If 2 players toss a fair die, one tosses it 999 times and the other tosses it 1000 times, what is the probability that one gets more even numbers than the other? It's clearly comparing binomial distributions but I'm not sure how to do it.
First player rolls a 6-sided die 100 times and the second player 101 times. What are the odds of the second player getting more odd numbers than the first one?  I tried solving this via listing the options which led to a nasty sum that I wasn't able to solve neither by hand nor by using wolfram alpha. Clearly there has to be a more clever solution using the properties of a binomial distribution, but so far I haven't come across anything useful.
Question : $$\begin{align}a+b+c &amp;= 0\\ a^3 +b^3 +c^3 &amp;= 12\\ a^5 +b^5 +c^5 &amp;= 40\end{align}$$ Then , $a^4 + b^4 + c^4 = ?$  My try : As a common perspective I just went to find any trick related to it and my first step is as usual as a common man will think $3abc = 12$ and got $abc = 4$ After that I am unable to link it with any other equation What do I need to get photos with a unifom black background (not with post)? Often i see portraits such as the one below and like the way they look. I wanted to know can I get this look in my shots.  I am sure there are professional materials you can get to outfit a studio that will deliver such results. I have no clue what these items would be, can i get clued in?  Also, I was wondering if there is a DIY way to achieve this as well. Maybe with a black bed sheet or something?  I want to set something up in a room in my house for studio type shots. I am looking to get an idea of what items I need to get this done and how the setup should go. How to automatically keep user remembered in Devise Gravitational field intensity inside a hollow sphere How do I pre-install Ubuntu for someone (OEM install)? Show the post's comments on the "Suggested Edits" review interface Why do I get a black rectangle instead of my external picture? I write some of my network notes in TeX. I wanted to make a table, and inside it insert an image:   .   I wrote a document:  \documentclass{article} \usepackage[demo]{graphicx} \usepackage{booktabs} \usepackage{array}  \begin{document} \begin{tabular}{m{3cm}m{3cm}m{5cm}} \toprule Configuration for ... &amp; Commands &amp; Topology\\ \midrule PPP: CHAP, PAP &amp; \#sh int se 0/0/0  &amp; \includegraphics[width=4cm, height=4cm]{chappap.png}\\  &amp; &amp; \\ \bottomrule \end{tabular} \end{document}   But instead of my network topology image, I see only this: .   I don't know what is wrong, cause my image is 100% in the same dir, as my *.tex file.  EDIT:  Changed my code as @Jubobs suggested, now everything is as I wanted:  \documentclass{article}     \usepackage{graphicx}     \usepackage{booktabs}     \usepackage{array}      \begin{document}     \begin{tabular}{m{3cm}m{3cm}m{5cm}}     \toprule     Configuration for ... &amp; Commands &amp; Topology\\     \midrule     PPP: CHAP, PAP &amp; \#sh int se 0/0/0  &amp; \includegraphics[width=4cm, height=4cm]{chappap.png}\\      &amp; &amp; \\     \bottomrule     \end{tabular}     \end{document} What is the name of the horizontal division line? What is the name of the horizontal division line? Is there a name for it other than "division line? Where are packages stored installed with APT in Ubuntu 16.04? What is the explicit meaning of “corresponding author”? Problem  This question was triggered by the comments on , but comments and answers on   confirm the existence of conflicting ideas.  There seem to be two prevalent interpretations of the word corresponding author:   The corresponding author is the one who corresponds the (unpublished) paper to the journal, i.e., who performs the technical process of submitting the paper, is available for requests on the review and publishing processes, and so on. The corresponding author is the one who corresponds on the (published) paper with anybody who likes to do so, e.g., because they found an error or like some clarification.   What I found out so far  Until today, I was only aware of the first interpretation, which was confirmed by my experience: For all papers I submitted, I automatically became the corresponding author and cannot remember that I could choose one of my co-authors for this role. Some of these journals do not even indicate a corresponding author to the reader. Also, there already is a way to indicate that an author is available for questions on the paper (interpretation 2): stating the e-mail address.  On the other hand, other journals do indicate corresponding authorship, which makes little sense for interpretation 1. Also, the value put into this role by some funding agencies for its  makes more sense for interpretation 2 (not that I completely agree with it).  