Suppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in the end, we did manage to pick $x$, despite its probability of 0.  Does this mean that a 0% chance is actually possible, or is there some flaw in this logic?
I read a comment under :     There are plenty of events that can occur that have zero probability.   This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering    if zero probability and impossibility mean the same? if an event with zero probability doesn't mean that the event is impossible to occur, how probability theory represents/describes impossibility?   Thanks and regards!
How do you find the directory of a command? How to format a USB drive? What EXACTLY cause Voldemort's soul piece to split off when he killed Lily and tried to kill Harry? Story about criminal being "adjusted" which stops a man's commute I'm trying to remember the name of a story, I think the author was Larry Niven, but it might possibly be from Heinlein's "Future History" series (but a check of the stories included in the Future History indicates it's not in there), about a man whose commute is stopped because a criminal has to be 'adjusted'.  In this case, the criminal was a hit-and-run driver who left his victim untreated on the roadside, and was sentenced to suffer in the same manner. So his leg was carefully broken and he was left, untreated and in pain, on the side of the road in the same place for an hour. All the time ambulances were standing by, and every effort was made to ensure that the criminal did not suffer unduly - but that he did suffer, in a manner as close to the suffering he inflicted on his victim. Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$. Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.   I do not understand how to go about completing this problem or even where to start. How do I clean the inside of my viewfinder? My viewfinder (not LCD) has spots of dust on it.  What is the best way to clean the inside?  I have cleaned the outside with a QTip. Android SQLite database: slow insertion This is a homework problem and I'm having trouble just getting a basic understanding.  I understand that $S_6$ is a symmetric group of degree $6$.  I'm not sure how to start looking for elements. Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$:   $$f(f(x)f(y))+f(x+y)=f(xy).$$   It was six hours ago in IMO 2017 (problem 2).  I tried  the standard way: $x=0$, $x=y$, $x=1$,etc. but without any success.   Let $f(0)=a$. Hence, for all $x\in\mathbb R$ we have $f(a(f(x))+f(x)=a$. What is the rest?   Also we have $(x-1)(y-1)-1+x+y-1=xy-1$ and  $1-(1-x)(1-y)+1-x-y=1-xy$. How can I find color codes on Mac? I used to find color codes which I want to use for texts or background colors on htmls with color edit tool of MS paint which pre-installed on Windows. But now I use Mac instead of Windows. How can I find color codes with Mac? Is an software like MS paint pre-installed on Mac? Is it possible to smooth or generalize a polygon in QGIS, perhaps by using a plugin like the generalizer-plugin for lines?  I could convert my polygons to lines, smooth them and convert them back to polygons but this is not working for my polygons because of several reasons. I have a web service in C# and would like to have a nested inner class, that abstracts away the session collection, something like this:     public class Service : System.Web.Services.WebService {     [WebMethod]     public string Foo(string ticket)     {         SessionPool.getSession(ticket);     }      private class SessionPool      {         public static Session getSession(string ticket)         {             // this is what i want to do, but I can't access Context             return (Session)Context.Session[ticket];         }     } }    Is it possible to access the HTTP context of the WebService class via a nested class? If not, is there way I can store the reference to it? Regain of reputation from downvoted and deleted answers - update FAQ Krull dimension of this local ring I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases. I want to monitor the output of a command and whenever it contains a certain string I want to run another command.  Example: when command1 outputs 1234 I want to run command2 Gravitational slingshot of light using a black hole/massive object Wikipedia has this page on  using planets. In some cases this effect was used to accelerate the spacecraft to a higher velocity. This diagram shows this in a very oversimplified manner.   That got me thinking that if light is affected by gravity, and if it slingshots around the black hole/massive object, can't it gain a higher speed than $c$? What limitations are stopping it from doing this?  Forgive me if the answer to this question is pretty straight-forward or staring-you-in-the-face kind. I haven't fully understood the mechanics of the gravitational slingshot fully yet, but I couldn't wait to ask this. Note: If it is possible (I highly doubt that!), could you provide an explanation using Newtonian mechanics, I'm not very familiar with general relativity because I'm in high school. Chrome is unable to connect to the Internet I'm getting the error code: DNS_PROBE_FINISHED_NO_INTERNET in Google Chrome. The thing is, I have a working internet connection. I am posting this using that connection, using Internet Explorer. It's only Chrome doing this.  What happened was my laptop was refusing to go to sleep mode, so it ended up running out of battery power. I charged it, and when I subsequently booted the computer, Chrome stopped working. I don't see the connection...  I added Firewall rules (inbound/outbound) to allow Chrome to connect...so unless I'm missing something.  UPDATE: Chrome portable is working for me...just not the regular chrome.  UPDATE 2 Now I realize it had to do with Covenant Eyes which I have installed. See the accepted answer How does the ternary operator work? GCD of rationals Disclaimer: I'm an engineer, not a mathematician  Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also:    $$ \gcd\left(\frac{13}{6}, \frac{3}{4} \right) = \frac{1}{12} $$  I can do this for a number of cases on sight, but I need a method. I tried Euclid's algorithm, but I'm not sure about the end condition: a remainder of 0 doesn't seem to work here.    So I tried the following:    $$\gcd\left(\frac{a}{b}, \frac{c}{d} \right) = \frac{\gcd(a\cdot d, c \cdot b)}{b \cdot d}$$  This seems to work, but I'm just following my intuition, and I would like to know if this is a valid equation, maybe the correct method. (It is in any case consistent with $\gcd$ over the natural numbers.)  I'm not a mathematician, so please type slowly :-) I'm having the strangest issue since updating to OS X El Capitan. My "Open File" dialogs show the sidebar but all my main (and useful) locations are missing. Take a look at this screenshot. This screenshot shows Keychain Access but this happens in Safari when I'm trying to attach a file to Gmail for example, or uploading an image to Facebook.  What I would like is for the sidebar to show my home directories, such as Desktop, Documents, Downloads, etc.  Help?