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===Solutions using conditional probability===
The [[Monty Hall Problem#Simple Solutions|simple solutions]] above show that a player with a strategy of switching wins the car with overall probability 2/3
====Refining the simple solution====
▲The [[Monty Hall Problem#Simple Solutions|simple solutions]] above show that a player with a strategy of switching wins the car with overall probability 2/3, i.e., without taking account of which door was opened by the host ([[#refGrinsteadandSnell2006|Grinstead and Snell 2006:137–138]] [[#refCarlton2005|Carlton 2005]]). In contrast most sources in the field of [[probability]] calculate the [[conditional probabilities]] that the car is behind door 1 and door 2 are 1/3 and 2/3 given the contestant initially picks door 1 and the host opens door 3 ({{Harvtxt|Selvin|1975b}}, [[#refMorganetal1991|Morgan et al. 1991]], [[#refChun1991|Chun 1991]], [[#refGillman1992|Gillman 1992]], [[#refCarlton2005|Carlton 2005]], [[#refGrinsteadandSnell2006|Grinstead and Snell 2006:137–138]], [[#refLucasetal2009|Lucas et al. 2009]]). The solutions in this section consider just those cases in which the player picked door 1 and the host opened door 3.
If we assume the host opens a door at random, when given a choice, then which door the host opens gives us no information at all as to whether or not the car is behind door 1. In the simple solutions, we already observed that the probability that the car is behind door 1, the door initially chosen by the player, is initially 1/3. Moreover, the host is certainly going to open ''a'' (different) door, so opening ''a'' door (''which'' door unspecified) does not change this. 1/3 must be the average probability that the car is behind door 1 given the host picked door 2 and given the host picked door 3 because these are the only two possibilities. But these two probabilities are the same. Therefore they are both equal to 1/3 ([[#refMorganetal1991|Morgan et al. 1991]]). This shows that the chance that the car is behind door 1 given that the player initially chose this door and given that the host opened door 3 is 1/3, and it follows that the chance that the car is behind door 2 given the player initially chose door 1 and the host opened door 3 is 2/3. The analysis also shows that the overall success rate of 2/3, achieved by ''always switching'', cannot be improved, and underlines what already may well have been intuitively obvious: the choice facing the player is that between the door initially chosen, and the other door left closed by the host, the specific numbers on these doors are irrelevant.
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