Adam Kucharski

Do you want to switch doors?

And a lot of the, I think, intuition people have, and certainly when I first came across this problem many years ago, is, well, you've got two doors left, right?

You know, you've picked one, there's another one, it's 50-50.

And even some quite well-respected mathematicians, people like Paul Erdosch, who has really published more papers than almost anyone else,

That was their initial gut reaction.

But if you work through all of the combinations, if you pick this door and then the host does this and you switch or not switch and work through all of those options, you actually double your chances if you switch versus sticking with the door.

So it's something that's counterintuitive.

But I think one of the things that really struck me, and even over the years trying to explain it, is

convincing myself of the answer which was when i first came across it as a teenager i did quite quickly is very different to convincing someone else and even actually paul erdos one of his colleagues kind of showed him the what i'd call proof by exhaustion so go through every combination and that didn't really convince him so then he started to simulate and said let's do a computer simulation of the game a hundred thousand times and again you know switching was this optimal strategy but erdos wasn't um

wasn't really convinced because I accept that this is the case but I'm not really satisfied with it and I think that encapsulates for a lot of people their experience of proof and evidence it's kind of it's a fact and you kind of have to take it as given but there's actually quite a big bridge often to really understanding why it's true and feeling convinced by it.

Yeah, this was a fascinating situation that emerged in the late 19th century where a lot of maths, certainly in Europe, had been derived from geometry because of a lot of the ancient Greek influence on how we shape things.

And then Newton and his work on rates of change and calculus, it was really the natural world that provided a lot of inspiration, these kind of tangible objects, tangible movements.

And as mathematicians started to build out the theory around rates of change and

and how we tackle these kinds of situations, they sometimes took that intuition a bit too seriously.

And there was some theorems that they said were intuitively obvious, some of these French mathematicians.

And so one, for example, is this idea of kind of how things change smoothly over time and how you do those calculations.

But what happened was some mathematicians came along and showed that when you have things that can be infinitely small,

that intuition didn't necessarily hold in the same way.

They came up with these examples that broke a lot of these theorems.

A lot of the establishment at the time called these things monsters.