You have 6 green chips, 6 black, and 6 yellow. You randomly select a sample of 3 chips without replacement. What is the probability that the sample contains at least one chip of each color?
You have 6 red balls, 6 blue, and 6 white. You randomly select a sample of 3 balls without replacement. The balls, save for the color, are indistinguishable from each other. What are the odds that your sample contains at least a ball from each color?  I thought about finding the chance that the sample will not contain at least one color and subtracting that from one:  $$P(\text{sample is missing at least one color})=P(\text{sample is missing one color})+P(\text{sample is missing two colors}) $$  Essentially the second probability statement on the right hand side is the probability that all the balls in the sample are the same color. The case where the sample does not contain three colors doesn't make sense, so I didn't include it. $$=\binom{3}{1} \left[\left(\frac{12}{18}\right) \cdot \left(\frac{11}{17}\right) \cdot\left(\frac{10}{16}\right)\right]+\binom{3}{2}\left[\left(\frac{6}{18}\right) \cdot \left(\frac{5}{17}\right) \cdot\left(\frac{4}{16}\right)\right] \approx 0.882353$$  When all of this is done, we simply subtract the probability from one to get $0.117647$.  What am I doing wrong?
This is a  about hostname-awareness and proxying.   I know that some protocols are hostname-aware; that is, when I connect to the HTTP server at www.example.com it knows I want www.example.com's HTTP service, not www.example.net's, even though they are on the same IP address.  How can I do this for protocol foo?  (Temporary note: this question appears pursuant to .) My questions is very simple. How can I set security setting for a PDF file created with PDFLatex?  I should set some password or print permissions, etc. I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)  In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each term after the first two is the sum of the two previous terms. What is the sum to infinity of the series: $$\frac{1}{2} + \frac{1}{4}+ \frac{2}{8} + \frac{3}{16} + \frac{5}{32} +\frac{8}{64} + \frac{13}{128} +\frac{21}{256} +\frac{34}{512}+ \frac{55}{1024} + \cdots$$  Now, I solved this using an infinite geometric matrix series (incorporating the matrix version of the relation $a_n= \frac{a_{n-1}}{2}+ \frac{a_{n-2}}{4}$), and my students, after much hinting on my part, googled the necessary string to stumble across Binet's formula (which allows one to split the series into two simple, if rather messy, geometrics). Both of these are good methods, but neither really seems plausible for a challenge set for 15-18 year olds under exam conditions. So how is one supposed to do it? Why does the Moon not revolve around the Sun directly? Polygon intersection using turf js Is there a way to absolutely force a Log Out, Shut Down, or Restart with open tabs and unsaved documents? I could not find any relatively simple way to log out of a public use Mac quickly with many tabs open and unsaved document in the background without resorting to sudo shutdown -h now.  The reason I ask is that sometimes a user will have his/her account logged into the iMacs here at school and they would try to log off and then leave, thinking it has completed but the process is interrupted by Safari and Office apps, leaving their account, mounts, and websites still accessible to the next person who sits down.  Obviously a shortcut string would be nice instead of posting a note telling the person to type arcane things in a terminal. For example, just having them try to find Terminal.app is going to be a chore.  This doesn't seem to make sense because the user can just quit each app and click a few dialog boxes but unlike you and I to whom repeatedly pounding Command-Q while hitting the spacebar and D keys are probably second nature. Do I need to change my ESTA address when visiting the US? I have an ESTA for my visit last Sept to NY and I know it is valid for 2years.  I am going to Las Vegas in May, do I have to update the address at which I will be staying? Distribute objects optimaly So I've got a file with loads of simple objects. This will later be used to cut out these shapes out of plastic. So, using the material optimally is of outmost importance. I was wondering if there is any script or plugin that could automaticly arrange the objects within the artboard so they could fit best. Why does energy have to be emitted in quanta? I was reading some popular science book and there was this sentence. It says that energy has to be emitted in discrete portions called quanta. Otherwise the whole energy in the universe would be converted into  high frequency waves.  I'm not a physicist, so this conclusion seems to me like a huge leap.  First, we assume that energy is emitted in a continuous way (not in quanta).  And how do we get to the statement "the whole energy in the universe would be converted into  high frequency waves"? Solenoid Driving Circuit I'm trying to drive a 12v (14W) solenoid and have problems figuring out the components for the driving circuit.  The solenoid draws about 1.166A and the MCU is being operated at 3.3V.  I've seen many driving circuits and they all look something like this:         &ndash; Schematic created using   My questions:   What datasheet parameters should I look for when designing this circuit? Is the IRF530PBF-ND a suitable MOSFET for driving this solenoid?    Also, how do I calculate R1's value?  I don't want to blindingly copy a circuit that may or may not work, I want to understand it.  