Home page: https://www.3blue1brown.com/ For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation. *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support ... https://www.youtube.com/watch?v=BaM7OCEm3G0

Bayes factors, aka Likelihood Ratios*, offer a very clear view of how medical test probabilities work. Home page: https://www.3blue1brown.com Brought to you by you: https://3b1b.co/bayes-factor-thanks The book by my friend Matt Cook about paradoxes mentioned at then end: https://amzn.to/3aBrEzg On the topic, I can't help also mentioning another paradox book I'm rather fond of by Bunch: https://amzn.to/3mBDSKE *As mentioned in the on-screen note at the end, while the terms "Bayes Factor" and "Likelihood Ratio" refer to the same term in this setting, where Bayes rule is used on the probability of an event with only two possible outcomes (you either have the disease or you don't), they do take on divergent meanings in more general contexts. Namely, if you have a continuous parameter you are trying to estimate, the two terms reflect two alternate approaches you can use in comparing hypotheses. In fact, some people take the phrase "Bayes factor" to _specifically_ refer to its use in this more continuous context. If you want more details, Wikipedia actually has a really nice example discussing the difference: https://en.wikipedia.org/wiki/Bayes_factor#Example This post has some nice discussion of the distinction: https://stats.stackexchange.com/questions/27345/likelihood-ratio-vs-bayes-factor ------------------ These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown ... https://www.youtube.com/watch?v=lG4VkPoG3ko

Full video on the central limit theorem: https://youtu.be/zeJD6dqJ5lo Thanks to Dawid Kołodziej from long-to-short editing ... https://www.youtube.com/watch?v=GOSezO0CHss

A tale of two problem solvers. Numberphile video on Bertrand's paradox: https://youtu.be/mZBwsm6B280 Help fund future projects: https://www.patreon.com/3blue1brown​ Special thanks to these supporters: https://3b1b.co/lessons/newtons-fractal#thanks An equally valuable form of support is to simply share the videos. I first heard this puzzle in a problem-solving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy. Mémoire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy https://ia600208.us.archive.org/27/items/bub_gb_EomNI7m8__UC/bub_gb_EomNI7m8__UC.pdf The artwork in this video was done by Kurt Bruns ------------------- Timestamps 0:00 - The players 5:22 - How to start 9:12 - Alice's initial thoughts 13:37 - Piecing together the cube 22:11 - Bob's conclusion 29:58 - Alice's conclusion 34:09 - Which is better? 38:59 - Homework ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown ... https://www.youtube.com/watch?v=ltLUadnCyi0