The Monty Hall question and Marilyn vos Savant

Just got directed to an interesting article by Priceonomics about The Monty Hall question and Marilyn vos Savant

Curious one, this one, and a lot relies on unspoken semantics.

The lady is question, once considered the smartest woman in the world, had a magazine column, answering puzzles et cetera, often for academics.

One day someone posed the following question, and chaos ensued:

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

What do you think? Would you swap, or doesn’t it matter?

Think about it a minute. Take away one goat, what’s left?

Ms Savant said you are 66% better off swapping! Pttt!

Naturally she got over 10,000 irate letters from top scientists, physicists, mathematicians and sundry other academics who all called her an idiot, or worse.

The arguments, counter-arguments and name calling went on for YEARS!

Touchy lot, pedants. I get like that when I see grammatical errors on Facebook. So many errors (*sobs*), but I digress.

Clearly, absolutely, beyond any doubt if you know 100% guaranteed that the goat is behind one door, with the car behind the other, it is a 50:50 chance, cannot be anything else. The academics were right, the world’s smartest woman was wrong, and that on a simple stats question.

Here, look:

[Goat OR Car] [Goat OR Car] [Goat OR Car]
so (1/3) chance


[Goat OR Car] [Goat OR Car]
which is now a 1/2 chance (or, technically, still 1/3 OR 1/3), but you only have to look, it HAS to be 50:50

But, here’s where it gets squirrely.

Here’s the world’s smartest woman (maybe) says ‘Nope’, and the rest of the (academic) world says, ‘You muppet’, so naturally I’m going to change my mind and agree with her, ‘cos, reasons.

Actually, at first I did say 50:50, and I stand by my 50:50, except, umm, no, it depends. If you have an INTP friend or relative, you’ll be used to the funny look we give when we say, ‘it depends…’

As I said at the start, it’s all about semantics. You have to look deeper: the answer is iterative.

Took me a few seconds to work out that she was in fact right saying you are better off swapping, so it begs the question why 10,000 or so of the smartest people in America etc took over a decade to agree. Those that did, at any rate; apparently some still cannot see it. Why not? It’s simple!

Remember, the question wasn’t simply ‘what is best NOW?’, it remains more enigmatic: ‘what is best?’

Let’s look at the question and choices again:

Three doors
Two goats
One car

You pick one door.
Another door is opened – and a goat is revealed, leaving a car, or a goat
You can swap your door, but should you?

So, as a grid, the potential options are:

Car | Goat | Goat
Goat | Car | Goat
Goat | Goat | Car

Doesn’t matter how many times look at it, if you KNOW one of the goats is gone, the choice HAS to be a goat or a car.

BUT you are limited to one door. See it yet? You have a 1 in 3 chance of it being a car, 2/3 chance of being a goat.

But, we may argue, you know that one door reveals a goat, the other a car. QED, it makes no difference!

Aha! But it does, because:

[Door 1: Goat OR Car] [Door 2: Goat OR Car] [Door 3: Goat OR Car]
[Door 1: Goat OR Car] [Door 2: Goat OR Car] [Door 3: Goat OR Car]
[Door 1: Goat OR Car] [Door 2: Goat OR Car] [Door 3: Goat OR Car]

Becomes, for instance:

[Door 1: Car] [Door 2 (or 3): Goat OR (not Goat)]
[Door 1: Goat] [Door 2 (or 3):Car (or not Goat)]
[Door 1: Goat] [Door 2 (or 3):(not Goat) or Car]

Simplifying it further:


Which, focused on the car, becomes C[CC]

Seen like that, given the possible combinations, if you pick door #1 you will always have a 33% of it being in door #1, but a 66% chance of it being in door #2 (OR door #3). Whether it is 2 or 3 is irrelevant, statistically. Same applies if your door is 2 or 3.


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