Problem statement : +++++++++++++++ Given two identical unbiased dice, determine the probability of getting sum as 7. Event  = Sum of dots on the top face of both dice is 7. $E = {(1,6),\ (2,5),\ (3,4),\ (4,3),\ (3,4),\ (5,2),\ (6,1)}$ $|Sample Space|$ = $36$. Hence, $P(E) = 1/6$ I have a doubt here. As the two dice are given identical, why do we have to consider ordered pairs? Shouldn't it be unordered consisting of only 3 possible pairs $\{(1,6),\ (2,5),\ (3,4)\} $? Hence, $|S| = 21$ and $P(E) = 3/21$.
In either case, one coin flip resulted in a head and the other resulted in a tail. Why is {H,T} a different outcome than {T,H}? Is this simply how we've defined an "outcome" in probability?   My main problem with {H,T} being a different outcome than {T,H} is that we apply binomial coefficients (i.e. we count subsets of sets) in some common probability problems. But if we take {H,T} and {T,H} to be different outcomes, then our "sets" are ordered, but sets are by definition unordered...   I feel as though the fact that I'm confused about something so basic means that I am missing something fundamental. Any help or insight whatsoever is greatly appreciated!
When quoting something said by someone else, should I use a comma before the quote? Suppose I quote the definition given in a dictionary. Am I correctly using the the punctuation marks in the following sentences? (I am using them in the way I think correct in American English.)     One of the meanings of cat is, "a small animal with soft fur that people often keep as a pet."        A cadet is, "a young person who is training to become an officer in the police or armed forces."   Should I use a comma only when I explicitly quote somebody?     Groucho Marx once said, "Although it is generally known, I think it's about time to announce that I was born at a very early age."   Should I use a comma in this case too?     As Groucho Marx said,            No one is completely unhappy at the failure of his best friend.      On Comma Sense, A fun-damental guide to punctuation (Richard Lederer and John Shore), I read the following about using the comma:     Use commas to set off complete quotations:           The great general George S. Patton once said, "No, no—the war is this way, you idiots!" I have a trained logistic regression model that I am applying to a testing data set. The dependent variable is binary (boolean). For each sample in the testing data set, I apply the logistic regression model to generates a % probability that the dependent variable will be true. Then I record whether the acutal value was true or false. I'm trying to calculate an $R^2$ or Adjusted $R^2$ figure as in a linear regression model.  This gives me a record for each sample in the testing set like:  prob_value_is_true         acutal_value    .34                          0    .45                          1    .11                          0    .84                          0     ....                        ....             I am wondering how to test the accuracy of the model. My first attempt was to use a contingency table and say "if prob_value_is_true > 0.80, guess that the actual value is true" and then measure the ratio of correct to incorrect classifications. But I don't like that, because it feels more like I'm just evaluating the 0.80 as a boundary, not the accuracy of the model as a whole and at all prob_value_is_true values.  Then I tried to just look at each prob_value_is_true discrete value, as an example, looking at all samples where prob_value_is_true=0.34 and measuring the % of those samples where the acutal value is true (in this case, perfect accuracy would be if the % of samples that was true = 34%). I might create a model accuracy score by summing the difference at each discrete value of prob_value_is_true. But sample sizes are a huge concern here, especially for the extremes (nearing 0% or 100%), such that the averages of the acutal values are not accurate, so using them to measure the model accuracy doesn't seem right.  I even tried creating huge ranges to ensure sufficient sample sizes (0-.25, .25-.50, .50-.75, .75-1.0), but how to measure "goodness" of that % of actual value stumps me. Say all samples where prob_value_is_true is between 0.25 and 0.50 have an average acutal_value of 0.45. Is that good since its in the range? Bad since its not near 37.5% (the center of the range)?  So I'm stuck at what seems like should be an easy question, and hoping someone can point me to a resource or method to calculate an accuracy stastic for a logistic regression model. How to install wps office via terminal? It's pretty complicated and sometimes the system freezes when I try to download and install wps office. I think it is much more convenient to use the terminal. But I never used installed WPS Office from the command line. That would be great if I could update it using software manager or terminal. Third party deb files are from separate locations. I specifically want to know about WPS Why is this command deleting all entities? I'm trying to remove a specific, named, item if it is dropped on the floor. I have the following on my command block:  kill @e[type=Item] {Item:{tag:{display:{Name:"RedFlag"}}}   But it's removing all items, not just the one I want. What am I doing wrong?  I also tried  kill @e[type=Item] {Item:{id:"red_flower",Damage:2}}   But that didn't work either, it's still removing all items, even things that aren't red flowers! Square bracket notation for dimensions and units: usage and conventions Why does Stack Overflow only allow one @User per comment?  This restriction lead me to a non-comment like :     @MiMo - seem my previous comment (Only one @ user per comment)   Which left me wondering why Stack Overflow is blocking me from notifying two users of a response.  I am sure there is a reason for this, can some one share it with me? Convert UTC/GMT time to local time We are developing a C# application for a web-service client. This will run on Windows XP PC's.  