The story below was submitted at their request to the Malta Skeptic's magazine. I do not claim originality, only another approach to the correct solution.Please try to restrain yourself from reading my version at the end of the article until you have had a go yourself at Baysian statistics..

Three doors

by Bias     

And there you stand! Center stage! The winner of the TV-quiz! Clammy hands, strangled voice, but the winner!

The evening started off with 8 contestants. Some taxing questions like 'The Isle of Malta lies in the a) Mediterranean b) Pacific', and 'The present pope is called a) Johannes b) Leo' quickly thinned the field. The next set of questions to further reduce the number of participants were clearly more difficult. Things like 'Peanuts grow a) on trees b) underground'.

I was suddenly reminded of a test that went as follows:
a number of students were confronted with a long list of questions. Some examples:

1. Three quarters of all cocoa is grown in a) Africa b) South America
2. More people die of a) appendicitis b) childbirth
3. Kahlil Gibran was inspired by a) Buddhism b) Christianity
4. Adonis was the Greek god of a) love b) vegetation
5. The potato originated in a) Ireland b) Peru

The challenge now was this: be absolutely honest - not too difficult for a student - and bet against yourself on the strength of your conviction that you know the correct answer. In other words: when in complete doubt, bet 1:1. When absolutely sure bet 100:1. And everything in between.

There was one catch: you had to put your money where your conviction lay. Betting 5:1 meant that you had to put 5 p in the kitty when you lost, while winning only paid out 1 p. You see where the emphasis on honesty comes in.

The students in question must have been very honest indeed, for the result was devastating!

Oh yes, the percentage of correct answers showed (in direction) perfect correlation with the height of the bets. This was to be expected. But expressed in money things looked a lot less optimistic. Stacking the odds showed clearly the danger of being too sure of your facts:

Odds: % correct answers
needed to play quits
actually correct on
average (%)
loss in p on 100
bets placed
1
2
5
10
20
50
100
50
67
83
91
95
98
99
53
63
72
76
78
79
80
6 (gain!)
9
68
164
362
971
1920

All these considerations helped me very little when I faced the last remaining opponent, and the outcome of the quiz depended on who was the first to give the correct answer to the last question. Live. On camera. It requires nerves of steel. And, as a friend remarked, a lot of luck. Sometimes I wonder what friends are for.

But, as I started to tell you at the beginning of this column, I made it. And without the help of my fans, for that would have disqualified me.

To surmount the final hurdle however, anybody is permitted to assist. The challenge?

At the rear of the stage there are three closed doors. Behind one of the doors stands a brandnew cross-country motor-bike. Exactly the thing you have tried to convince Santa Claus for years that it would fulfill a dire need. Behind the other two doors are hidden a box of matches and a teaspoon. Not worth having, not really. All you have to do is to choose the right door and the bike is yours. Whatever encouragement and advise comes from the public, it matters not a damn which door you pick. There are no clues. Your chances are 1 in three, no more, no less.

Let's assume that you finally pick the door on the left. For the other two doors the same sequel can be constructed. This is the moment the quiz-master comes to the rescue. He insistently asks whether you don't want to change your mind. The center door? No? The right door? No? Are you quite, quite sure? Yes, you are. You stick to your guns. The left door it is.

With a huge smile the quiz-master now dramatically opens the center door, revealing a box of matches. And then asks you once more, whether you wish to change your mind, to the door on the right.

Pandemonium. The public shouts. Yes, no, yes, no! Change, don't change! Opinions appear to be divided evenly. Should they be?

Or are you well advised to change your mind, and to switch to the door on the right?

Yes you are. And if you disagree, let me know, care of the editor. And why.

But don't say you were not warned!

My version of the solution

Imagine that you were initially given the choice between picking one door or two doors. Pretty obvious what your choice would be, isn't it? Two doors of course!

But the quiz-master gives you exactly that opportunity, and never mind whether it was a bit belated. Nothing has in fact changed between the moment he confronted you with your first opportunity to choose and the second one: the fact that one of the two doors you should have chosen turns out to be empty is irrelevant.