I ran across an apparent paradox which I then located in the paper  as such:     Imagine that you are shown two identical boxes. You know that one of   them contains. \$b and the other \$2b. Picking one at random and opening   it, you must decide whether to keep it (and its contents), or exchange   it for the other box.   In short, when you find $x dollars in a box, the expected value of the other box is .5*.5x + .5*2x = 1.25x, meaning that it's always better to switch. This appears to violate the symmetry of the problem and the fact that you still know nothing meaningful about either box.  The paper goes another direction with it, talking about how having prior knowledge of expected values gives a more meaningful analysis (along with other discussions). However, what if there's no prior knowledge, and we have the original problem as stated. Can anyone give me some intuition to make sense of this?  EDIT: Someone found  which asks a slightly different formulation, but an identical problem. The accepted answer there just points to , and I'm having difficulty understanding the paper. It explains away the paradox by noting the expectation is based on an infinite sum, and the value depends on the order the sum is evaluated. I'm not familiar with how the order of a sum can change a value, and I also don't see talking about a different way to evaluate the expectation explains the strange result outlined above. My math understanding is primarily based on reading textbooks as a hobby, and I haven't yet worked up to fully understanding math academic papers, so a simpler explanation would be helpful.
Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card.  If you choose to swap, your expected value should be the same, as you still have a $50\%$ chance of getting the higher card and $50\%$ of getting the lower card.  However, the other card has a $50\%$ chance of being $0.5x$ and a $50\%$ chance of being $2x$. If we keep the card, our expected value is $x$, while if we swap it, then our expected value is: $$0.5(0.5x)+0.5(2x)=1.25x$$ so it seems like it is better to swap. Can anyone explain this apparent contradiction?
There are $k$ penguins, $k\ge 3$. They are all different heights. How many ways are there to order the penguins in a line, left to right, so that we cannot find any three that are arranged tallest to shortest (in left to right order)? The penguin triples do not have have to be adjacent.  This problem is courtesy of someone I know only as "DaMancha." Windows 8 Setting an image on the login screen In Windows 8.1 when you log in there's the prelogin screen that has an image you can change, but then there's also the login (password entry) screen that as far as I can see is just a solid color.  How could I replace that with an actual picture? (ie, if I wanted a company logo to be displayed in the large space underneath the password entry box. How to change IP address to point to localhost? How to reset main user account? My main account got messed up, as I tried to fix it things went downhills.   keyboard mess - tried unity --restore and deleting .gconf2*, fail deleted .* in ~, total havoc. Chromium and Firefox crashes etc.   Now, another user account is fully functioning, but I really want my regular username back.  Is there any way to completely wipe the settings for the messed up account or copy the profile from the functioning user? Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ For the $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0,$ my text book asks the following questions.  (b) Why is $f$ uniformly continuous on any bounded subset of $\mathbb{R}$?   (c) Is $f$ uniformly continuous on $\mathbb{R}$??  The graph for the function is this.    For the question (b), if I take subset between $[0.2,0.6]$ or the subset where the slope is steep, I don't think the function is uniformly continuous because I think for a given $\epsilon&gt;0$, there is no unique $\delta &gt;0$ for the bounded subset. Therefore, it also cannot be uniformly continuous on $\mathbb{R}.$ However, the questions sounds like the function is uniformly continuous and the book says that it is uniformly continuous. The answer on the book says something but I need more explanation. Thanks. Are delphi variables initialized with a value by default? I'm new to Delphi, and I've been running some tests to see what object variables and stack variables are initialized to by default:  TInstanceVariables = class   fBoolean: boolean; // always starts off as false   fInteger: integer; // always starts off as zero   fObject: TObject; // always starts off as nil end;   This is the behaviour I'm used to from other languages, but I'm wondering if it's safe to rely on it in Delphi?  For example, I'm wondering if it might depend on a compiler setting, or perhaps work differently on different machines.  Is it normal to rely on default initialized values for objects, or do you explicitly set all instance variables in the constructor?  As for stack (procedure-level) variables, my tests are showing that unitialized booleans are true, unitialized integers are 2129993264, and uninialized objects are just invalid pointers (i.e. not nil).  I'm guessing the norm is to always set procedure-level variables before accessing them? Does Professor X have any responsibility for the state of the world? Throughout the course of Logan, it's obvious that Professor Xavier is having issues controlling his powers, and at the end of the movie, just before      X-24 kills him   he claims to "remember" what he did and expresses regret. We also hear about an incident in Westchester that killed several mutants, that sounds like Xavier may have been involved. Is Xaiver the reason there are no more mutants? Is that why he's sorry, because he knows he killed them all? PHP regex delimiter, what's the point? Sequence converges iff $\limsup = \liminf$ I want to prove that a sequence of real numbers $\{s_n\}$ converges to $s$ if and only if $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$.  