I'm so, so done with this now that I am actually going to render inoperable my only HN account so I cannot possibly come back to this, or any other thread.

But I am going to leave this to someone else to explain. I'm tired out now.

A goat behind a door the contestant did not choose is eliminated. That is all that matters. It could have been done by a Heath-Robinson machine, or a passing lonely shrubber, or elves.

Monty isn't a floating variable in the puzzle who makes choices. His choice is fixed. Which I imagine is why vos Savant adds the actually extraneous information that Monty is fully aware what is going on behind the scenes -- to underscore the concept that Monty isn't a variable.

The fact that a goat behind an unchosen door was revealed is what is crucial to the setup of the entire puzzle.

And that -- despite jncfhnb's protestations -- is information that makes the puzzle determinate.

https://www.youtube.com/watch?v=AWvefaN8USk

Enjoy.

Monty isn't a contestant. Monty's action and outcome is part of the fixed description of the problem. He opens one of the doors and reveals a goat. Full stop.

If you think revealing one of the two goats (the other of which might or might not be behind your current selection) isn't information of value in assessing whether your chances in that scenario are improved by switching, I'd encourage you to consider why.

And now I really must leave it to someone else to help you, if they will. But I very much appreciate your politeness.

Yes. It is. It is a fixed part of the scenario. Monty opens a door and shows you a goat. He knows it is going to be a goat (he is "well-aware of what is going on behind the scenes"). He's showing you a goat as part of the problem which is: should you switch?

Again: think through what the problem actually SAYS:

> Imagine that you’re on a television game show and the host presents you with three closed doors. Behind one of them, sits a sparkling, brand-new Lincoln Continental; behind the other two, are smelly old goats. The host implores you to pick a door, and you select door #1. Then, the host, who is well-aware of what’s going on behind the scenes, opens door #3, revealing one of the goats.

> “Now,” he says, turning toward you, “do you want to keep door #1, or do you want to switch to door #2?”

No matter what: the goat Monty shows you is not a matter of chance. It is a fixed part of the problem. It's there in writing: he shows you a goat. Full stop.

There's still a goat you could show me. And it is a fact that you show me a goat. Nowhere in the problem does it suggest there is a chance you show me a goat.

I do honestly admire your dogged commitment, and I think the way you are committed shows up an important point about the article and the history of the problem.

Which is that one can quite clearly fairly argue the point, as you are doing, without resorting to misogynistic or patronising rudeness as so many did at the time!

But you're still wrong. :-)

And I'm going to leave it here.

What do you mean?

It's specified in the scenario. He shows you a goat. It's right there. This isn't a variable. It's a fact.

Your only job is to work out whether, given the scenario described, it makes sense to switch. Given all your possible choices.

This is precisely where you are wrong.

It matters what he did in the problem as it is described. Because that is what you're solving.

It doesn't matter. Because the problem explains what happens: he opens a door and reveals a goat.

That is crucial information. There's now only one goat and one car left. But the choices have not been shuffled: you're definitely still pointing at your original choice.

What were the odds that your original choice is a goat? Two in three.

If you picked a goat, what is guaranteed to be behind the other door? A car.

So what are the odds that behind the other door is a car? Two in three.

You should switch.

Showing you the goat is the event that, as you say, guarantees that the other door has the opposite of your original choice behind it. Because nobody closed the curtain to shuffle the choices.

One of the things I think people struggle with -- and I struggle with -- is that probability isn't about hypothesising about a single event that happened and how it might have happened. It's about encapsulating all the possible ways a single specified scenario can play out in a single expression.

Monty shows you a goat this time, but this means Monty always shows you a goat. There's no scenario where he is unable to show you a goat. Just like you always only pick one door. And there's always only two goats and one car.

(Full marks for username choice)

That is:

- the setup of the system matters.

- The state of the system at the point of the decision to switch matters.

- The choices don’t get re-randomised.

So the probabilities assigned to the original choice (and the remaining alternative) still count.

If the host had closed a curtain over the stage and randomised the remaining doors, then it would be 50:50.

But he didn’t. So you’re still in the probabilities of the original choice.

One of the goats has been removed. The car and one goat remain: you know this for sure.

You are being offered a door knowing that behind it must, necessarily, be the opposite of your original choice, and the probabilities have not been reset.

If you originally picked the goat, that door absolutely has a car behind it. And there's a 2/3 chance you picked the goat originally. So by inference there's a 2/3 chance the door has a car behind it. You should switch.

I didn’t get it until I had written a simulation to see it for myself though!

The only problem you are being asked to solve is the one described only in the words of the problem.

That's actually what this is all about. This is a learning exercise, not just about probability, but about comprehension of what the problem as described exactly says.

That's what makes it so enormously valuable.

Nobody is saying she was criticised for being a woman, though. What they are observing is that it was the basis of and the tenor of attacks on her character, intelligence, certainty etc.

It was nasty and it really happened, is all I'm saying. Are we better? I hope so. But I don't know that I think so.

People who get it wrong are usually projecting some implied understanding onto it, which is why they get it wrong.

Which is the point of the problem. It's designed to reveal this tendency in analytical thinking leading to unexpected outcomes. You get it wrong, you're gobsmacked, you understand why, you gain some enlightenment from it. It's fun to be wrong in ways you later understand. (This is one of the thrills of programming: that moment during debugging when you shriek with joy: "YES, IT BROKE!")

The fact that it also exposes the way some overly confident people lash out with anger that exposes other biases is the point of the article, I think.

It was years before I understood the actual lesson here -- both the probability lesson and also the way it hints at the fundamental truth that in life, binary choices may have been weighted in ways you don't understand, perhaps by the people asking you to make the choice.