7 ms·

Here's how I was able to understand it. There are 100 doors, you choose 1, then host opens 98 doors with goats. Do you keep your door or take the 1 remaining?

Here's how I was able to understand it. There are 100 doors, you choose 1, then host opens 98 doors with goats. Do you keep your door or take the 1 remaining? In the original, the 3 doors muddles the odds. But with 100, you're choosing between the 1 you chose on the first try, or the 99 you didn't choose.

vitus 4y ago
Or, an even more telling variation: You pick door #1. The host goes and, in order, one by one, opens doors 2-52, conspicuously skips door 53, and then opens 54-100. Would you like to stick with your original choice?
- cgriswald 4y ago
  I think this variation makes it more intuitively clear one should switch in the case with 100 doors, but I think it makes the explanation of why less clear because of the extra, unnecessary information.
  - lupire 4y ago
    I disagree. This version more vividly explains what's happening, making it easier to visualize, similar to illustrating an algebra problem by using easy small numbers instead of juggling variables.
    - cgriswald 4y ago
      I think that's probably true for you and it's definitely true for some, but I've tutored many students in both algebra and discrete and am very familiar with the failure modes of students' thinking. The "Door 53" example makes it very easy for people to understand that in the case of 100 doors they should switch. It doesn't, however, always get them to see that the issue is fundamentally choosing between odds of 1:X and (X-1):X (or even that they should also switch with only three doors).
jwilk 4y ago
This doesn't help my intuition at all.
- quesera 4y ago
  This is how I think about it: * Your first choice had a natural chance of 1% to be correct. * After eliminating the 98 other options, a random choice of the remaining two is 50% likely to be correct. * BUT, you know that the reason the door you picked was not opened is because you picked it! Its inclusion in the set-of-two is not random. * So your first choice is isn't any more likely to be correct that it was when you first you chose it.
  - patrickdavey 4y ago
    The way I think about it (which I hope is correct): when you first chose your probability of guessing correctly was 1 in 100. It doesn't magically become 1 in 2 when the other 98 doors are opened.
    - lijogdfljk 4y ago
      Funny, i think (but am not arguing i'm correct) it does change probability when the doors change. As i explained[1] in the sister post. Perhaps this is some mathematical concept? To me i'm viewing every choice as a dice roll. I have no question that the first dice roll was worse than the 2nd. However to me the 2nd is a die with two faces. Two choices, of which both are equally possible. To look at it differently. The doors changing probability to me sounds like.. imagine two people, the PersonA has this primary scenario. They had 100 doors, chose one, and are then asked if they'd like to switch. PersonB then is asked the second question, of the 2 doors, which do they want? PersonA and B are standing next to each other. The doors are the same for both of them. Why would PersonA have different odds than PersonB? And how could PersonB's odds be any different than 50/50 with no prior history of the doors? Is there some fundamental difference between the grand idea of probability and "reality" as i'm trying to describe it here? [1]: https://news.ycombinator.com/item?id=31535557 https://news.ycombinator.com/item?id=31535557

vitus 4y ago

Or, an even more telling variation: You pick door #1. The host goes and, in order, one by one, opens doors 2-52, conspicuously skips door 53, and then opens 54-100. Would you like to stick with your original choice?

cgriswald 4y ago
I think this variation makes it more intuitively clear one should switch in the case with 100 doors, but I think it makes the explanation of why less clear because of the extra, unnecessary information.
- lupire 4y ago
  I disagree. This version more vividly explains what's happening, making it easier to visualize, similar to illustrating an algebra problem by using easy small numbers instead of juggling variables.
  - cgriswald 4y ago
    I think that's probably true for you and it's definitely true for some, but I've tutored many students in both algebra and discrete and am very familiar with the failure modes of students' thinking. The "Door 53" example makes it very easy for people to understand that in the case of 100 doors they should switch. It doesn't, however, always get them to see that the issue is fundamentally choosing between odds of 1:X and (X-1):X (or even that they should also switch with only three doors).

jwilk 4y ago

This doesn't help my intuition at all.

quesera 4y ago
This is how I think about it: * Your first choice had a natural chance of 1% to be correct. * After eliminating the 98 other options, a random choice of the remaining two is 50% likely to be correct. * BUT, you know that the reason the door you picked was not opened is because you picked it! Its inclusion in the set-of-two is not random. * So your first choice is isn't any more likely to be correct that it was when you first you chose it.
- patrickdavey 4y ago
  The way I think about it (which I hope is correct): when you first chose your probability of guessing correctly was 1 in 100. It doesn't magically become 1 in 2 when the other 98 doors are opened.
  - lijogdfljk 4y ago
    Funny, i think (but am not arguing i'm correct) it does change probability when the doors change. As i explained[1] in the sister post. Perhaps this is some mathematical concept? To me i'm viewing every choice as a dice roll. I have no question that the first dice roll was worse than the 2nd. However to me the 2nd is a die with two faces. Two choices, of which both are equally possible. To look at it differently. The doors changing probability to me sounds like.. imagine two people, the PersonA has this primary scenario. They had 100 doors, chose one, and are then asked if they'd like to switch. PersonB then is asked the second question, of the 2 doors, which do they want? PersonA and B are standing next to each other. The doors are the same for both of them. Why would PersonA have different odds than PersonB? And how could PersonB's odds be any different than 50/50 with no prior history of the doors? Is there some fundamental difference between the grand idea of probability and "reality" as i'm trying to describe it here? [1]: https://news.ycombinator.com/item?id=31535557 https://news.ycombinator.com/item?id=31535557