I dare say that most people who work with statistics on a regular basis are familiar with the Monty Hall problem. To recap,
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, 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?
Lots has been written about how this caused an uproar, with many folks with serious technical abilities disagreeing on what the right answer is. I rather like Jeff Atwood’s write up on this, so suggest those interested in the relevant history take a look at this.
The reason I’ve been thinking about this recently is that I think it’s not an example of how ambiguous statistics can be, or how hard probabilities are, but in fact, an elegant demonstration of how a poorly posed problem can derail a simple question. Let’s pose this question in a different way,
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. The host gives you two options, “Either you can pick blind, or I can reveal one of the goat doors before you pick”. Which one of these two scenarios is advantageous?
This is exactly the same problem, just re-stated slightly. It’s basically a linguistic slight of hand. The question makes you focus on which door you’re going to pick, but that’s not what this is about, it’s about which route to picking a door should you take. In my opinion, this is the cleanest explanation as to why “switching” is always the right decision. You’re not switching because you never “had” the first door in any meaningful sense, you’re just choosing to select a 50:50 option, instead of a 33:66 option.