More on Bertrand's Paradox (with 3blue1brown) - Numberphile

Brady's certified method to finding answers to unanswerable mathematical questions: find a mathematician and threaten them to give you the answer. Love it :D

Brady: "I'm going to burn your house down if you don't tell me the answer" Grant: "That's a great question."

15:04 The lack of even a chuckle at the premise of that question tells me that Grant 100% has laid awake at night thinking about Bertrand's paradox.

18:55 it's just amazing how quickly Grant grasps the definition Brady provides and is able to flawlessly lay bare its weaknesses.

I think this paradox can be partially summed up with saying "Math can answer questions for you, but it can't ask your questions for you."

I'm in awe of Grant's ability to speak, unscripted, with perfect clarity. (Both in the sense of what he's saying, and incidentally, his flawless diction.)

I've already been a fan of 3blue1brown's videos, but seeing him explain/answer/clarify all his points in "real time" was nothing short of amazing. In his videos, you assume preparation, practice, etc., but here he's just talking to someone else and making a whole bunch of sense on the fly.

I think Grant is amazing as well, for all the reasons in the top comments here - but I think Brady deserves an enormous amount of credit as an interviewer of mathematicians. He always succeeds in asking the questions (be it scripted or on the fly) I want to hear asked, as well as a couple I didn't even think of, but which sends the interviewee on exactly the tangent they need to be on for a great video. This one was epic.

Grant's on-the-fly analogy of the multi-sided die really got me there with this paradox. Absolutely nailed it!

Actually got chills when Grant so quickly and eloquently explained the failings of Brady’s proposed method.

Our friend, Danish mathematician Piet Hein, offers a take on this: "When you're desperately trying to make up your mind and bothered by not having any; you'll find that the simplest solution by far is to simply try spinning a penny. No, not that chance should decide the event while you're passively standing there moping; but once the penny is up in the air you'll suddenly know what you're hoping" Aaaah his powerful Gruks.

This reminds me of the problem of choosing a random location using latitude and longitude. If you don't correct the distribution used for selecting latitude, the polar regions are over sampled

"When you lie in bed at night, and think about Bertrand's Paradox..." has got to be one of the best quotes from numberphile

15:26 "One of the biggest misconceptions is that maths shows us truths, but it doesn't. It tells you 'given certain assumptions, what are the necessary links to consequences'" - Grant Sanderson

I love the way Grant smiles throughout - he is having such fun, and so are we!

Grant's reaction after Brady saying "draw every possible chord, each one is numbered... just wonderful

This should be part 2 on the main channel, too important to miss

Grant is a uniquely gifted communicator. I can't think of anybody with greater ability to explain mathematics clearly. Absolutely love this guy.

I really like that he is able to engage with Brady’s questions in a sensible way. A lot of people would deflect Brady’s “burn your house down” question trying to be clever, but Grant actually gives a thoughtful response to its intent.

Possibly another (equivalent but mathier) way of explaining the “paradox”: When you talk about “random cords”, doing the calculation requires a procedure for constructing random cords. That procedure is essentially a measurable function from a parameter space (such as the space of pairs of points on the circle, or the space of angles and midpoints) into the space of cords. If you assume a uniform distribution on your parameter space, the measurable function into the space of cords carries that distribution forward—but there’s no reason to assume that different measurable functions from different parameter spaces will carry forward to the same distribution in the space of cords.