The Monty Hall problem is a very well-known brain teaser in statistics.
In the problem, you are a contestant on a game show, hosted by Monty Hall. Monty shows you three doors —A,B, and C— behind one and only one of which is a new car (the other two doors have goats).
If you guess correctly, the car is yours; otherwise, you get a goat.
You guess A at random.
Monty, who is forbidden from revealing where the car is, then opens door C, which, of course, has a goat behind it.
He tells you that you can now switch to Door B or stick with Door A. Whichever you pick, you’ll get what’s behind it.
Are you better off opening Door A or switching to Door B?
Many people when they first encounter the problem, reason that, since the location of the car is independent of the door you first choose, switching doors neither gains nor loses you anything the probability that the car is behind Door A is equal to the probability that it is behind Door B.
But the correct answer —as decades of statistics students have found to their consternation‚— is that you are twice as likely to win the car if you switch to Door B as you are if you stay with Door A.
Why is that?
Well, here comes an explanation that I came up with and which managed to convince my girlfriend. Which in my opinion was more than enough to convince the entirety of hacker news.
I believe the problem arises from the small numbers that are being used. 3 doors is too little. Let’s spice it up with 1.000.000.000 one billion doors (or a unicorn of doors as many VC’s would like to call it).
Now it’s your turn to pick one of those doors and win a car.
Your chances are 1/1.000.000.000 one in a billion. Meaning that the chance you got it wrong is 999.999.999/1.000.000.000
So then Monty opens the remaining 999.999.998 doors. Leaving only one door closed (apart from the one you picked).
Now, if you decide to keep your original pick, you are also keeping your one in a billion chance that you got it right.
This is because you already knew that Monty would open 999.999.998 doors with nothing behind them.
BUT if you now decide to change your pick and instead choose the only door that Monty didn’t open, you are changing your bet and now betting on the 999.999.999/1.000.000.000 chance that you got it wrong on the first try.
So you’re most likely to win if you change your pick!
Funny huh?
Credits to the book Causal Inference in Statistics by Judea Pearl which made me review this problem and actually understand what was happening.