Waste less time on Facebook — follow Brilliant.Excel in math and scienceMaster concepts by solving fun, challenging problems.It's hard to learn from lectures and videos Learn more effectively through short, conceptual quizzes.Our wiki is made for math and science Master advanced concepts through explanations, examples, and problems from the community.Used and loved by 4 million people Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.Sign up Existing user?Sign in Problem Loading... Note Loading... Set Loading...Busts of Digital Idols - Shop the Collection!A series of pieces from my Facets Project.Illustrations for Nike's 2011 Basketball Apparel Artwork for Koan Sound's Dynasty EP A series of symbols from my Facets Project.Illustration for Nike x Acclaim Work created for Depthcore's "ECHOES" ChapterSource: Jane Street Capital Interview (Quantnet) Problem: You are given a 100 sided die.

After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll.What is the expected value of the game?(There is no limit on the number of rolls.)Update (22 June 2014): Solution posted by me, Felix, JDGM, Sebastian, Fu Shi and Dumb Phoenix in comments!The Bernoulli trials process is one of the simplest, yet most important, of all random processes.It is an essential topic in any course in probability or mathematical statistics.The process consists of independent trials with two outcomes and with constant probabilities from trial to trial.Thus it is the mathematical abstraction of coin tossing.The process leads to several important probability distributions: the binomial, geometric, and negative binomial.Bernoulli trials appear in many chapters in this project, further evidence of the importance of the model.Much has been written about Zeno’s paradoxes.The link above states the paradoxes, as well as the various resolutions offered over the years, and the problems with some of the various resolutions.

We have a special guest with us for this article, Zeno himself.He’ll explain why Achilles will never catch the tortoise.Take it away, Zeno.Zeno: Assume Achilles takes a step of length 1/2, then a step of length 1/4, etc. Will he actually ever touch the line in the sand one unit away from where he began?No, he never will.1 is the Least Upper Bound of his steps.No matter how many steps he takes, even if he keeps taking steps until the Universe freezes over, he will never touch the line at distance 1.Smiling Dave: Yes, you are right.But Achilles does not walk that way.He takes steps of the same length each time.Zeno: Achilles does not actually walk that way, I agree, but we can analyze his walk, dividing it in our minds into an infinite number of steps.Then, when we try to put the infinite number of steps back together, we cannot, because Achilles is mortal and can only take a finite number of steps.Same thing with my Arrow paradox.The arrow does not fly before our eyes through an infinite number of positions.

It is our mind that analyzes the flight of the arrow as being made up of an infinite number of steps.And then comes the same punchline as with Achilles.Having cut the path up into an infinite number of steps, we cannot put it back together, because we have no way of adding an infinite number of things together.SD: You understand infinity, Zeno, as well as anyone.bitcoin still existsBut infinity is a very tricky thing.bitcoin easyTwo infinite sets A and B can be equal, even though A has everything B has, and more [Galileo’s Paradox].bitcoin fake or realA hotel with an infinite number of rooms, all of which are filled, can accommodate an infinite amount of new guests, such that there there is only one person in each room [Hilbert’s Grand Hotel].bitcoin federal judge

The unit sphere, because it has infinitely many points, can be cut up into finitely many pieces, and put together into a sphere ten times the area [Banach Tarski Paradox].And some infinite sets are “more infinite” than others [Cantor’s Diagonal Argument].SD: All that has been proved by modern mathematics.It’s after your time.Zeno; And your point is?bitcoin gemSD: Nobody has ever seen the infinite.bitcoin furnitureWe grasp it not by direct experience, but by mental processes.Since we have no experience to guide us, we have to be very careful when using this tricky thing, which can be proven to behave so paradoxically, as a model of reality.If our attempt at modelling the flight of the arrow as passing through an infinity of points, or of Achilles passing through an infinity of points, leads us to patently false conclusions, then all we have shown is that our model assumes something about infinity that is incorrect.

Zeno: Dave, that’s not good enough.Yes, infinity is tricky.But you have to show me exactly where my mistake is.SD: Both your paradoxes assume you cannot take an infinite amount of steps in a finite amount of time.Wikipedia got it right, and I quote.Simplicius has Zeno saying “it is impossible to traverse an infinite number of things in a finite time”.This presents Zeno’s problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a “last event”?SD: And who told you that you cannot take an infinite amount of steps in a finite amount of time?SD: Sorry, Zeno, you can’t use “common sense” when it comes to infinity.You either prove it from first principles, or you don’t.And you certainly haven’t proven it.Zeno: I see that now, Dave.Thank you for clearing away millennia of confusion.

What really brought it home to me was what you wrote about being able to fill an already full hotel, over an over, if there are an infinite number of rooms.That really shocked me, and I see now that you can’t make any unproven assumptions about infinity.Dave: You are welcome.BTW, all of those supposed solutions over at Wikipedia are really silly.Zeno: I know, right?They just don’t get it.For instance, what’s the point of running to quantum mechanics and a finite universe and discrete time and all that other nonsense.That’s just throwing in the towel, admitting Newtonian physics is logically flawed.And those guys you see on Youtube, who think they discovered America because they can sum an infinite series on paper.How badly they missed the boat!Just because an infinite series has a sum, that doesn’t mean Achilles can take an infinite number of steps.SD: I thought you died before Newton.Zeno: You have anything about my other arrow paradox?SD: Which one is that?Zeno: Here’s Wikipedia about it: If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.[11]