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?
Counterintuitively (to me, anyway), the correct choice is to always switch. Lots has been written about how, when this was first published as a brain teaser in Parade in 1990, the proposed solution caused an uproar. Thousands of readers wrote in to claim this was not correct, and even the likes of Paul Erdos were unconvinced until he was shown a numerical simulation. 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. In this scenario, switching is clearly the superior choice.
Of course, the Monty Hall problem has the additional factor that, in fact, the host opening a door PROVIDES extra information. This means that, in reality, switching gains you a 2/3 likelihood of getting the right door. Why? Simply put - if you guessed the car door to start, the host has two possible doors to open, BUT if you guessed the wrong door they only have one option. The key is you are MORE likely to guess the wrong door than the right door. Taken to an extreme, imagine there are now 100 doors; you pick one, and the host then opens 98, leaving one door closed. Now, the chance you got it right on the first try is 1/100 (low!). But now you have two doors, you KNOW the chance of getting it right at first was 1/100, and you know there are only two outcomes - so, if you stick, chance of being right is 1/100, but if you switch, chance of being right is 100/100 - 1/100 = 99/100. I'd switch.