User talk:Summers999
I'm new at contributing to Wikipedia and hoped to shed some clarity on the Monty Hall problem based on a conversation I had trying to explain it to others. Some confusion can be cleared up by pointing out the importance of Monty Hall responding to the game show contestant's selection. This knowledge and consideration of the contestant's pick underlines the difference between this scenario and a blind, random selection. I give an example of how we think of that case and how it can be confused with the Monty Hall case as a way to illuminate it.
It's by no means perfect:
The problem also arises due to the idea that the contestant is asked to make a second choice among the remaining two options, leading them to think their odds at that point are 1 in 2. The choices appear to be equal, but they are not because Monty Hall acts on the contestant's choice. His behavior relies on the contestant's intial choice. This is not the same as being offered a choice between two equivalent choices. What happens in a seemingly similar case when there are six contestants in a singing contest and one will be selected to go home? If all things are equal, each contestant puts his/her odds of going home at 1 in 6. But as the MC declares one contestant after another to be safe, the remaining contestants no longer rely on those 1 in 6 odds. Had I placed an initial bet that one of the final two contestants remaining would go home, I would no longer view the odds of my winning that bet as 1 in 6, but I'd think I had a fifty-fifty chance at that point. I wouldn't imagine that the other contestant had a 5 in 6 chance of going home that day just because I made an initial random choice of a contestant. The difference in these scenarios is that the judges and the MC have no knowledge of my bet and take no actions based on it; whereas, Monty preserves the gameshow contestant's choice when he acts.
To look at it from a slightly different angle, the contestant has a 2 in 3 chance of choosing a door with a goat. That means the remaining pair of doors have only a 1 in 3 chance of containing two goats. When Monty reveals a goat from the remaining two doors, then the other door will have a goat one-third of the time. Therefore, the contestant halves the odds of getting a goat when switching doors.
---Summers999
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