structural:scholar_edition, structural:summarization
Criticism of the simple solutions

As already remarked, most sources in the topic of probability, including many introductory probability textbooks, solve the problem by showing the conditional probabilities that the car is behind door 1 and door 2 are 1 / 3 and 2 / 3 (not 1 / 2 and 1 / 2) given that the contestant initially picks door 1 and the host opens door 3; various ways to derive and understand this result were given in the previous subsections.

Among these sources are several that explicitly criticize the popularly presented "simple" solutions, saying these solutions are "correct but... shaky", or do not "address the problem posed", or are "incomplete", or are "unconvincing and misleading", or are (most bluntly) "false".

Sasha Volokh (2015) wrote that "any explanation that says something like 'the probability of door 1 was 1 / 3, and nothing can change that...' is automatically fishy: probabilities are expressions of our ignorance about the world, and new information can change the extent of our ignorance."

Some say that these solutions answer a slightly different question – one phrasing is "you have to announce before a door has been opened whether you plan to switch".

The simple solutions show in various ways that a contestant who is determined to switch will win the car with probability 2 / 3, and hence that switching is the winning strategy, if the player has to choose in advance between "always switching", and "always staying". However, the probability of winning by always switching is a logically distinct concept from the probability of winning by switching given that the player has picked door 1 and the host has opened door 3. As one source says, "the distinction between seems to confound many". The fact that these are different can be shown by varying the problem so that these two probabilities have different numeric values. For example, assume the contestant knows that Monty does not open the second door randomly among all legal alternatives but instead, when given an opportunity to choose between two losing doors, Monty will open the one on the right. In this situation, the following two questions have different answers:

The answer to the first question is 2 / 3, as is shown correctly by the "simple" solutions. But the answer to the second question is now different: the conditional probability the car is behind door 1 or door 2 given the host has opened door 3 (the door on the right) is 1 / 2. This is because Monty's preference for rightmost doors means that he opens door 3 if the car is behind door 1 (which it is originally with probability 1 / 3) or if the car is behind door 2 (also originally with probability 1 / 3). For this variation, the two questions yield different answers. This is partially because the assumed condition of the second question (that the host opens door 3) would only occur in this variant with probability 2 / 3. However, as long as the initial probability the car is behind each door is 1 / 3, it is never to the contestant's disadvantage to switch, as the conditional probability of winning by switching is always at least 1 / 2.

In Morgan et al., four university professors published an article in The American Statistician claiming that Savant gave the correct advice but the wrong argument. They believed the question asked for the chance of the car behind door 2 given the player's initial choice of door 1 and the game host opening door 3, and they showed this chance was anything between 1 / 2 and 1 depending on the host's decision process given the choice. Only when the decision is completely randomized is the chance 2 / 3.

In an invited comment and in subsequent letters to the editor, Morgan et al were supported by some writers, criticized by others; in each case a response by Morgan et al is published alongside the letter or comment in The American Statistician. In particular, Savant defended herself vigorously. Morgan et al complained in their response to Savant that Savant still had not actually responded to their own main point. Later in their response to Hogbin and Nijdam, they did agree that it was natural to suppose that the host chooses a door to open completely at random when he does have a choice, and hence that the conditional probability of winning by switching (i.e., conditional given the situation the player is in when he has to make his choice) has the same value, 2 / 3, as the unconditional probability of winning by switching (i.e., averaged over all possible situations). This equality was already emphasized by Bell (1992), who suggested that Morgan et al ' s mathematically-involved solution would appeal only to statisticians, whereas the equivalence of the conditional and unconditional solutions in the case of symmetry was intuitively obvious.

There is disagreement in the literature regarding whether Savant's formulation of the problem, as presented in Parade, is asking the first or second question, and whether this difference is significant. Behrends concludes that "One must consider the matter with care to see that both analyses are correct", which is not to say that they are the same. Several critics of the paper by Morgan et al., whose contributions were published along with the original paper, criticized the authors for altering Savant's wording and misinterpreting her intention. One discussant (William Bell) considered it a matter of taste whether one explicitly mentions that (by the standard conditions) which door is opened by the host is independent of whether one should want to switch.

Among the simple solutions, the "combined doors solution" comes closest to a conditional solution, as we saw in the discussion of methods using the concept of odds and Bayes' theorem. It is based on the deeply rooted intuition that revealing information that is already known does not affect probabilities. But, knowing that the host can open one of the two unchosen doors to show a goat does not mean that opening a specific door would not affect the probability that the car is behind the door chosen initially. The point is, though we know in advance that the host will open a door and reveal a goat, we do not know which door he will open. If the host chooses uniformly at random between doors hiding a goat (as is the case in the standard interpretation), this probability indeed remains unchanged, but if the host can choose non-randomly between such doors, then the specific door that the host opens reveals additional information. The host can always open a door revealing a goat and (in the standard interpretation of the problem) the probability that the car is behind the initially chosen door does not change, but it is not because of the former that the latter is true. Solutions based on the assertion that the host's actions cannot affect the probability that the car is behind the initially chosen appear persuasive, but the assertion is simply untrue unless both of the host's two choices are equally likely, if he has a choice. The assertion therefore needs to be justified; without justification being given, the solution is at best incomplete. It can be the case that the answer is correct but the reasoning used to justify it is defective.