One of the classic examples of un-intuitive probability is the Monty Hall problem whereby an individual is presented with a choice of three doors — behind one is a prize and behind the other two are nothing. After the contestant selects one door, he is shown one of the other two doors which hides nothing behind it. The contestant is then given a chance to change his/her mind. The question is: should the contestant change his mind?
Many people who are faced with this problem oftentimes see the choice as irrelevant — reasoning that they have even odds between the door they have currently selected and the remaining unknown door. This, however, is the incorrect answer. The optimal strategy is to switch. (You can test this yourself if you don’t believe me)
The simplest explanation I’ve heard (courtesy of this New York Times article, hat tip: A. Garvin) for this is that the probability of winning the prize if you don’t switch is 1 out of 3 (the probability that you guessed right the first time). On the other hand, the probability of winning the prize if you do switch is 2 out of 3 (the probability that you guessed incorrectly the first time — in both cases where you guess wrong, the other incorrect door will be shown meaning that switching will get you the right door).
The NYT Article I linked proceeds to talk about economist M. Keith Chen’s use of this simple toy case to debunk a large swath of psychology experiments around the concept of cognitive dissonance (that people forced to choose between things they are indifferent between will rationalize that they actually preferred what they wound up choosing) by pointing out with this test case how the initial choice changes the odds involved in subsequent choices.
Very interesting, very well done, and I think most people would agree it casts a new light on much of the theory behind cognitive dissonance.