Looking for some guidelines from publishers covering multiple fields, I found the following  from Elsevier¹, which kind-of covers both interpretations:     Clearly indicate who will handle correspondence at all stages of refereeing and publication, also post-publication.   While there are other questions on this site touching this issue, they focus on consequences of the interpretation, such as , , or .  Actual question  What is the explicit meaning of the word corresponding author? Interpretation 1 or 2 as listed above, both, something else, or is this an extreme example of  and the answer depends on the field or publisher? Please answer only if you have more to offer than personal experience and individual journal guidelines. In particular, please address the possibility of a varying meaning.     ¹ which is separate for each journal, but seems to be identical in content for many journals How to add Windows to GRUB I have my encrypted Archlinux installed on my SSD. Now I have one HDD with Windows installed. But in my GRUB just Archlinux is shown (no Wonder, because I just installed Windows on the other HDD, GRUB doesn't know of that by now.).  My Question is, how do I properly add my Windows-HDD to GRUB, so that I can choose which to boot when starting?  I am a little bit afraid of destroying my GRUB so I can't boot my encrypted Archlinux anymore.  Thanks! Simplest way to send one-line mail out via command line using gmail? Using Raspbian and Ubunntu 16.04 LTS so need a generic Linux solution.  Requirement is simple:  I need a way to send one-line email messages from the command line.  I have set up a gmail account just for this particular Rpi3, with the address of rpi3abc@gmail.com - with no 2FA  So now I need to be able to send one-line mail messages from anywhere (including cron) without user intervention.    I also would like it to be able to send text files; basically, anything from stdin. Evaluate $\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$ So I really like the tag cloud on Climate Deal. Is there anyway we could get something similar to this on StackOverflow? I don't necessarily mean that we should replace the interesting/ignored tags with this but something like this on the front page would be nice. How to get the environment variables of a subprocess after it finishes running? About free quarks and confinement I simply know that a single free quark does not exist. What is the reason that we can not get a free quark?  If we can't get a free quark then what is single-top-quark? How to grep for text in a file and display the paragraph that has the text? Below is the text in the file:  Pseudo name=Apple Code=42B state=fault  Pseudo name=Prance Code=43B state=good   I need to grep for "42B" and get the output from the above text like:   Pseudo name=Apple Code=42B state=fault   Does anyone have idea on how to achieve this using grep/awk/sed? Union of the conjugates of a proper subgroup Let G be a finite group and H be a proper subgroup. Prove that the union of the conjugates of H is not the whole of G.  Thanks for any help I've been writing in English for many years, but there are certain recurring issues that come up in my writing, which I can't seem to resolve permanently. Primarily I have a problem with punctuation. In my native language, commas are very commonly used. One would place a comma to separate any portion of the sentence that serves a specific grammatical role. At least, this is the case according to the rules I was taught in school. I can see when I write in English that there are extraneous commas, but I can't always figure out which ones, and why. When dependent clauses require commas, when do they allow them, and when are they definitely incorrect? Can Compound Topographic Index (CTI) / Topographic Wetness Index (TWI) produce negative values? I am calculating Topographic Wetness Index for a study area using SRTM DEM (90 and 30m).   I've seen several versions of TWI (CTI).  I have been using: TWI = ln(As +1 / tanB)       where As = catchment area, B = slope (degrees/radians??)  When I run this function, the output contains negative values (using slope in degrees and radians).  Are negative values acceptable? Or does this indicate an error in the analysis??  I understand the output should be scaled to 1-10, but negative values seems counter intuitive. Could someone explain what negative values could mean (due to flat areas, no data values?)  On a side not, the calculations are being run on on a very large area, which overaps several watersheds. Is it appropriate to run TWI over such a large extent?