A worker carries two bags. Each of the bags initially contains N nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point the worker picks an empty bag. Find the probability that the other bag contains exactly m nails.
A worker carries two bags. Each of the bags initially contains N nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point the worker picks an empty bag. Find the probability that the other bag contains exactly m nails.  My reasoning:  The desired probability: $C_2^1 \cdot P(|A| = 0\ and\ |B| = m)$, where A, B - sets of nails in each bag. Then we can "divide" this probability (I suppose that I deal with independent events):  $P(|A| = 0\ and\ |B| = m) = P(|A| = 0) \cdot P(|B| = m);$  $P(|A| = 0) = (\frac{1}{2})^N$, where $|A| = |B| = N$ (initial condition).  $P(|B| = m) = (\frac{1}{2})^{N - m}$  Result: $C_2^1 \cdot P(|A| = 0\ and\ |B| = m)$ = $C_2^1 \cdot (\frac{1}{2})^{2N - m}$  Am I wrong or it is correct?
When are infinities okay? I'm a little confused about how to handle infinities in physics. On the one hand, I always here things like infinities are not physically tenable answers. GR can't answer anything about the origin of the universe because of singularities. Throw away the radial solution that gives an infinite potential when solving the Laplace equation in spherical coordinates. But then there are other times that the infinities are okay. For instance, it is usually okay to set the reference point for your potential at infinity. There are certain times in David Tong's quantum field theory lecture notes where he seems to almost to hand wave away infinities. So, philosophically, physically, and mathematically when are infinities okay, and when are they not okay? Evaluate the sum $\sum_{k=1}^n {n \choose k-1} {n \choose k}$ Evaluate the summation of: $$ \sum_{k=1}^n {n \choose k-1} {n \choose k} $$ I have been trying to solve this problem by translating C(n, k-1) into C(n, k), because I know that: $$ \sum_{k=1}^n {n \choose k}^2={2n \choose n} $$ However, I haven't been having any luck. Am I on the right track? What am I doing wrong? Conjugate to $\cos z$ is not equal to $\cos z$ conjugate $\overline{\cos(z)}\neq\cos(\overline{z})$ I have a dilemma. I have a task where I'm supposed to show that $|\cos z|^2+|\sin z|^2=1$ if and only if $z\in \mathbb{R} $. In my argument I have that $$\cos z\cdot\overline{\cos z} + \sin z\cdot\overline{\sin z}=\cos z\cdot \cos\overline{z} + \sin z\cdot \sin\overline{z}$$ but my teacher said that $\overline {\cos z} \neq \cos \overline{z}$.  Is this true?  But $$\overline{\cos(z)} = \overline{\frac{e^{iz}+e^{-iz}}{2}}=\frac{e^{-i\overline{z}}+e^{i\overline{z}}}{2}=\cos(\overline{z})$$ Take for example  both at SO and SU. Now that there's , (new) users asking a LaTeX question should at least be informed that there is a new Stack Exchange site dedicated to TeX and thus increases their chance of an answer. This should happen before the question is actually posted and ideally offer the option to post the question directly there instead (without the need to copy-paste).  LaTeX is just an example, there's also , ,  etc., and it is obvious that this can only be decided on a case-by-case basis which is why I certainly do not suggest migrating questions automatically when these tags are used. But informing users of that possibility would probably help the SE network in general. I understand that you can (in principle) measure whether "free particles" (no forces) experience accelerations in order to tell whether a frame is inertial. But fundamentally, what determines which frames are inertial (i.e. what principle selects in which frames free particles will not appear to accelerate)? I've been told that the cosmic microwaves determine the ultimate rest frame of the universe, but that doesn't make sense to me, since one can still ask why that frame is an inertial frame.  Also, I understand that there are no real inertial frames in general relativity, but it seems like there certainly exists approximate inertial frames and we can ask why those frames are approximately inertial and not others. For example, in the frame of a person riding a merry go round, free particles appear to greatly accelerate; while in the frame of someone standing next to the merry go round there are no such great accelerations. Why does the guy (or gal) on the merry go round see free particles accelerating while the other guy doesn't.   And if you're gonna tell me that it's "the rest of the stuff in the universe" that determines whether the person on the merry go round sees free particles accelerate, I'll ask how you know that all that stuff is not spinning.  