One of the fields returned by the web service is a DateTime field. The server returns a field in GMT format i.e. with a "Z" at the end.  However, we found that .NET seems to do some kind of implicit conversion and the time was always 12 hours out.  The following code sample resolves this to some extent in that the 12 hour difference has gone but it makes no allowance for NZ daylight saving.  CultureInfo ci = new CultureInfo("en-NZ"); string date = "Web service date".ToString("R", ci); DateTime convertedDate = DateTime.Parse(date);               As per :     UTC/GMT Offset       Standard time zone: UTC/GMT +12 hours   Daylight saving time: +1 hour   Current time zone offset: UTC/GMT +13 hours     How do we adjust for the extra hour? Can this be done programmatically or is this some kind of setting on the PC's? Let $f(x)$ a irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be a polynomial in $F[x]$. Prove that every irreducible factor of the composition $f(g(x))$ has a degree which is divisible by $n$.  I don't know even how to begin. I really need help. Thanks Why is Euclid's proof on the infinitude of primes considered a proof? I've expressed Euclid's proof on the infinitude of primes on Mathematica:  f[x_] := Product[Prime[n], {n, 1, x}] + 1 TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]   Which results in:  $\begin{array}{ll}  3 &amp; \text{True} \\  7 &amp; \text{True} \\  31 &amp; \text{True} \\  211 &amp; \text{True} \\  2311 &amp; \text{True} \\  30031 &amp; \text{False} \\  510511 &amp; \text{False} \\  9699691 &amp; \text{False} \\  223092871 &amp; \text{False} \\  6469693231 &amp; \text{False} \\  200560490131 &amp; \text{True} \\  7420738134811 &amp; \text{False} \\  304250263527211 &amp; \text{False} \\  13082761331670031 &amp; \text{False} \\  614889782588491411 &amp; \text{False} \\  32589158477190044731 &amp; \text{False} \\  1922760350154212639071 &amp; \text{False} \\  117288381359406970983271 &amp; \text{False} \\  7858321551080267055879091 &amp; \text{False} \\  557940830126698960967415391 &amp; \text{False} \\ \end{array}$  The proof flaws for all those values, how is it considered a proof then? I guess that there might be infinite prime numbers according to the proof, but what is the guarantee that at some point it won't fail indefinitely? How to render an ASP.NET MVC view as a string? I'd like to manage my home machines and laptops with Landscape, how can I do this? I'm new to this website. I've checked the other topics and couldn't find the answer to my question. I've also tried to find the result with TI-CAS but the only thing it does is simplify the inequality which really doesn't help at all.  the equation is : $1000n^3 &lt; 2^n$ Now I've been trying to use log but it gets me nowhere (I haven't seen this form before), the answer is supposed to be 24 but I'm not sure. So my question is how to solve step by step this inequality for n. Could you guys help me with this one? Sort Order on Close Vote Queue The Close-Vote queue is large and growing. I searched for definitive information on how that queue is sorted for reviewers, and the nearest I could find is that it is randomized - but that post was some years old.  Would it be possible to sort on order of votes already received? That is, all questions with 4 votes to close are surfaced first (in random order) to reviewers? Questions with 3 votes would come up before ones with 2, etc. In that manner a given question gets out of 'will it close?' limbo faster.   This feature request would, in my opinion, be supportable if the majority of reviewer votes were going to single-vote questions. I cannot (I think) actually find these numbers. If this is the case, the majority of the work is going to an ever-increasing queue, and proportionally fewer votes are being resolved.  Note there are other distributions that could help get in this direction without being a strict ordering, like weighting the chance a given question is surfaced to a reviewer by the number of votes. However, for discussion purposes I thought I'd keep it simple.  Usual apology if this has actually been addressed. Search is sad-making. Can a virtualized machine have the CPU and RAM resources of multiple underlying physical machines? I'm trying to wrap my head around this for a while now. It's a simple question for what I think will be a not so simple answer. How would you go about modeling one leg of this stool? I would like to do it without using sculpt mode because I can't get things quite clean enough. Universe from nothing? The default PS1 variable on my machine (Kubuntu 13.10) is this:  \[\e]0;\u@\h: \w\a\]${debian_chroot:+($debian_chroot)}\u@\h:\w\$   I'm looking for a reference for how the $PS1 variable works that, at a minimum, allows me to understand the above PS1 variable. Simple extension fields If I am correct simple extension fields are extensions generated by one element. I have learned that this means that elements of a simple extension can be written as powers of that element as long as the power is greater than the degree of that element. Also this means that all elements of the simple ext. Can be written in terms of decending powers of the generating element where the highest power is the degree of the generator minus 1. This concept seems to work for generating elements whose irreductable in a certain field is a descending degree polynomial. Looking at the construction of the complex numbers by starting with $\Bbb R$ (and doing $\Bbb R [x]/\langle x^2+1\rangle$), it is said that all elements are of the form $a+b\cdot \text{generator}$. But when the generator is evaluated for the irreductable, $x^2+1$, we see... $(\text{generator})^2=-1$. Thus there is no first power of gen. Can someone explain where all elements of the extension just explained are of the form $a+b\cdot \text{generator}$ comes from or explain this situation for anytime where the irreduc ( gen, F)  is not a descending degree polynomial. I hope my question is clear.  Thank you How to interpret a quantum circuit as a matrix? Expanding electromagnetic field Lagrangian in terms of gauge field