Please read Re^7: Marilyn Vos Savant's Monty Hall problem carefully. You'll find that it is not enough to say how much knowledge Mr. Hall has - to make the answer unambiguous you also need to specify how Mr. Hall will use that knowledge.
On the spooky math problem, there is no divide by an infinite number problem, but there is a lot of subtle behaviour with infinity lurking around. I strongly recommend against coming to any opinion without thinking it through very carefully. It took me about a week to really understand what was happening when I first ran into the problem. (It didn't help that Laurie let me get myself well and truly convinced that there was no way that you could beat 50% before he gave me the answer and told me to figure out for myself why it worked...)
Given any pair of numbers in the envelopes, and given the distribution from which you pick a random number, your probability of winning is well-defined and will be bigger than 50%. Depending on those three factors, the actual odds can be anywhere between (but not including) 50% and 100%. But you are not provided with sufficient information to figure out what the odds are. | [reply] |
Please read Re^4: Marilyn Vos Savant's Monty Hall problem carefully, especially where I wrote the host knows and will not open the door with the car (emphasis mine) The knowledge Mr. Hall has and how he uses it is unambiguous. He always opens a door, he never opens the door with the car. and he always offers a switch. What's more important than the information he has is the information he provides by using what he knows in a predictable way.
I fully understand your previous node, and actually we are in agreement about the necessary information to calculate the probablilities. We just differ on what the original problem statement is and/or what assumptions can be safely made for this specific problem.
Re the spooky math problem: what I wrote were my initial thoughts when I read that root node. I'll reserve judgement and not argue either side until I investigate further (depends on life, kids, the missus etc. if I ever get to it). From what I've seen, if you are this sure you are right then you probably are. I've seen stranger things with my own eyes, so I won't rule out any possibility.
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Please read Re^4: Marilyn Vos Savant's Monty Hall problem carefully, especially where I wrote
the host knows and will not open the door with the car (emphasis mine)
The knowledge Mr. Hall has and how he uses it is unambiguous. He always opens a door, he never opens the door with the car. and he always offers a switch. What's more important than the information he has is the information he provides by using what he knows in a predictable way.
Your memory is fooling you.
As is pointed out at Re^4: Marilyn Vos Savant's Monty Hall problem, the original statement of the problem when Marilyn Vos Savant gave it was, Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?
Note that how the host uses his information is ambiguous in that problem statement.
Furthermore most of the time when this comes up for discussion, the problem is even less carefully stated than that. It is typically not even specified that the host knows where the car is. For example look at the node that started this discussion, Marilyn Vos Savant's Monty Hall problem.
Therefore no matter how much you'd like to believe that you've never seen it poorly stated, you're wrong. In fact you've seen it poorly stated at least twice already today.
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