I hope this question sort of makes sense, it's been bothering me for a while and my study of relativity (most of special relativity and just the outline of general relativity) hasn't really clarified it for me much. How do I know I have the correct exposure when shooting in manual mode? I have a Canon XTI which I've been using for a few years now. Most of the time I'm been shooting in one of the "semi-manual" modes like aperture priority or shutter priority, but now I'd like to try using full manual mode. I should have a new EF-S 17-55 f/2.8 delivered when I get home today. Before I was using a variable aperture Sigma 17-70 which made manual mode more difficult with the aperture changing arbitrarily.  I understand the effect of changing aperture and shutter speed, but right now I'm accustomed  to only controlling one at a time. The camera is usually setting the other to get the right exposure automatically for me. So if I'm controlling both how do I know I have the correct exposure? I understand I need to use the metering in my camera but I'm not sure exactly how do do that.   First when should I be using each of the metering modes (Evaluative, Partial, Center weighted, or Spot)? These just change the area of the scene which is metered correct?  How do I know what the meter should read off a specific object? Obviously if I just happen to be shooting a 18% gray wall then it should be at zero, but that can't apply to everything? If I'm shooting a black cat and take a spot meter reading off of the cat then the meter should read less then zero right? But how do I know how much less then zero? Are there rules of thumb I can follow? I've ordered a couple gray cards with my new lens. So experimenting with those should help.  How often should I be changing these settings? Say I'm at a party indoors which is consistently lit. I should be able to set the aperture, shutter speed, and ISO and just forget about it unless the lighting changes correct? Does the same apply to outdoor scenes? I've been hesitant to use manual mode in the past because I felt like I was spending a lot of times messing with settings before each shot. But not I'm realizing that this is really only the case when the lighting changes, or I want a different style of photo such as a different depth of field.  Finally, does exposure change with distance to the subject? For example say I want to take a picture of a statue with correct exposure using a gray card. I'd place the card on the statue, and while being careful not the block the light on the card find the correct exposure by spot metering the card. If I were then to step back 20' to get the whole statue in frame would I still have the correct exposure? What tool do I need to install to make my terminal show the autocompletion of the command that I'm writing? Bluetooth Broadcom 43142 isn't working Bluetooth is on but can't find or be found by any other devices.  Output of sudo service bluetooth status:  ● bluetooth.service - Bluetooth service    Loaded: loaded (/lib/systemd/system/bluetooth.service; enabled; vendor preset: enabled)    Active: active (running) since ხუთ 2015-06-04 22:33:18 GET; 13min ago  Main PID: 26678 (bluetoothd)    CGroup: /system.slice/bluetooth.service            └─26678 /usr/sbin/bluetoothd -n  ივნ 04 22:39:14 Rangoo bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/HFPAG ივნ 04 22:39:14 Rangoo bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/HFPHS ივნ 04 22:39:14 Rangoo bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/A2DPSource ივნ 04 22:39:14 Rangoo bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/A2DPSink ივნ 04 22:39:14 Rangoo bluetoothd[26678]: bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/HFPAG ივნ 04 22:39:14 Rangoo bluetoothd[26678]: bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/HFPHS ივნ 04 22:39:14 Rangoo bluetoothd[26678]: bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/A2DPSource ივნ 04 22:39:14 Rangoo bluetoothd[26678]: bluetoothd[26678]: Endpoint registered: sender=:1.63 path=/MediaEndpoint/BlueZ4/A2DPSink ივნ 04 22:39:14 Rangoo bluetoothd[26678]: Adapter /org/bluez/26678/hci0 has been enabled ივნ 04 22:39:14 Rangoo bluetoothd[26678]: bluetoothd[26678]: Adapter /org/bluez/26678/hci0 has been enabled   Output of lsusb:  Bus 003 Device 005: ID 0a5c:21d7 Broadcom Corp. BCM43142 Bluetooth 4.0   How can I fix it? Probability that $m$ balls will fall into first box Suppose we have $n$ balls that are randomly distributed into $N$ boxes. Find the probability that $m$ balls will fall into the first box. Assume that all $N^m$ arrangements are equally likely.  