Thank you so much in advance! Are the tags shown in a particular order? I noticed that, sometimes, the tags I add to a question are not shown in the same order I wrote them, when the question is saved. Are they shown in a particular order? If they are, in which order they are shown? Requesting raw data from previously published research On Windows 10 Professional, with the November update (Build 10586, Version 1511) , is there any possibility to completely remove Windows Defender ?  I don't want to just disable Windows Defender.  I want to completely uninstall from my system. The "New Contributor" dialog is masked The New Contributor dialog is masked from where the comment section ends. Here are some screenshots just to be more clear: With no comments:    With one comment:    No masking with more comments: Can a willing lycanthrope be True Polymorphed? Can a lycanthrope or other shapeshifter be willingly affected by the spell True Polymorph? I know that it can be affected by regular Polymorph, but the text for the enhanced version seems to differ. Object not visible in render I have a model which you can see here (camera perspective).    In the second image, you can see how the render turned out.    I'm at my wit's end trying to figure our why it's not showing up in the render. If anyone needs more information, just ask and I'll get it to you. I've already checked the normals and they're not the cause. What is the 'molecular' origin of the viscosity? The molecular origin of elasticity is almost clear for me: at the very bottom the 'elasticity' comes from the attraction and repulsion between atoms when they are reasonably far from each other or very close (like Lennard-Jones potential). Is there a similar description for the viscosity as well? Where does it come from? 3D arrows with TikZ Asymptote does give 3D arrows when drawing three dimensional figures (they even have shade!). Could we fake it with TikZ? Here's an example, manually written, with just one arrow worked:  \documentclass{scrartcl} \usepackage{tikz} \usetikzlibrary{arrows.meta} \begin{document} \begin{tikzpicture}[-&gt;,scale=2]   \draw[&gt;={Triangle[width=3pt,length=5pt]}] (0,0,0) -- (1,0,0);   \draw[&gt;={Triangle[width=3pt,length=5pt]}] (0,0,0) -- (0,1,0);   \draw[&gt;={Circle[sep=-.9pt,width=3.08pt,length=2pt]Triangle[width=3pt,length=5pt]}] (0,0,0) -- (0,0,1); \end{tikzpicture} \end{document}     But that's (wrongly) manually calculated by trial and error (apart from the fact that it's not perfect, seeing how the other two axis —x and y— are oriented, it should have certain slant). It would be nice to have an interface like the other arrows  &gt;={3D[length=⟨real length⟩,width=⟨real width of the base⟩]}   Which would then calculate from those values (and, of course, also taking into account x=⟨x vect⟩,y=⟨y vect⟩,z=⟨z vect⟩)  Circle[sep=⟨value⟩,width=⟨value⟩,length=⟨value⟩,slant=⟨value⟩] Triangle[width=⟨value⟩,length=⟨value⟩] % And I'm not sure if `slant=⟨value⟩`                                        % is necessary in this Triangle case   Example  Using the code from the answer (thanks to Symbol1), for instance, this code (taking the idea from another question, link in the comments)  \begin{tikzpicture}[x={(0cm,1cm)},                     y={({cos(30)*1cm},{sin(30)*-1cm})},                     z={({cos(20)*1cm},{sin(20)*1cm})},                     thick,                     scale=3]   \pgfmathsetmacro{\ax}{.4}   \pgfmathsetmacro{\ay}{.6}   \pgfmathsetmacro{\az}{-.6}   \pgfmathsetmacro{\bx}{.4}   \pgfmathsetmacro{\by}{-.7}   \pgfmathsetmacro{\bz}{-.8}   \draw[densely dotted,thin] (\ax,\ay,\az) -- (\ax,0,\az);   \draw[densely dotted,thin] (\ax,\ay,\az) -- (0,\ay,\az);   \draw[densely dotted,thin] (\ax,0,0) -- (\ax,0,\az) -- (0,0,\az);   \draw[densely dotted,thin] (0,0,\az) -- (0,\ay,\az) -- (0,\ay,0);   \draw[densely dotted,thin] (\bx,\by,\bz) -- (\bx,0,\bz);   \draw[densely dotted,thin] (\bx,\by,\bz) -- (0,\by,\bz);   \draw[densely dotted,thin] (\bx,0,0) -- (\bx,0,\bz) -- (0,0,\bz);   \draw[densely dotted,thin] (0,0,\bz) -- (0,\by,\bz) -- (0,\by,0);   \draw[-{Cone3}] (0,0,0) -- (1,0,0) node[right] {$x$};   \draw[-{Cone3}] (0,0,0) -- (0,1,0) node[above right] {$y$};   \draw[-{Cone3}] (0,0,0) -- (0,0,1) node[below right] {$z$};   \draw[-{Cone3}] (0,0,0) -- (0,0,-1) node[above left] {$-z$};   \draw[thin] (0,0,0) -- (0,-.9,0);   \draw[-{Cone3}] (0,0,0) -- (\ax,\ay,\az) node[below left] {source};   \draw[-{Cone3}] (0,0,0) -- (\bx,\by,\bz) node[above right] {sim}; \end{tikzpicture}   gives this figure Can we make it more plain to recruiters that we're not open to relocation? The preference for being open to relocation is a checkbox, and if you don't check it, nothing about relocation shows up in your profile.  Evidently recruiters can't understand that a lack of "I am open to relocation" is the same thing as "I am not open to relocation", and so every single contact I've gotten through Careers is for a job that I would have to relocate for.  Can we make it more obvious to the recruiters (especially ones from Amazon) that we have not checked that box on purpose? error in running bash script as root I have a simple shutdown script which i want to run as root  shutdown -h +30; echo "succesfull";   I have followed these answer, to make my script root.  first I run this command  sudo chown root.root $HOME/test/test.sh sudo chmod 4755 $HOME/test/test.sh   and then made changes in sudoers  sudo visudo   added this command after this line %sudo   ALL=(ALL:ALL) ALL  eka ALL=(ALL) NOPASSWD: $HOME/test/test.sh   But when i executed my script its showing this error  shutdown: Need to be root succesfull