Assertion: door's host | /r/CapableOf | open door
Facets:
value: always
type: TEMPORAL
value: given chose door
type: OTHER
value: to reveal never the car
type: PURPOSE
value: randomly
type: MANNER
value: to reveal a goat
type: PURPOSE
Source sentences:
text: On those occasions when the host opens Door 2,
text: You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.
text: If the car is behind door 1 the host can open either door 2 or door 3, so the probability the car is behind door 1 AND the host opens door 3 is 1/3 × 1/2 = 1/6.
text: On those occasions when the host opens Door 3,
text: The host must always open a door that was not picked by the contestant (Mueser and Granberg 1999).
text: The host must always open a door to reveal a goat and never the car.
text: This would be true if the host opens a door randomly, but that is not the case; the door opened depends on the player's initial choice, so the assumption of independence does not hold.
text: Before the host opens a door there is a 1/3 probability the car is behind each door.
text: The host opens a door, the odds for the two sets don't change but the odds move to 0 for the open door and 2 / 3 for the closed door.
text: 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.
text: 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.
text: If we assume that 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.
text: The answer follows if the car is placed randomly behind any door, the host must open a door revealing a goat regardless of the player's initial choice and, if two doors are available, chooses which one to open randomly (Mueser and Granberg, 1999).
text: Adams did say the Parade version left critical constraints unstated, and without those constraints, the chances of winning by switching were not necessarily two out of three (e.g., it was not reasonable to assume the host always opens a door).
text: If the car is behind door 1 the host can open either door 2 or door 3, so the probability the car is behind door 1 AND the host opens door 3 is 1/3 × 1/2 = 1/6.
text: If the car is behind door 2 (and the player has picked door 1) the host must open door 3, so the probability the car is behind door 2 AND the host opens door
text: The key is that if the car is behind door 2 the host must open door 3, but if the car is behind door 1 the host can open either door.
text: As already remarked, most sources in the field 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.
text: 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.
text: What is the probability of winning the car given the player has picked door 1 and the host has opened door 3?
text: 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 that the player initially chose door 1 and the host opened door 3, is 2/3.
text: 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 that the player initially chose door 1 and the host opened door 3, is 2/3.
text: You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.
text: The version of the Monty Hall problem published in Parade in 1990 did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car.
text: The key is that if the car is behind door 2 the host must open door 3, but if the car is behind door 1 the host can open either door.