In the lead-up to Christmas, most of my readers move off to fun places that usually involve either beaches or snowboarding, alongside many new instagram pictures and much facebook envy. So I always feel like Christmas-week is a freebie for me, where I don’t have to be as current (unless something dramatic happens), and I can write about some of the stuff that I’ve been thinking about in my free time.

One of those things is “math” – and, specifically, how much it can tell us about the accuracy of our gut feelings and intuition. Personally, I find this whole side of human nature to be particularly interesting, as most economic theory is built on the premise of “rational” people. Which isn’t to say that people have to be rational all of the time – just that we generally have to be rational. As a group. For the most part.

Except that, as a group, we’re forced to mostly rely on intuitions and gut feelings. And my favourite counter-argument to this is an old probability brainteaser (it’s often called the “Monty Hall” problem):

- There are three doors.
- There is a prize behind one of them.
- You pick one.
- I then open one of the doors that has nothing behind it.
- Question: should you stick with your original choice, or should you switch?

Every time I read this, my gut feeling tells me that it should make no difference. Before, there were three doors. Now, there are two. I’ve picked one of those two. My odds may have been two-to-one when there are three doors – but now, they’re fifty-fifty that I picked the right door.

And when someone says to me “No, the right answer is that you must always switch”, my gut feelings knot up in a kind of febrile rage. Because *how*. And *how dare you*. There are two doors. The odds are fifty-fifty that I’m picking the right one – how can you be so damn prescriptive? Fool*.

*Apparently, my mathematical ego is a megalomaniac that cannot be dealing with criticism. Awkward.

The trouble is, *even *when I draw out the scenarios, and *prove *that switching is the right answer, my gut feelings remain unconvinced. Mostly, they conclude that all probability math must be hokum. Even when Mythbusters proved it empirically.

https://www.youtube.com/watch?v=FAljAvR3L4s

I even made my own infographic (involving wine) to show that you should always switch:

If I’m forced to articulate a reason for my gut feelings, the trouble is that I’m guilty of a bit of myopia here. I am so familiar with the number 2, and being binary, and dealing with fifty-fifty odds, that I can’t help but look at a situation where there are two doors and see it as an isolated game of chance where the odds are pretty much even.

It’s so familiar that I can’t help but ignore* *the fact that the two doors that I’m being presented with are *not random*. The only random choice that took place was the first one – and thereafter, my original choice *directly* affected which subsequent non-prize door was opened.

To repeat what mythbusters concluded:

- When I choose my first door, there is a 33% chance that I’ve picked the winning door.
- Equally, there is a 67% chance that one of the other two doors is the winning door.
- I then get told which of the two doors that I didn’t choose had nothing behind it.
- That’s not a random thing – that is specific additional information about the original scenario that I was presented with.
- And therefore, there is still a 67% chance that one of the other two doors is the winning door – only, now I know which one of them is definitely not a winner.
- So if I switch, I now have a 67% chance that the other unopened door is the right one.

Even as I write that, my gut feelings are grumbling.

But still: deductive math, and empirical evidence.

Which makes me wonder about how much else is both irrationally and myopically defended and believed.

*Rolling Alpha posts about finance, economics, and sometimes stuff that is only quite loosely related. Follow me on Twitter @RollingAlpha, or like my page on Facebook at www.facebook.com/rollingalpha. Or both.*

## Comments

## Philip Sterne December 21, 2015 at 16:43

The solution became a lot more intuitive to me once the game was changed to have 1,000,000 doors and the host opens 999,998 of the doors that don’t contain prizes.

Intuitively it doesn’t feel like 50/50 odds anymore.

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