Monty Hall.

This is a continuation of the previous posts under this title.

Just what do we mean when we say that picking Door C gives you a two-in-three chance of winning the car? After all, the car is either behind Door C or it isn't. Well, what it means is that if you were to play this game over and over again, always using the strategy of swopping doors, you would win the car on average about two times in every three. It in no way means that you will win the car on this particular occasion - just that it's twice as likely as not that you will. So now consider Contestant II. They have different (that is to say less) information than you and as far as they are concerned the car is just as likely to be behind one door as the other. They will therefore pick a door at random, and again, if they played the game over and over again using that stategy, they would pick the right door about half of the time. So you see it's a bit like a self-fulfilling prophesy - your assessment of the probabilities governs your strategy, and your strategy in turn produces results which reflect your assessment. We could imagine another contestant - Contestant III - who knows all about the game and is therefore going to use the swopping-doors strategy, and who originally picked Door C and has now swopped to Door A. They, like you, have a two-thirds chance of winning the car, and yet clearly one of you is going to lose. The only way this can only be explained is that, playing the game over and over, you would both win the car about two times in three. This of course supposes that you each make your original choice of door independently, and that if you both pick the correct door, you both win a car. More to come...