These two introductory points bring us to the so-called Monty Hall Problem (or Monty Hall Paradox as some people call it.) The Monty Hall Problem was first explained in popular terms by Marilyn von Savant in Parade magazine in 1990. She explains it thus
It turns out, despite the apparently counter-intuitive nature such a problem, it is indeed to your advantage to which door if you are given the choice. There are a number of ways to explain this claim which at first glance seems so patently false. But I am no mathematician and so I choose an easier way to explain it that makes sense to those of us who cannot do it with numbers and equations. The explanation is simple - in the initial choice you have a one in three change of winning the car. After one door has been opened, if you choose not to switch doors, you still have a one in three chance of winning the car. However, if you switch doors you have a fifty-fifty chance of winning the car. The initial problem is so counter-intuitive that I did the experiment myself. I tried the experiment two hundred times, one hundred of which I didn't change my door and one hundred of which I did. When I switched doors I won 14 more times than when I didn't change doors.Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Besides the fascinating issue of the problem itself, what I find interesting about the Monty Hall Problem is that, as pointed out by von Savant in her book The Power of Logical Thinking, "even Nobel physicists systematically give the wrong answer, and they insist on it, and they are ready to berate in print those who propose the right answer."
I will let people draw their own conclusions concerning the implications of these interesting issues. One thing that I think von Savant's problem has demonstrated is that there is more to the work of Thomas Kuhn than many people might think.