Here are my definitions:  For any sequence of real numbers $\{s_n\}$, let $E$ be the set of all subsequential limits of $\{s_n\}$, including possibly $+\infty$ and/or $-\infty$ if any subsequence of $\{s_n\}$ diverges to infinity.  Then $\limsup_{n \to \infty} s_n = \sup E$, and $\liminf_{n \to \infty} s_n = \inf E$.  I know the theorem that a sequence converges to a point if and only if every one of its subsequences converge to that same point, so one direction of this proof is easy:  If $\{s_n\}$ converges to some point $s \in \mathbb{R}$,  then every subsequence of $\{s_n\}$ converges to $s$.  So the set $E$ of every subsequential limit of $\{s_n\}$ consists of the single point $s$, so $$\limsup_{n \to \infty} s_n = \sup \{s\} = s = \inf \{s\} = \liminf_{n \to \infty} s_n$$  But the other direction seems more tricky...  If $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$, then every convergent subsequence converges to the same point $s$.  Also, there can be no subsequences which diverge to infinity (otherwise $\limsup_{n \to \infty} s_n$ would be $+\infty$, or $\liminf_{n \to \infty} s_n$ would be $-\infty$).  But can't there be subsequences which diverge otherwise?  And wouldn't that throw off the convergence of $\{s_n\}$?    EDIT:  I'd also be willing to accept a solution which makes use of the "Pinching Theorem" (if $a_n \leq s_n \leq b_n$ for every $n \in \mathbb{N}$, and if $a_n \to s$ and $b_n \to s$, then $s_n \to s$). Can I boil milk unattended (and still keep the pan clean)? Inequality vs. Inequation, Equation vs. Equality When talking about statements that two terms are mathematically (un-)equal (e.g., 1 = 1, 1 &lt; 2, or 1 != 2), what is the correct notion for such mathematical statements? Are there special cases w.r.t. "mixed" relations like x &lt;= y? Can some notions be used interchangeably?  It seems many online sources use "equation" and "inequality" but there is also a . Apart from personal posts in forums, I have not found any sources using "equality" in such a situation, yet. Systematically, I would think that "equation" and "inequation" would make more sense than "equation" and "inequality", but not everything (probably very few things) in natural languages have developed according to a purely logical system.  Note that this question is more general than  for that it particularly asks for the relation between the notion for equations and the one for inequalities. Equations are not part of the existing question at all. Trigonometric fraction Inequality question I have a custom script and I want to send a desktop notification (the one that appears in the top right corner of the screen) with a custom message. How do I do that? Heres a picture. im making a body, and it seems to have flash in the middle. I need this date format in a visualforce page: "December 14, 2013". I can't seem to find the format options for visualforce. I read that it's the same as the java messageformat class options, but I cannot find a list for that either.  A little help would be much appreciated I need to purchase my first ever lens. So far I've only used lenses that were built into a camera. This specific lens is for the BlackMagic Deign Pocket Cinema Camera (a video only camera, but one that uses photography lenses). I would prefer this to be a prime lens.  I have tried to find the exact way to calculate the focal length of the lens I need but I have not been able to find a good resource to make this simple enough for my limited optics knowledge. Can you please explain a simple way to calculate the focal length of a lens given the following inputs as an example:    I would prefer a simplified mathematical explanation so that I may use it to make similar determinations in the future using a "" method.  I have . Using that method, I came up with:  Required lens focal length = 7.02mm sensor height x (77in / 36in) = 15mm   Is that correct? std::auto_ptr to std::unique_ptr Adjective for "terrified yet calm and capable" What is an adjective to describe the duality of a first responder (such as an EMT responding to a terrorist attack) who is terrified for the wounded, yet simultaneously calm, detached, and capable enough to give aid? Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one. This is a recent homework bonus question assigned in my Proofs and Conjectures class. It (evidently) includes and evaluates our understanding of elementary-set theory and how to determine and prove that the function f is onto and one-to-one. I haven't been able to really grasp firmly how it is we do this. Help would be much appreciate, thank you. Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” Reproduced from Exercise 2.2 of Nielsen &amp; Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):     Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\rangle$, and $A$ is a linear operator from $V$ to $V$ such that $A|0\rangle = |1\rangle$ and $A|1\rangle = |0\rangle$. Give a matrix representation for $A$, with respect to the input basis $|0\rangle, |1\rangle$, and the output basis $|0\rangle, |1\rangle$. Find input and output bases which give rise to a different matrix representation of $A$.   Note: This question is part of a series attempting to provide worked solutions to the exercises provided in the above book.