
The Monty Hall Problem: A Statistical Illusion - garyclarke27
https://statisticsbyjim.com/fun/monty-hall-problem/
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herodotus
The intuitive explanation I like is this: Suppose there are 10000 boxes. You
pick one. Monty Hall knows where the prize is. He opens 9998 boxes, leaving
just two closed: your original choice, and one other one. Should you switch?
Before Monty Hall opened any boxes, the chances of the prize being in one of
the boxes other than the one you chose was 9999 in 10000. This probability
does not change when he opens 9998 of these. So you should choose the
remaining box, not your original choice.

The same argument applies with 3 boxes, but intuition is much less reliable in
this case.

~~~
justinsaccount
I always found that one makes things more complicated. I like this explanation
better:

You win by switching as long as you were wrong in the initial guess. You have
a 2/3 chance of being wrong in the initial guess. Therefore you will win by
switching 2 times out of 3.

Can also phrase it in reverse: You lose by switching if you were right in the
initial guess. You have a 1/3 chance of being right in the initial guess.
Therefore you lose by switching 1 time out of 3.

~~~
indigodaddy
This was the key for me as well in having a real understanding of it.

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bhk
This problem is confusing partly because the premises are often incorrectly
stated... as in TFA.

If it is given that Monty _always_ picks a door that does not have a prize
behind it, then indeed the correct answer is to switch doors. But otherwise,
there is no advantage in switching: 1/3 of the time he would expose the prize,
and in the other 2/3 of cases you are just as likely to win without switching.

~~~
tromp
For an article focussing on assumptions, it's rather sloppy to state the
situation as

"Monty opens one of the other two doors, and there is no prize behind it."

rather than

"Monty opens one of the other two doors, one which he knows has no prize
behind it."

which rules out the interpretation of Monty opening a door that by chance
happens to have no prize behind it, leading to a different answer.

~~~
BoiledCabbage
I'm not convinced him knowing where the prize is before opening the door
matters. If he opens a door and knew there was no prize you now get the same
info as if he opened the door and was surprised there was no prize. In both
cases switching its better. And if he unwittingly opens the door and there is
the prize, again switching is better.

His knowledge before hand doesn't matter, switching its always better.

That said, I feel like there is a limitation in our modeling of probability
that this problem isn't easier. Yes anyone who knows stats can solve it, but
somehow we're using the wrong "language"that the problem is still tricky.
Geometry isn't tricky becsuse we have the right language for it. Geometry is
"hard" when problems are hard/complex. Monty Hall the problem isn't complex,
but solving it correctly is still difficult.

To take a programming term, probability applied to this problemfeels like it
has accidental complexity.

~~~
teahat
What if Monty only opens a door when he knows you've chosen the prize door?

Extension: what if Monty tosses a coin to decide whether to open a door,
conditioned on you having chosen the prize door?

Specifying Monty's behaviour matters, I think.

~~~
mojomark
Monti's a priori knowledge absolutely matters and should have been stated in
the article. If he had no knowledge beforehand about which door the prize was
behind, then you would have to expand the probability table to include cases
in which Monti opens one of the remaining two doors and accidentally reveals
the prize, in which case the user just loses without getting the opportunity
to switch or not switch. Monti knowing the door is hiding the prize and
actively selecting the non-prize door versus just guessing and sometimes
accidentally revealing the prize, are two different statistics problems.

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dang
Many previous threads:
[https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...](https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=%22monty%20hall%22%20comments%3E10&sort=byDate&type=story)

(Sharing for curiosity purposes.)

