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
Imagine 100 doors. Pick one, now have Monty Hall open 98 others.
You gonna keep your original choice? Or are you going to switch?
They create bias with the door they select to show you. Imagine this
> The contestant has chosen door number 1. But lets look behind door number 2. Behind door number 2 is the grand prize that you can’t win any more. Do you want to keep the goat you know you have because of the process of elimination or switch to the other goat behind door 3?
That option never happens. It doesn’t happen. If you pick a goat they will always show you the other door with a goat. Its that bias that make your switch more likely to win an not 50/50
Pretend there are 100 doors with only one prize. You pick a door at random. Chances are 1:100 you picked the right door. Next, 98 doors are removed from the game (guaranteed to be without a prize).
The odds you picked the correct door are still 1%. The odds the other door has the prize is 100%-1%=99%.
Ok pick a door out of 1000 doors
How confident are you that’s the door with the prize?
Not very?
Okay now I’ll open every other mf’n door except ONE, and they’re all not the prize.
So now, either the one one you picked has the prize. Or the door I left closed does.
If I were you, there’s no way I’m assuming I picked the right door out of ONE THOUSAND doors, so I’d choose the other door.
The prize has stayed in the same place but the amount of knowledge about the conditions has increased
For someone who knew nothing and walked in and saw two doors and had no explanation it’d be 50/50. But luckily, for you it’s not.
It would be as if the host farted in front of one of two oors and said if you can point to the door I farted in front of, you can have the new Nintendo VR x Sony Sparkle Pony video game console. He pretty much just gave you a hint at which door has the prize
The host knows which door has the prize, and can never open that one. The host is not random in any way.
The way it makes sense to me is:
Switching will only get you a goat if you picked the door with the car first (1/3 chance).
If you picked either goat first, switching will get you the car (2/3 chance).
It works because the host always reveals a goat after you pick.
I think the thing that helped me understand better is this:
if you picked the RIGHT door originally, and then switch, you will end up with a bad door; that’s clear: you have the right door, thus by switching you are for sure going to a bad one. So if you chose right and switch, you end up with a bad door.
If you picked the WRONG door, and then switch, you will end up with the RIGHT door. You have a bad one, Monty eliminates the other bad one, this the only one left is the right one and if you switch, you will get it. So if you chose wrong and switch, you end up with the good door.
Now, originally there were 2 bad doors and one good door. So you had 2 or 3 odds of picking a bad door and 1 of 3 of picking the good one. Since switching will always take you to the opposite result (see above), then in 2 out of 3 cases switching results in winning and in 1 or 3 cases switching results in losing.
The most intuitive way I’ve figured this out.
You’re not the contestant. You’re the host.
The contestant picks the first door- they get it wrong 2/3 times, you then pick the other losing door and the one remaining is the winner. in those 2/3 times, they get the door right by switching. Between their wrong guess and the elimination you’ve narrowed it down to 1 door.
In the 1/3 times they guess it right, you have to pick one of the two losing doors randomly and they’ll get it wrong by switching. Showing them a losing door didn’t make the original odds higher than 1/3 because they guessed it right anyways.
The problem here is that most people look at it as a contestant rather than the host.
The odds of picking the prize door is 1/3. That means the odds of the prize being in the other doors is 2/3.
You have a 2/3 chance to pick the goat on the initial pick, but switching in this situation is a guaranteed win since the host will remove the other goat door always leaving the prize as the other door. By always switching, the time you lose is 1/3 time you actually picked right from the start. If you stick with your choice, it’s 1/3 chances to win. If you always switch, it’s 2/3 chances to win.
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