A few years ago a controversy raged in the Sunday supplements of the newspapers of our country--or at least as much of a controversy as can be generated by a syndicated column regarded principally as a source of entertainment. Marilyn Vos Savant answered a question, and for what may have been months the argument ran, as readers wrote to argue or support the position she took. One group actually developed a computer model to prove that she was correct. She was not incorrect; however, her opposition was not incorrect, either. The argument stemmed entirely from what assumptions one made--or did not make--concerning the situation.
The question, as well as I can reproduce it, concerns a game or contest with three doors. Behind one of those three doors is the grand prize; behind the other two are prizes of little or no value. You are given the opportunity to choose one of those three doors. Once you have chosen, but before you know the results of your choice, the game referee will choose one of the remaining two doors, and give you the opportunity to trade your door for his. According to Marilyn Vos Savant, you should always take the trade; according to her opponents, taking the trade does not improve your odds.
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She reasons thus. Your chance of selecting the right door is one in three; therefore the chance that you have not chosen the right door is two in three. By trading your door for the one selected by the host, you double the chance that you have the right door.
Her opponents argue that the odds of any door being the right door are always one in three. Therefore the door which the host offers you is no more likely to be the correct door than the one you already have, and you might as well stay with your initial choice.
Ms. Savant attempt to support her position by increasing the number of doors. If there are a hundred doors, she says, then the chance that you have the right door is only one percent, and the chance that the right door is one of the others is ninety-nine percent. Therefore your chance of trading the wrong door for the right door is ninety-nine, against one chance of changing the right door for the wrong door. This is very compelling. But it relies on certain assumptions which were never stated to be part of the problem--and Ms. Savant makes a point of stating that this is being approached as a logic problem, not a discussion of the particular television game show on which it is clearly based. Therefore, the assumptions should be revealed.
Two questions have not been asked. The first is, does the referee know which door has the grand prize behind it? And the second is like it: if you have not chosen the door concealing the grand prize, is the referee required under the rules to offer it to you?
Note how important these two assumptions are for Ms. Savant's position. If there are a hundred doors, and you choose one, and the referee chooses one, and the other ninety-eight are eliminated, then there are ninety-eight chances that the grand prize is no longer in the game at all! Without both of these assumptions, the opposition is correct: the door that the referee offers is no better than the door you have already chosen, and given that the odds are even, there is no reason to change.
To examine the odds in the fashion which Ms. Savant most commonly uses in other situations, after you and the referee have both chosen one of the three doors, there are three possible situations. First, you might have the right door, and the other two doors are both wrong. Second, the referee may have the right door, and you have one of the two wrong doors. Third, the right door may already have been removed from the game, and both you and the referee are holding wrong doors. Only in the first of these scenarios do you win with the door you have--a chance of one in three--but only in the second of these scenarios do you win by trading--again, a chance of one in three.
Given that this scenario is taken from an actual televised game show, it is likely that in the game show Ms. Savant is correct--the referee almost certainly offers you the right door if you have chosen the wrong door. But her insistence that no it was not an argument concerning the game show precludes making any assumptions based on the rules or practices of that show.
So why does the computer model support Ms. Savant's position? That should be obvious: the programmers included the same assumptions which were her downfall: the computer was instructed not to eliminate the right door when moving to the second phase. Were the program designed such that the computer would randomly choose between the remaining doors, the results would have shown that the referee's choice was no better than the player's choice.
Perhaps Ms. Savant will recognize that her mistake lies in unwarranted assumptions, and print a retraction. Those of us who did not share her assumptions would appreciate having our position validated. In all logic problems, that which you assume is as important as that which you declare, and a conclusion based on facts not included in the problem is invalid.