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Comment by andischo | Hacker News Reader

The easiest explanation I've ever heard and which immediately made me understand it was the following:

Instead of 3 doors, imagine there are 100. 99 of which have a goat and only one of which has a price behind it. Now blindly choose a door and the host opens 98 of the other doors which have a goat behind it. Would you switch your door now, given the choice?

It's easy to see that your probability of choosing a "wrong" door when you had 100 doors to choose from was much higher than choosing the right door when you only have two doors to choose from.

This method of thinking, i.e. increasing or decreasing the problem space by some orders of magintude has helped me a lot in thinking about problems and their solutions in general.

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This does not (at least to me) explain why a switch is interesting.

The probabilities are different but this does not make my first choice "worse"

I also do this, among other things to find subtle bugs in code and math. For every unbounded variable, what happens if it is 10^100, or 10^-100? It's a good habit that also makes you view division and logarithms with great suspicion :)

One of my first computer science teachers pronounced "/" (in C) as "divide and throw away the remainder", which sounds awkward at first but turns out to be quite helpful. It's one of those operators which, unfortunately, looks just like one from math class but acts very differently, even in common cases.

Good idea, but it's not intuitive why the original problem is equivalent to the host revealing N-2 (98) goats rather than, say, (N-2)/2 (49) or 1.

Interesting. I was explained this question the exact same way (100 doors) some years ago and it was instantly intuitive to me. Maybe it's how every brain works differently.

I understand that it makes sense intuitively, but how do you justify it to yourself mathematically? As far as I can tell there is nothing in the original formulation that explains why it would scale that way.

Sure but making it 99/100 just increased your odds significantly from 2/3.

I'm the host, and I also happen to be blind. I chose 98 doors and none of them have the car - terrible luck on my part! But probably this lends some credence to your initial guess?

That is also a very interesting way to think about it, if I understood you correctly. Seeing the host as another player, any "bad luck" he has, should translate into myself having a higher chance of success if used correctly.

With the host as another blind player, his opening of 98 goat doors only increases the probability you were right from 0.01 to 0.5, so still makes no difference for you to change. But of course the original version of the problem is predicated on the host knowing where the car is and only revealing goats.

No, if the host is acting randomly no information was revealed. It is equally likely that the car is behind your door or the other door.

The player are indifferent to switching under your scenario. Why not, I suppose?

This is correct, but based on specific assumptions. If the host flipped a coin and selected a door and 1/Nth of the time it had the car then you switching would not change your odds.

Similarly, an adversarial host who sometimes opens a door and sometimes just shows you your choice can similarly mess with the odds.

It’s really due to the host both being forced to show a door and knowing which door to choose that you gain from leveraging that knowledge.

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