I feel like there's no link between P(A) and P(A|B), where A and B are random events. For example, P(A) can be 0.3 and P(A|B) can be 0.9.  However, I'm not so sure of this when P(A) is either 0 or 1. I mean:   P(A)=0 => P(A|B)=0, for every random event B ? P(A)=1 => P(A|B)=1, for every random event B ?
$\newcommand{\P}{\mathbb{P}}$Is it valid to interpret $\P(B|A)$ as "infinite probability" when $\P(A) = 0$?
COUNT(*) vs. COUNT(1) vs. COUNT(pk): which is better? PHP replaces spaces with underscores Prove or disprove the uniform continuity of $f: \mathbb{C} \to \mathbb{C}, z \mapsto f(z)=\frac{z^2}{1+|z|}$ I tried using the definition of the continuity to try and find a $\delta$ but I got stuck here $$ |f(w) - f(z)| = |\frac{w^2 \cdot (1+|z|) - z^2 \cdot (1+|w|)}{(1+|w|)\cdot(1+|z|)}|$$ What can I do to find a suitable $\delta$ that would end my proof? Changing data type in ArcGIS attribute table? Which process scheduler is my linux system using? I'm not quite sure if stackoverflow or serverfault is the forum to use for this question, but ill try here.  Is there some command I can use or a file I can check in order to find out which process scheduler I'm using on my Linux system? I am not looking for the I/O scheduler, but the process scheduler, and please, do not refer to guides or howtos for what a scheduler is, what I'm simply asking is if there is a command or file I can run/check to see this. I could check the doc for my kernel but again I'm just wondering if there is a command or file i can check.  I might have been a little redundant above but that's because the posts I've seen people seem to confuse process scheduler with the i/o scheduler, and they don't seem to answer the actual question, just provide links to what a scheduler is which is not what I'm after here, so apologies if the above comments sounded rude. Warning or confirmation on new tag creation In retrospect of this , I would like to ask the community how they feel about a warning or confirmation (or just special highlighting) on new tag creation.  Everybody hates stock "are you sure?" confirmation, but a lot of tags get created by mistake (and then used ad nauseam).   For example, I think I might have created the  tag by accident. Assuming the right tag was , there would be two tags for c# 4.0, until they are made synonyms. With a warning, some accidental tags could be prevented. What is the rule for adjective order? Curved space-time VS change of coordinates in Minkowski space Email from IOS with attachments larger than 7.5Mb gets stuck in outbox and constantly tries to resend. Costs us hundreds in data fees This is the first part of the first half of the a proof in the Axiom of Choice. The proof that the Axiom of Choice is equivalent to the choice function. REsources: Keith Devlin's Joy of Sets: pg. 58. F.R. Drake and Singh Intermediate Set Theory: pg. 110).   I am confused on proving disjointness and will ask further questions of proof once I get this first part v. clear for myself.   AC1: Let $\mathcal F$ be a family of pairwise disjoint, non-empty sets. Then there is a set $\mathcal M$ that consists of precised one element from each member of $\mathcal F$  AC2: For every family $\mathcal F$ of non-empty sets, there is a choice function $\mathcal f$. (I.e. $\mathcal f(X)$ is a member of X, for each X in $\mathcal F$.   Disjointness: Two sets are disjoin if Q $\cap$ R = $\emptyset$. Meaning that they have zero elements in common. These sets do not overlap.   Show: AC1 $\leftrightarrow$ AC2   (1) ($\rightarrow$) Let $\mathcal F$ be a set of nonempty sets.   (2)For each X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X}) = {(x, X) | x $\in$ X})  (3) Show Disjointness.  An outline of how book proves  disjointness:  (a) if u = {(X $\times$ {X}) | x $\in$ X} then u is of the form $\lt  z, t \gt$ where z = X and t=x, i.e. x $\in$ X.   (b) then if u $\in$ ({X} $\times$ x) $\cap$ ({Y} $\times$ Y) $\rightarrow$ u = $\lt z, t \gt$ with z = X and z=Y, i.e. X=Y. Which means ({X} $\times$ x) = ({Y} $\times$ y).  Questions: (1) Apparently we cannot prove disjointness in $\mathcal F$ but must prove in a different set. If so, why is that?  (2) why is the case that if u $\in$ ({X} $\times$ X) $\cap$ ({Y} $\times$ Y) then z =X and z=Y? I am not following that step at all.   (3) why does the axiom of choice have to do with ordered pairs? Why are we setting X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X})?    Thank you! Renaming accessor/mutator methods in Eclipse? Is there any way to automatically rename accessor/mutator when a variable they get/set gets refactored -> renamed (Eclipse 3.4)? Can a continuous function from the reals to the reals assume each value an even number of times? Time Capsule cryptography? Does there exist any cryptographic algorithm which encrypts data in such a way that it can only be decrypted after a certain period of time?  The only idea that I can think of, is something like this: Seed a PRNG with a public value. Run the PRNG for a week and use the final value it produces to encrypt the message. Now anyone who runs that PRNG for a week starting from the seed value you made public can decrypt the message. Obviously this breaks down since they could use more computational power than you; if the time span was years then Moore's Law would apply, etc.  Is there anything like this other than physically burying a private key on a USB stick in a literal time capsule? What is vector division? My question is:  We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors? Uniform Scattering with vector I want to power my DIY stuff with a MacBook Pro USB C charger. The label on a charger says that it can provide three different power profiles:    20.2V - 4.3A 9V - 3A 5.2V - 2.4A   There is a ton of information about what USB-C power distribution is capable of, but I can't find any examples of how exactly to do it.   Is there an easy way or workaround to request one of those power profiles without using a microcontroller?  For example, I got apple MacBook Pro charger, apple USB-C cable, and a breakout board like this one: Note that this is only focusing on Minecraft Java Edition, and for versions 1.13+. I am predominantly looking for a solution that does not require any external programs (like MCEdit or NBTExplorer) and will work in just plain vanilla Minecraft, although if the only solution doesn't fit these criteria, I'll still take it (since having a solution is better than no solution!) Essentially, I am looking for a way to put a phrase (meaning essentially there are two or more words and spaces are involved) on a Minecraft scoreboard. This is what I mean by 'phrase': One two three, 1 2 3 These are not 'phrases': OneTwoThree, One-Two-Three, One_Two_Three, 1-2-3, etc. However, it seems like the only way to add text to an objective on the scoreboard is to add a &quot;fake player&quot;, via /scoreboard  players add [playername] [objectivename] [score], which shows the [playername] and the [score] part of the command on the scoreboard. However, since player names can't have spaces at all, it means that you can't have a phrase shown on the scoreboard. I have seen on many servers scoreboards that do have phrases listed on the scoreboard, but I have no idea how they do this (probably because of plugins that can do this sort of thing).  I did notice that teams can be given display names that can have spaces, so is there a way to display the team name on the scoreboard (thus being able to put phrases on there?)? Overall, my main question would be &quot;Is there a way to use spaces on the scoreboard (excluding the objective name) in vanilla Minecraft? Prove that $X^\ast$ separable implies $X$ separable There are several ways to execute a script. The ones I know are: /path/to/script # using the path (absolute or relative) . script        # using the . (dot) source script   # using the `source` command  Are there any other way? What are the differences between them? Are there situations that I must use one and not another? Windows installer for Ubuntu 14.04 LTS onwards