Attempt: First, we notice that for the first box, we have $n$ choices, and for the second box we also have $n$ choices, and so on. Thus, we have $n^N$ ways to place the balls into the boxes. Pick $m$ balls out of the total $n$ balls, that gives ${n \choose m}$. In how many ways can these $m$ balls go into the first box? Well, in just ${1 \choose 1 }= 1 $ ways. Thus, $$ P = \frac{ {n \choose m } }{n^N } $$ Is this correct? Prove that $T$ has an orbit of period 3 Verbless sentence How do I make ditto marks? In France, students are taught not to repeat the same text twice when there is little difference between a line and the one below. It's a quick way to take notes.  I reproduced this effect by tinkering:    The code is as follows:  \documentclass[aspectratio=1610]{beamer}  \usepackage[T1]{fontenc} \usepackage[french]{babel}  \usepackage{amsmath, amsfonts, amssymb} \usefonttheme[onlymath]{serif}  \begin{document} \begin{frame} Lorsque le \og \textit{carreau carré} \fg{}   mesure: \begin{itemize} \item $\rm 1\,cm$ de côté, l'unité d'aire est appelée le \og \textrm{centimètre carré} \fg{} et est noté $\rm cm^2$; \item $\rm 1\,dm$ \hspace{1mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}''\hspace{8mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{décimètre carré} \fg{} \hspace{2mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1.5mm} $\rm dm^2$; \item $\rm 1\,m$ \hspace{3mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}'' \hspace{7mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{mètre carré} \fg{}  \hspace{9mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1mm} $\rm m^2$; \item etc. \end{itemize} \end{frame} \end{document}   How can I get the same effect more simply than by tinkering like I just did? Word for a temporary project name I often see output from TeX with the warning overfull hbox, badness 10000. What does this message mean? During Winter Bash, users can earn hats. But sometimes it seems that a hat was awarded even though its requirements weren't (yet) met, or vice versa.  What could be the reason for this? I bought a Mac (13", early 2011) almost 5 years ago. When I first got it, it was running under OS X 10.7 and it was pretty fast. I've being upgrading it ever since and it's now running under Mac OS X 10.11.  I guess it didn't totally survive the upgrades because, today, there's a lot of bugs, freezes, lags, "spinning beach ball" appearing,... for example, when I write something on a normal program/editor, the cursor freezes for a few seconds, then catches up to what I'm writting.  As you can guess, that's really annoying! I've been thinkning about upgrading the hardware but I don't know what to do: should I change the RAM to 8 GB - 1 333 MHz (currently my computer has 4 GB - 1 333 MHz)? Or should I upgrade to an SSD (which is pretty expensive and difficult to install)?  Thank you very much. Updating VS Code in ubuntu 18.04 I'm trying to typeset SuperCollider code using the listings package.  Any "freestanding" identifiers that begin with a capital letter are class names, and should be highlighted; by "freestanding", I mean outside of strings or comments. I would be tickled pink if listings could let you define identifiers in terms of regexp, e.g. [A-Z][A-Za-z0-9_]* but the manual doesn't suggests that as a possibility.  SuperCollider has some 2300 classes... I really don't relish listing all of them literally in my preamble. I guess I could extract them by hand for each individual listing, but I'd rather not have to.  Is there a way to do this without a really massive morekeywords expression? (Perhaps using another package?)  Sample SuperCollider code:  p.clear;  ~grains.addSpec(\tfreq, [1, 40, \exp]); ~grains.addSpec(\overlap, [0.1, 10, \exp]); ~grains.addSpec(\pos, [0, b.duration]);  // 3.43 is nice! ~grains.addSpec(\rate, [0.5, 2, \exp]); ~grains = { |tfreq = 25, overlap = 6, pan = 0, amp = 0.2, pos = 3.43,    rate = 1|    var trig = Impulse.ar(tfreq);    TGrains.ar(2, trig, b, rate, pos, overlap / tfreq, pan, amp) }; ~grains.play;   Impulse and TGrains should be highlighted in blue. I have dozens of other listings using different capitalized keywords. I'm a web developer, started to become a web designer. However, I think my understanding of colors is not practical yet. There are tools out there that can give you color harmonies, and more stuff, like , but they don't give you the start point. I mean, how should I know that for this web site, I'd better use green and for that website, choose red. How should I know that?  I think color selection is more like an art, instead of being a science, and I'm like "you should become an color artist, in order to become a web designer".  How can I improve my color selection skill? Do Wine Viruses only work while Wine is running?