Monty Hall Revisited

A few weeks ago while sitting on a sunny Greek island I got into a conversation with a friend about the Monty Hall Problem. During the conversation I came up with a hand waving explanation that seemed to satisfy him that the correct solution was to always change your answer. Let me start with a brief explanation for those that may not be familiar with the problem.

Monty Hall hosted a game show in which the contestant was given the choice of 3 doors. Behind one of the doors was a fabulous prize. Behind the other 2 were things that you'd most likely never want. Upon picking one, Monty would show the contestant what was behind one of the other doors. He would then ask if the contestant wanted to change their minds and pick the other door. Most will answer, well it doesn't matter, the chance of getting the door you want is 50/50 at that point. The correct answer is; you should always change your answer because with one door exposed, you will on average do better if you pick the other door.

The reasoning is, when you made your first choice you had a 33% chance of being right and a 67% chance of being wrong. Having one door opened for you doesn't change that equation. Thus if you have a 67% chance of being wrong, you will do better to change your guess. Here is my hand waving alternative problem. If you were given 1,000,000 doors to pick from and then after picking one, Monty opened 999,998 of them (all wrong) and you were given the chance to change you guess, would you? My bet is everyone would see the value in changing their guess.

I mentioned this on the train back home from Aarhus to Attila and he mentioned that he wrote a monte carlo simulation. He of course found by random trials what we all expected to be true. If he always changed his mind after the initial guess, he would get the "prize" 67% of the time. Now just to confuse matters, if he chose randomly from the remaining two doors, he won the prize 50% of the time. And this makes perfect sense because in randomly choosing between the two doors he is ignoring the extra information that he got when working with three doors and thus the odds changed from 67% to 50%.

While this maybe a just for fun problem, there are many serious problems that affect peoples lives on a daily basis where "experts" make the wrong assumptions and consequently get the wrong answers and then come to the wrong conclusion. For example, statistics surrounding genetic evidence where one is often left with overwhelming odds either for or against some how seem suspect. Indeed bad maths being used as evidence in court has convicted more than 1 person. What Monty Hall should teach us is that statistics there are lies, dam lies, and bad statistics.