The simple solutions above show that a player with a strategy of switching wins the car with overall probability 2 / 3 , ., without taking account of which door was opened by the host ( Grinstead and Snell 2006:137–138 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 ( Selvin (1975b) , Morgan et al. 1991 , Chun 1991 , Gillman 1992 , Carlton 2005 , Grinstead and Snell 2006:137–138 , 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.