Bertrand's Paradox (with 3blue1brown) - Numberphile
Featuring Grant Sanderson, creator of 3blue1brown.
Extra footage from this interview: • More on Bertrand's Par...
3blue1brown video on the shadow a cube: • A tale of two problem ...
More links & stuff in full description below ↓↓↓
3blue1brown: / @3blue1brown
Grant on The Numberphile Podcast: • The Hope Diamond (with...
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Special thanks to our friend Jeff for the accommodation and filming space.
Extra footage from this interview:
It must be nice to collab with Grant since he did his own animation
4:52
Every other person: This is because of my bad drawing.
I think this paradox is a perfect example of how slippery a lot of concepts in probability theory can be. Even Erdos got the Monty Hall problem wrong.
A triple cheers for bringing this important example up and making it known to the greater public! And so wonderfully rendered and explained, too!
This ended in such a cliffhanger, never felt so compelled to see the extra footage
"It's implicit when you're told to choose a random number between 0 and 1 you would use some kind of uniform distribution"
I didn't know where it was going exactly, but immediately when he said that you pick a "random chord" I instantly thought "Define random. What's the distribution?"
This collab is a better Christmas gift than anything I have ever got from my family
Any point outside the circle will have two tangent lines to the circle; the intersection points of these lines to the circle will uniquely define a chord of the circle, thus any point outside the circle uniquely defines a chord. There is a finite region around the circle (bounded by a circle of radius 2r centred at a origin) where the chord defined by any point within will be shorter than s; any point outside this region defines a chord that is longer than s. The probability that a randomly selected point outside the circle will fall within the region where the chord would be shorter than s is 0; therefore the probability that a randomly selected chord will be longer than s is 1.
Most people don't even know about probability density functions, so this sort of topic will really challenge them. As someone once said, "The generation of random data is too important to be left to chance."
It seems to me that the biggest clue here was that the second method looked sparser in the centre than the first.
Great example of why it’s important to state your assumptions! The problem isn’t in dealing with infinite spaces, it is in how you state what you know about one aspect of the space vs another.
This reminds me a little of one of my math exams questions back at uni, where it was about hitting a dart in the inner 2/3 of the target. I simplified it to 1D (2/3 of a line vs 1/3) not taking into account the fact that I cannot just get rid of polar coordinates like that. Here the different distribution of cords in the circle remind me of that. That some simplification that lead to a wrong distribution took place...
Reminds me of the unsaid random (and non-random) distributions in the Let's Make A Deal / "Monty Hall Problem", the one with three doors and two goats and a car. It's (usually) just implied that Monty knows what's behind each door, and always chooses to open a door with a goat behind it, rather than choosing randomly and sometimes opening one with a car "by chance".
ah yes, how fitting that a man bearing my family name came up with a paradox in which you can never be sure that what you're doing is the correct thing and you are doomed to paranoia forever
It's fascinating: each method is a different way of seeing the topology of the circle. The first is the classic S1, the second is the disc filling in S1, and the last is the disk as [0,1]×S1. (In particular, quotienting out the S1 part).
Back to back videos by Grant. He just released one video on his channel 3b1b. What an amazing day!
I think the core assumptions of the second and third methods are actually just flawed.