A game show host always plays the following game: First he shows you 3 doors and informs you there is a prize behind one of them. After allowing you to select one of the doors, he throws open one of the other doors, showing you that it’s empty. He then offers you a deal: Stick to your original guess, or switch to the remaining door?
What is the most important piece of information in this problem statement? I claim that the bit that ought to shock you is that the host plays this game all the time, and the door he throws open ALWAYS turns out to be empty. Think about it: If the host randomly throws open a door, then in every third show, the door he opens would have the prize behind it. That would ruin the game!
The host knows which door has the prize, and in order not to lose the interest of the spectators, he deliberately opens an empty door every time. What this means is that the door you chose was selected randomly, but the door that the host DIDN’T choose is selected on the basis of a predictable algorithm. Namely, having the prize behind it.
This is the real reason why you would do better if you switched your guess to the remaining door.
What do you think? Is that clearer than the usual explanations?
Yeah, I think it’s better. It highlights the flow of knowledge: where the prize is → host’s knowledge → which door he opens → player’s knowledge.
I’d maybe change the phrase “predictable algorithm”, since the host’s actions aren’t predictable to the player. Maybe
but the door that the host DIDN’T choose is selected on the basis of a predictable algorithm. Namely, having the prize behind it.
could be replaced by
but the door that the host DIDN’T choose is selected on the basis of the host’s knowledge of where the prize is. His choice can therefore give you information about where the prize might be: namely, it’s more likely to be the door he avoided.
Thanks. You’re right, that part should be expanded. How about:
At this point, you have two choices: Either 1. one randomly selected door, or 2. one door among two doors, chosen by the host on the basis of the other not having the prize.
You would have better luck with option 2 because choosing that door is as good as opening two randomly selected doors. That is twice as good as opening one randomly selected door as in option 1.
A better explanation of the Monty Hall problem:
A game show host always plays the following game: First he shows you 3 doors and informs you there is a prize behind one of them. After allowing you to select one of the doors, he throws open one of the other doors, showing you that it’s empty. He then offers you a deal: Stick to your original guess, or switch to the remaining door?
What is the most important piece of information in this problem statement? I claim that the bit that ought to shock you is that the host plays this game all the time, and the door he throws open ALWAYS turns out to be empty. Think about it: If the host randomly throws open a door, then in every third show, the door he opens would have the prize behind it. That would ruin the game!
The host knows which door has the prize, and in order not to lose the interest of the spectators, he deliberately opens an empty door every time. What this means is that the door you chose was selected randomly, but the door that the host DIDN’T choose is selected on the basis of a predictable algorithm. Namely, having the prize behind it.
This is the real reason why you would do better if you switched your guess to the remaining door.
What do you think? Is that clearer than the usual explanations?
Yeah, I think it’s better. It highlights the flow of knowledge: where the prize is → host’s knowledge → which door he opens → player’s knowledge.
I’d maybe change the phrase “predictable algorithm”, since the host’s actions aren’t predictable to the player. Maybe
could be replaced by
or something similar?
Thanks. You’re right, that part should be expanded. How about:
At this point, you have two choices: Either 1. one randomly selected door, or 2. one door among two doors, chosen by the host on the basis of the other not having the prize.
You would have better luck with option 2 because choosing that door is as good as opening two randomly selected doors. That is twice as good as opening one randomly selected door as in option 1.
Yeah, I like that.