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You'll often hear people talk about thinking probabilistically as a superpower. If you can just learn to do it, you'll level up your brain, unlock Pandora's box, succeed at trading and finally get that Lambo.
On some level, that's true. Well, the first bit is. If you can understand probability, you're better prepared to face this complex, uncertain chaos that we call 'life'.
The rest of it's... ridiculous. As is thinking probabilistically.
The Monty Hall Problem is the perfect example. It's from an old game show with a simple premise 👇
Monty Hall asks you to choose one of three doors. One of those doors hides a prize and the other two doors have no prize. You state out loud which door you pick, but you don’t open it right away.
Imagine at this point that you chose door number 1. Then...
Monty opens one of the other two doors (door 3), and there is no prize behind it. 👇
At this moment, there are two closed doors, one of which you picked.
The prize is behind one of the closed doors, but you don’t know which one.
Monty asks you, “Do you want to switch doors?”
Our instinctive reaction is to think this question makes no difference. Stay with door 1 or switch to door 2... Who cares? It's just a 50/50 decision. A coin toss.
Cut scene to behavioural scientist Richard Thaler
Man knows his stuff. See, it's not a 50/50 choice at all. The door you originally chose (door 1) had a 1 in 3 chance of being the right door. Which means there was a 2 in 3 chance that it wasn't the right door.
As soon as one door is opened, your brain tells you that one door removed means the odds become 50/50. Your grey matter is jumping to conclusions while ignoring the rules of the game...
If this is the first time you've seen this problem, your brain's probably screaming FIFTY FIFTY at you. refusing to accept this 66% mathematical fact.
Monty the Meddler
Monty's the wildcard here. See, he KNOWS where the prize is. So as soon as he picks a door and reveals no prize, it's no longer random, (because the purpose of the game is to put the contestant in a stick or switch situation) 👇
The probability that your initial door choice is wrong is 0.66.
The following sequence is totally deterministic when you choose the wrong door. Therefore, it happens 66% of the time:
You pick the incorrect door by random chance (2 in 3 odds or 66% chance this will happen). The prize is behind one of the other two doors.
Monty knows the prize location. He opens the only door available to him that does not have the prize.
By the process of elimination, the prize must be behind the door that he does not open (in the more likely case that you originally chose the wrong door)
Because this process occurs 66% of the time and because it always ends with the prize behind the door that Monty allows you to switch to, the “Switch To” door must have the prize 66% of the time.
Now, it's REALLY hard to accept this. First time I heard this, I wrestled with it for hours.
To help, there's a little simulator here to prove the point. By switching every time, your odds of winning are higher.
Variance plays a part too. I switched every time, and at one point had a win rate of 93% (13 wins, 1 loss), then I remembered to take a screenshot...
I wondered if the simulation might have a bias built in, but the more you play it, the more the variance converges towards the actual odds...
My favourite thing about this game is how it exposes the flaws in all of our thinking. Human OS just isn't built to handle this stuff 👇
When Marilyn vos Savant was asked this question in her Parade magazine column, she gave the correct answer that you should switch doors to have a 66% chance of winning.
Her answer was so unbelievable that she received thousands of incredulous letters from readers, many with Ph.D.s! Paul Erdős, a noted mathematician, was swayed only after observing a computer simulation.
Which is one of (many) reasons to be wary of the most confident among us. Especially anyone with letters after thier name whose job is to 'be right'.
So, thinking in probabilities is ridiculous, but it's also a superpower. It opens our eyes to the realities of the world. Nothing is certain, except death, taxes & uncertainty.
Sticking with the topic of probablity, this from Brent Donnelly is superb. 👇
My favourites 👇
The longer you play, the more deeply you internalize variance and the inevitability of good decision/bad outcome.
You are playing a never-ending sequence of hands with incomplete information against financially-motivated, skilled opponents. Trading and poker are both textbook examples of decision-making under uncertainty.