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Message: <blockquote data-quote="Hawkman" data-source="post: 445578" data-attributes="member: 23715">Another way to understand the solution is to consider together the two doors initially unchosen by the player.  As <a href="https://en.wikipedia.org/wiki/Cecil_Adams" target="_blank">Cecil Adams</a> puts it, "Monty is saying in effect: you can keep your one door or you can have the other two doors". The 2/3 chance of finding the car has not been changed by the opening of one of these doors because Monty, knowing the location of the car, is certain to reveal a goat. The player's choice after the host opens a door is no different from if the host offered the player the option to switch from the original chosen door to the set of both remaining doors. The switch in this case clearly gives the player a 2/3 probability of choosing the car.<a href="https://en.wikipedia.org/wiki/File:Monty_closed_doors.svg" target="_blank"><img src="https://thumb.wikimedia.org/wikipedia/commons/thumb/7/79/Monty_closed_doors.svg/250px-Monty_closed_doors.svg.png?utm_source=en.wikipedia.org&utm_campaign=parser&utm_content=thumbnail" alt="" class="fr-fic fr-dii fr-draggable " style="" /></a>Car has a 1/3 chance of being behind the player's pick and a 2/3 chance of being behind one of the other two doors.<a href="https://en.wikipedia.org/wiki/File:Monty_open_door_chances.svg" target="_blank"><img src="https://thumb.wikimedia.org/wikipedia/commons/thumb/9/9e/Monty_open_door_chances.svg/250px-Monty_open_door_chances.svg.png?utm_source=en.wikipedia.org&utm_campaign=parser&utm_content=thumbnail" alt="" class="fr-fic fr-dii fr-draggable " style="" /></a>The host opens a door, the odds for the two sets don't change but the odds become 0 for the open door and 2/3 for the closed door.As <a href="https://en.wikipedia.org/wiki/Keith_Devlin" target="_blank">Keith Devlin</a> says, "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.'"Savant suggests that the solution will be more intuitive with 1,000,000 doors rather than 3.  In this case, there are 999,999 doors with goats behind them and one door with a prize. After the player picks a door, the host opens 999,998 of the remaining doors. On average, in 999,999 times out of 1,000,000, the remaining door will contain the prize. Intuitively, the player should ask how likely it is that, given a million doors, they managed to pick the right one initially. Stibel et al. proposed that working memory demand is taxed during the Monty Hall problem and that this forces people to "collapse" their choices into two equally probable options. They report that when the number of options is increased to more than 7 people tend to switch more often; however, most contestants still incorrectly judge the probability of success to be 50%.</blockquote>

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