proofs are offered in the wikipedia article linked above and those who remain skeptics about the problem there are a few simulations you can play with to confirm the results iteratively or just enjoy clicking on imaginary doors.
Why does this matter to psychologists or political scientists? It suggests that some theorist ordering preferences and behavior in laboratory experiments must be careful about generalizing about choices made after information is revealed. The New York Times piece offers a few of the classic experiments and the classic versus statistical interpretation of the results. In short, revealed and observed preferences in simulations may be more statistically related than innate. Observed violations of transitivity properties in the lab (often a fundamental assumption in our formal models), may not be as troubling if properly accounted for.