Bravo for bringing the notes. On a first glance, some of these feel like they require subjectivity (like, do we really believe the political spectrum is 1d?), but I agree I could run the computation myself from this.
Bravo for bringing the notes. On a first glance, some of these feel like they require subjectivity (like, do we really believe the political spectrum is 1d?), but I agree I could run the computation myself from this.
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Ah, so it’s the probability you win by playing randomly. Gotcha. That makes sense, it becomes a choice between 2 doors
Why do you have a P(x1) = 1/2 at the start? I’m not sure what x1 means if we don’t specify a strategy.
Oh that’s cool - I had heard one or two examples only. Is there some popular writeup of the story from Savant’s view?
An arithmetic miracle:
Let’s define a sequence. We will start with 1 and 1.
To get the next number, square the last, add 1, and divide by the second to last. a(n+1) = ( a(n)^2 +1 )/ a(n-1) So the fourth number is (2*2+1)/1 =5, while the next is (25+1)/2 = 13. The sequence is thus:
1, 1, 2, 5, 13, 34, …
If you keep computing (the numbers get large) you’ll see that every time we get an integer. But every step involves a division! Usually dividing things gives fractions.
This last is called the somos sequence, and it shows up in fairly deep algebra.
I now recall there was a numberphile with exactly that visualisation! It’s a clever visual
For the uninitiated, the monty Hall problem is a good one.
Start with 3 closed doors, and an announcer who knows what’s behind each. The announcer says that behind 2 of the doors is a goat, and behind the third door is a car student debt relief, but doesn’t tell you which door leads to which. They then let you pick a door, and you will get what’s behind the door. Before you open it, they open a different door than your choice and reveal a goat. Then the announcer says you are allowed to change your choice.
So should you switch?
The answer turns out to be yes. 2/3rds of the time you are better off switching. But even famous mathematicians didn’t believe it at first.
Note you’ll need the regions to be connected (or allow yourself to color things differently if they are the same ‘country’ but disconnected). I forget if this causes problems for any world map.
Would you then be posting your conclusions? Like, if you’re gonna do that work on some of these posts anyway… may as well share.