If the events $A$ and $B$ are independent and the events $B$ and $C$ are independent, does this necessarily mean events $A$ and $C$ are independent?  I used coin tosses to try to model this with $A = H$, $B = T$, and $C = H$ in separate fair tosses.  I get that they are independent, but am not sure if I am correct.  I also have problems determining if $B$ is independent to $A\cap C$ and also $A \cup C$.
Take any three random variables $X_1$, $X_2$, and $X_3$.   Is it possible for $X_1$ and $X_2$ to be dependent, $X_2$ and $X_3$ to be dependent, but $X_1$ and $X_3$ to be independent?  Is it possible for $X_1$ and $X_2$ to be independent, $X_2$ and $X_3$ to be independent, but $X_1$ and $X_3$ to be dependent?
What is the correct way to write 1.5 hours? I'm a bit confused in describing 1.5hrs in words.   Is writing one and a half an hour correct or should it be one and half hour?  E.g:     I'll see you there in one and a half an hour.     OR       I'll see you there in one and half hour.   Or is there any other correct way of writing this? Lightning: why can you not have an expression combined with text in an HTML attribute? So, I've been struggling with this for a while, trying to figure out how to get it to work. Here is what I originally did:  &lt;button class="btn btn-primary {!v.isLink1 ? 'active' : ''}"&gt;   And so on. Here's the documentation on it.   However, when you actually try to save this in the developer console, you get the error:  Field Integrity Exception: Failed to save undefined: Cannot mix expression and literal string in attribute value, try rewriting like {!'foo' + v.bar}: Source   I finally figured out that you have to put all of the classes inside of the expression, like so:  &lt;button class="{!v.isLink1 ? 'btn btn-primary active' : 'btn btn-primary'}"&gt;   This doesn't seem very intuitive. It's also not documented anywhere I could find. Could someone explain to me why it is this way and if it will ever change?   (and hopefully this question saves someone some headaches in the future) How do I remove duplicates from a C# array? I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array? I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? Do I need a transit visa for a layover in the United Arab Emirates? File l3backend-pdfmode.def not found when loading expl3 I updated my TeXlive distribution a couple times in the past week, and now I get this error when I include package expl3 (or any other package that includes it, such as fontspec). Before the updates it was working fine. Am I missing any package that should be installed? I tried searching for l3backend-pdfmode.def, but I found no results.  \documentclass{article} \usepackage{expl3} \begin{document} Hi \end{document}   λ lualatex a This is LuaTeX, Version 1.10.0 (TeX Live 2019)   restricted system commands enabled. (./a.tex LaTeX2e &lt;2018-12-01&gt;  luaotfload | main : initialization completed in 0.042 seconds (/opt/texlive/2019/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/opt/texlive/2019/texmf-dist/tex/latex/base/size10.clo)) (/opt/texlive/2019/texmf-dist/tex/latex/l3kernel/expl3.sty (/opt/texlive/2019/texmf-dist/tex/latex/l3kernel/expl3-code.tex)  ! LaTeX Error: File `l3backend-pdfmode.def' not found.  Type X to quit or &lt;RETURN&gt; to proceed, or enter new name. (Default extension: def)  Enter file name:  ! Emergency stop. &lt;read *&gt;   l.282   }   379 words of node memory still in use:    2 hlist, 1 rule, 1 dir, 3 kern, 1 glyph, 4 attribute, 48 glue_spec, 4 attrib ute_list, 1 write nodes    avail lists: 2:9,3:3,4:1,5:2,7:2,9:2 !  ==&gt; Fatal error occurred, no output PDF file produced! Transcript written on a.log. Consider this question on .  I put out a 200 bounty on the question, but did not get an answer from anyone other than myself that worked.    In fact, my answer is also the highest up-voted on it at this point.  I know that allowing OP's to retract a bounty can be a slippery slope; in fact I think that once the bounty is put up it should stay up.  But why not allow the OP to answer their own questions with a bounty and reward it to them if they receive the same up-votes as others?  Yes, I am fully aware that bounties are a risk.  People will use them for numerous reasons such as trying to draw extra attention to their issues.  However, in my specific question, the only valid answer is the one that I gave.  Had it been anyone else's question, I would have received the bounty, but since it was my own I did not. sed append after a line containing multiple unordered strings I'm trying to use sed to append appendstring after a line that contains both Hello and World.  Hello something here World World fsf Hello   This works if I do consider order:  sed -i '/Hello*World\|World*Hello/a appendstring' file   But if I have many strings to match and they are not in order, this is definitely not effective.  Then I tried solutions from another link which discussed deleting a line that contains both strings without considering their order.     However,  sed -i '/Hello/!b;/World/d' file   This does not even work for deletion.   