Okay, since there seem to be no more candidates for the Quizmaster’s Dilemma, here is the solution. In order not to spoil it for those who read this blog later, please click on ‘read more’ to read more…
The answer is that it is better to switch curtains. It will actually double the chances of winning – from 33% to 66%! Yes, I know this feels completely counter-intuitive, and that is the reason why, for many years, this statistical problem has been food for articles, polemics, even emotionally tense discussion. Try googling on ‘Monty Hall problem’ and you see what I mean.
Let me try to explain it in my own words – below I also give a link to a site that explains it in different words.
Say the candidate chooses curtain A. Initially, this means he has a 33% chance of winning, and a 66% chance of loosing (the chance that the prize is behind curtain B or C). Now suppose that he could choose options B and C together: he would have a 66% chance of winning. Let’s call this option B/C, which has two characteristics: 1) two curtains and 2) at least one empty curtain. In the game, the quizmaster asks the candidate to make his choice (in this example A), and then he opens one of the curtains of option B/C – one of which he is sure it is empty. There are three possibilities:
1. the prize is in B – the quizmaster opens C
2. the prize is in C – the quizmaster opens B
3. the prize is in A – the quizmaster opens either B or C
So, the quizmaster can always open an empty curtain. As this is always possible, the statistical chances (probability) on the combined option B/C do not change. However, what does change is the actual number of alternative curtains – there is only one left to open. The probability is thus transferred from the two curtains to the one curtain left. Key in this is the fact that the quizmaster does not act randomly, but bases his action on the foreknowledge of the winning curtain and the choice of the candidate.
Not convinced? Look again at the three options above:
1. option alternative is B: changing means winning
2. option alternative is C: changing means winning
3. option alternative is B or C: regardless of which one the quizmaster opens, changing to this option means loosing.
So, there you have it: the 3 options – changing means winning in 2 out of the 3 possibilities.
Now, there is another chance: that you do not believe me… if so, look here for another explanation and a simulator. And if you’re still not convinced – many went before you, even intelligent people. Robin just came in and said that he’d seen the light this morning – so there is hope 🙂