I’m going to make two objections to standard justifications of the correct answer to the Monty Hall problem. The conclusion to an unsound argument can still be true, so if I’m successful at showing that there is a problem with the reasoning supporting the accepted answer to the problem, this will not show that the accepted answer is false, just that the justification of the answer is faulty.
At a high level, the two objections are that the standard justification (1) begs the question, and (2) commits the fallacy of equivocation. I’m more confident of the first objection, but the second objection might well turn out to be the more interesting point. It is unclear to me, at this point, whether either objection will be successful.
It might be the case that the objections I raise point to unstated assumptions, and that the argument can be fixed simply by making explicit an unstated assumption. It has been pointed out by others that there are often various unstated assumptions in the presentation of the Monty Hall problem that are required to make the reasoning for the accepted conclusion deductively valid.
For example, one must assume that the placement of the car prior to the selection of a door by the contestant was done randomly, and that each door had an equal chance of having the car placed behind it. If, contrary to this assumption, the car was always placed behind door #1, then it would obviously be best for the contestant to always select door #1 and always stick to that initial selection.
Similarly, if the placement of the car was done in such a way that there was an 80% chance that it would be placed behind door #1, then always selecting door #1 and always sticking to door #1 would be the best policy for a contestant. Thus, in order to prove that switching is the best strategy, the assumption must be made that the placement of the car is done at random and that each of the three doors has an equal chance of having the car placed behind it.
One must also assume that there is no switching of the location of the car and goats after the initial placement of them prior to the game. One must assume that Monty Hall knows which door the car is behind. One must assume that the contestant does not have X-ray vision (like Superman) or infrared vision (to detect the body heat of the goats) or super-sensitive hearing (so that, as Jim Lippard pointed out, one could hear a goat behind one of the doors).
One must assume that Monty Hall will not lie to, or blatantly deceive, the contestant, for example by opening a door with a life-size picture of a goat that blocks the contestant’s view of the car behind the picture (although some misleading of the contestant is allowed). One must assume that God does not intervene and transform the car into a goat or vice versa. These very specific assumptions need not all be made explicit, because more general assumptions can cover a multitude of sins or, rather, preclude many odd ways of messing up the problem, so that the accepted answer will follow from the stated assumptions.
Let me start my first objection with a critique of the probability tree diagrams. The diagrams abbreviate a sequence of events. A more detailed sequence would look like this:
1. A car is placed behind one of the three doors, and a goat is placed behind each of the two other doors.
2. The contestant makes an initial selection of a door (in this case, door #1).
3. Monty Hall opens one of the other two doors, revealing a goat behind the door (in this case, door #3)
4. Monty Hall offers the contestant the option to switch to the other remaining door (in this case, to door #2).
5. The contestant makes a final selection of a door (in this case, choosing between door #1 and door #2).
6. The door chosen by the contestant in the final selection is opened, revealing whether the car is behind that door.
7. If the car is revealed to be behind the door chosen by the contestant in the final selection, the car is then given to the contestant.
There are thus, at least seven different events that occur in temporal sequence, and thus the entire event occupies at least seven different moments or points in time. Since each of the seven events requires a measurable period of time to occur, there are at least seven periods of time here.
There is no indication of the passage of time in the probability tree diagrams. However, with the passage of time, come the possibility of new information. Assuming that the contestant is a normal human being and is conscious during each of the seven events, the contestant is constantly having experiences during the seven events, and thus is constantly receiving new information throughout the duration of the seven events. As the information available to the contestant is constantly growing, the probabilities of various events are also changing, from the point of view of the contestant.
Most people recognize that at the time the contestant makes the initial selection of a door (in this case, selecting door #1) the probability that the car is behind that door is 1/3, and many (most?) people believe that the information received by the contestant after the initial selection changes the probability that the car is behind the door that was initially selected. This is, on the face of it, in keeping with the general principle that new information can affect the probability of an event (as with my example of the prediction that it will rain tomorrow).
This suggests to me that the probability tree diagrams are ambiguous, in that it is unclear at what point the probability of 1/3 is being assigned to the statement that “The car is behind door #1″. Was this probability assigned prior to the initial selection of door #1? immediately after the initial selection of door #1? or after Monty Hall has opened up door #3 to reveal a goat behind that door?
If the information the contestant gets from Monty Hall is irrelevant to the probability of the statement “The car is behind door #1″, then I suppose it does not matter which of the above three points in time is intended, since the probability would be the same whichever point in time is intended. But it seems to me to beg the question to simply assume that the information received by the contestant when Monty Hall opens door #3 is irrelevant to the probability of the statement “The car is behind door #1″. This is the point of disagreement between the many who are inclined to say that the probability of winning by sticking with door #1 changes from 1/3 to 1/2, and the few who insist that the probability of winning by sticking with door #1 starts out as 1/3 and remains 1/3 even after Monty Hall has revealed a goat behind door #3.
In other words, since the disagreement appears to be over whether the information received by the contestant when Monty Hall opens door #3 is relevant to, or has an impact on, the probability of the statement “The car is behind door #1″, it is incumbent upon a defender of the accepted answer to the Monty Hall problem to show that this information is irrelevant or has no impact on the probability of the statement “The car is behind door #1″.
Since the probability tree diagrams make no reference to the passing of time, and fail to distinguish between probability assessments made at different points in time during these events, I don’t see how the diagram can possibly address the main question at issue. I suppose a verbal explanation of the diagram could provide the missing temporal aspect of this problem, but such an explanation, I believe, would show the diagram to be ambiguous.
Now to address the verbal reasoning in support of the accepted answer. I thought I was going to object to the validity of the logic of the argument, but on a closer look, my objection seems to focus on a specific premise:
6. If the contestant sticks with the door that was initially selected, then the probability of the car being behind the finally selected door is equal to the probability of having initially selected the door with the car.
This premise appears to bridge the gap in time between the initial selection of a door by the contestant, and the final selection of the door by the contestant. What assumption warrants the bridging of this period of time? the idea that the probability remains stable through the period of time in question?
Is the assumption that the contestant receives no information during that period of time? That would be a bizarre assumption, and it would contradict any straightforward reading of the problem (since the contestant at the very least needs to hear Monty Hall offer the opportunity to switch to the remaining other door). Is the assumption that the new information received by the contestant–during the period of time between the initial selection and the final selection–is irrelevant to the probability of the statement “The car is behind door #1″? In that case, premise (6) begs the main question at issue.
It might well be the case that (6) is true, and I think I know how to defend the truth of (6), but as the argument stands, without further elaboration and justification, it appears to me to commit the fallacy of begging the question.