The following one, also one of the answers to that question:  sed -i '/Hello/{/World/d}' file   works, so I tried to implement it to append. Using:  sed -i '/Hello/{/World/a} appendstring' file   But received error:  sed: -e expression #1, char 0: unmatched `{'   Update: The post in the link stated the solution with !b, and it was not working in cshell because of the exlamation mark, but it is working in sh and bash. And it can also be easily extended to inserting many lines when running sed inside a script and the appending a large paragraph.  sed -i '/Hello/!b; /World/a \ addline \ addanotherline ' file iOS emails stuck in All Inboxes In the iOS Mail app, there are 4 emails in the "All Inboxes" section. Those 4 emails are NOT in the actual inbox on Gmail.  I thought, oh well I don't mind deleting them entirely since they're just newsletters. I deleted them directly on gmail, but they still appeared in iOS. So I tried deleting them via iOS and ... they keep coming back!  Even stranger, the 4 emails ONLY appear in "All Inboxes", they don't appear in any of the sub-account inboxes.  So, I disabled "Mail" in my gmail account. They're still here! I no longer have a "gmail" sub-account, but those 4 emails are still in All Inboxes. I tried to quit the app a few times, restart the phone. Still here...  When I tap on one of those messages, I get a black screen (dark mode), no errors. Proof concerning Stirling numbers of the first kind How do I show that   $${ n\brack 2} = \frac{n!}{2} \sum_{m=1}^{n-1} \frac{1}{m(n-m)}$$  using a combinatorial argument?  My definition of ${n \brack 2}$ is "the number of elements of $S_n$ which can be decomposed as the product of exactly $2$ disjoint cycles". How do I create a public view from a private view It seems like I can't modify a private view to be public, or duplicate a private view to be public.  Am I restricted to copying by hand all the settings of my private view, then creating a fresh public view and selecting all the right columns in the same order with all the sort and filter options? When I take the pth root, the bottom of the p comes very close to the \sqrt symbol.  In the New Century Schoolbook font (which I am using), the p actually intersects the \sqrt symbol, which looks even more terrible.  Is there a clean way to fix this?  \documentclass{article} \usepackage{fouriernc} % use the New Century Schoolbook font \begin{document} When I write $a^{(1/p)}$ as $\sqrt[p]{a}$, the bottom of the $p$ touches the top of the root symbol, which looks ugly. Even when the root symbol is larger, such as with $\sqrt[p]{\frac{1}{n}}$, it still looks bad (since the tail of the p almost hits that line). \end{document} Why doesn't an orbiting body keep accelerating? As a physics newbie I'm trying to get a grip on basic orbital mechanics. I think I'm beginning to get grasp on how bodies interact with each other. When a body approaches another body it accelerates due to gravity. It can reach a point where its velocity is high enough to keep falling but also keep missing the object it is falling towards. What keeps it from accelerating (because of gravity) and eventually reaching escape velocity? I feel like I'm either looking at things the wrong way or I have the entire thing wrong. How to overwrite only part of a file in c++ Calculate $\lim\limits_{x\to4} \frac {\sqrt {1+2x}-3} {\sqrt x -2}$ After I just learned there are ways to cancel out $0$s in the divisor of fractions onto limits I looked back at a task where I gave up when I got the result $\lim\limits_{x\to4} \frac {\sqrt {1+2x}-3} {\sqrt x -2}$ so I tried to find a way to get the $0$ out here, as well. Am I just not seeing the solution or is there no way to do it? Let $SL_n(\mathbb{R})$ be the set of $n\times n$ matrix with determinant one. Where can I find a proof that $[SL_n(\mathbb{R}),SL_n(\mathbb{R})]=SL_n(\mathbb{R})$? I've seen this result mentioned in several places, but couldn't find a proof.  For reference, the commutator $[SL_n(\mathbb{R}),SL_n(\mathbb{R})]$ is defined as the group generated by the set of all elements of the form $a^{-1}b^{-1}ab$, with $a,b\in SL_n(\mathbb{R})$. The Java documentation says:      It is not possible for two invocations of synchronized methods on the   same object to interleave.   What does this mean for a static method?  Since a static method has no associated object, will the synchronized keyword lock on the class, instead of the object? The problem is to prove the following equivalence:  $$2^{\aleph_0}=\aleph_1 \iff \exists\ A,B\in\mathbb{R}^2:\\ \textrm{a)}A\cup B=\mathbb{R}^2 \\\textrm{b)}\ \forall\ \textrm{lines } l\ \textrm{parallel to x-axis, } |A\cap l|\leq\aleph_0 \\\textrm{c)}\ \forall\ \textrm{lines } l\ \textrm{parallel to y-axis, } |B\cap l|\leq\aleph_0$$  I think I know how to prove "$\impliedby$":  Let's assume the RHS and that $2^{\aleph_0}\geq\aleph_2$. Now let's take a sequence $\langle x_{\alpha}:\ \alpha&lt;\omega_1\rangle$. Since $\bigcup_\limits{\alpha&lt;\omega_1}|B\cap x_{\alpha}|\leq \aleph_1\cdot\aleph_0=\aleph_1$ there is such a $y$-coordinate that neither of $(x_{\alpha},y)$ is in $B$. But hence we obtain $\aleph_1$ point which all lie on the same line parallel to $x$-axis. Since $A\cup B=\mathbb{R}^2$ they all must be contained in $A$ $-$ a contradiction as $A$ has at most countably many of those points.  But what about the other implication? Do I contruct these sets by induction? Over all lines or something else? I haven't come up with anything yet, perhaps you could help me. Linked and Related sections contain duplicate links to questions Coloring arrows in \xymatrix