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Monty Hall problem

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The answer to the problem is yes; the chance of winning the car is doubled when the player switches to another door rather than sticking with the original choice. At the point the player is asked whether to switch there are three... See more » The answer to the problem is yes; the chance of winning the car is doubled when the player switches to another door rather than sticking with the original choice. At the point the player is asked whether to switch there are three possible situations corresponding to the player's initial choice, each with equal probability (1/3): * The player originally picked the door hiding goat number 1. The game host has shown the other goat. * The player originally picked the door hiding goat number 2. The game host has shown the other goat. * The player originally picked the door hiding the car. The game host has shown either of the two goats. If the player chooses to switch, the car is won in the first two cases. A player choosing to stay with the initial choice wins in only the third case. Since in two out of three equally likely cases switching wins, the odds of winning by switching are 2/3. In other words, a player who has a policy of always switching will win the car on average two times out of the three. Another way of thinking about this explanation is by observing that the player begins the game with a 2/3 chance of picking a door with a goat behind it. In this case the host must show the other goat, so switching turns the initial 2/3 chance of picking a goat into an equivalent chance of winning the car. Switching loses only if the player picks the car first (1/3 chance). The total expected outcome for the player who always switches is therefore a 2/3 chance of winning the car. The problem would be different if the game host were permitted to make the offer to switch more often (or only) depending on knowledge of the player's original choice or if the host does not know what is behind each door. Some statements of the problem, notably the one in Parade Magazine, do not explicitly exclude these possibilities. For example, if the game host only offers the opportunity to switch if the contestant originally chooses the car, the odds of winning by switching are 0%. In the problem as stated above, it is because the host must reveal a goat and must make the offer to switch that the player has a 2/3 chance of winning by switching. [edit] See less »

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maths, philosophy, puzzles

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Monty Hall problem

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