I have read through around 10 articles and webpages on this problem, and still don’t understand. I’ve run simulations and yes, switching does get you better odds, but why?
In: Mathematics
Let’s say they don’t open any doors. They just say “Do you want to switch doors? If you choose to switch, we’ll switch to the right door! Well, unless you already got it right, of course, because then there won’t be any right door to switch you to.”
You’ll probably say “Sounds great! In that case, I’m probably going to benefit from switching.”
And they might ask “Want us to tell us which door that is? Which door is the right one, unless you already got it right?”
And you’ll probably say “I guess so, but it doesn’t really matter, does it? It’ll only tell me which door *would* be the right one if I picked the wrong one to start with. It won’t tell me what the odds are that I did pick the wrong one.”
They open one door, making sure to pick one that you didn’t choose and that is not the winner door. “Now that we’ve made it clear which door is the winner if yours isn’t, wanna switch?” they ask.
You might reply “Sure, but it doesn’t matter that you opened the door, because the odds that I picked the right one isn’t dependent on *which* other door might be the right one.”
**You gotta remember:** A lot of people will insist that if you pick randomly between the door you picked to start with and the only other remaining door, then the odds are 50%, right? And the answer is yes, but that’s not what the question is about. It’s about what the odds are if you *know* that you picked one specific door to start with, and you also *know* that the other door is guaranteed to be right if your first door was wrong.
**Shorter explanation:**
When should you switch?
When you picked the wrong door to start with.
What are the odds that you picked the wrong door to start with?
66%.
So the odds that switching works are also 66%, right?
Of course.
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