There are $n$ seats in a classroom and $n$ students who have been assigned seats on small slips of paper. Unfortunately, the first student to enter the classroom has lost his slip, so he just chooses a seat at random and sits in it. Each of the remaining students enters one at a time and either sits in their assigned seat if it is empty or if someone is sitting in their seat, chooses a seat at random from those that are empty. For $n\geq 2,$ show that the probability that the last student to enter will sit in their assigned seat is $\frac{1}2.$  I tried a few small examples and drew some tree diagrams to list out the possibilities, and got $\frac{1}2$ each time. I did make a few obvious observations though. Firstly, if the first student sits in the right spot, everyone else will too, by the premise. There is a $\frac{1}n$ chance for this. Secondly, if the first student chooses the last person's seat, the last person won't be able to sit in their assigned seat. And thirdly, the only way the last person can sit in their assigned seat is if everyone before him/her did not choose his/her seat. We can label the students $S_1,\cdots, S_n.$ I'm not quite sure how to determine the sample space for this problem (i.e. the total set of points or possibilities to consider).
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless  Imagine there are a 100 people in line to board a plane that seats 100. The first person in line realizes he lost his boarding pass so when he boards he decides to take a random seat instead. Every person that boards the plane after him will either take their "proper" seat, or if that seat is taken, a random seat instead.  Question: What is the probability that the last person that boards will end up in his/her proper seat.  Moreover, and this is the part I'm still pondering about. Can you think of a physical system that would follow this combinatorial statistics? Maybe a spin wave function in a crystal etc...
I'm missing all the usual processing tools in the Vector menu in QGIS 2.16 (eg vector overlay, analysis, etc). They do appear in the processing toolbox but there's no sign of them in the Vector menu. Has anyone come across this issue? Relativistic Mass and Dark Matter In a question  Ron Maimon comments that "relativistic mass makes gravity, not rest mass."  If so, does that mean that the faster that stars orbit the galaxy the larger the relativistic mass of the galaxy therefore more gravity therefore the stars orbit even faster therefore the gravity gets even more therefore the stars keep on getting faster ...  So the speed of stars and mass of galaxy would be correlated with age and history of the galaxy or galaxy clusters and that's why calculations of galaxies' masses don't add up. How to change the Play Store language? I'm having trouble understanding the wording of one of my homework questions. Since this question appears early in the chapter, I have a feeling that I'm misinterpreting it.  Give a finite automaton that that accepts the following language: The language over $\{a, b\}$ of any even number of $a$'s and any odd number of $b$'s.   I'm having trouble with the usage of "and". Is it asking for an FA that accepts any string that is either an even number of $a$'s or an odd number of $b$'s, e.g. $aa$, $bbb$, $aaaa$, etc, but not accept strings like $aabbb$, $abbab$.   Or, is it asking for an FA that accepts any string that contains an even number of $a$'s or an odd number of $b$'s, e.g. $aabbb$, $ababb$, etc.   The first case is relatively simple, but I've been struggling with the second. I haven't even managed to convince myself that it is a regular language (because I haven't been able to come up with a regular expression that matches it).   My questions: Which interpretation seems most likely. Is the second case a regular language, if so, give a regular expression that recognizes it. The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$ Difference between "theoretical computer science" and (beta) "computer science" Possible Duplicate:          Title says it all. From what I can see, "theoretical computer science" questions seem to be slightly more isolated to discussing algorithms, whereas "computer science" seems to have questions that are slightly broader and higher-level. Is there any place on the Stack Exchange sites where they give a longer description?  I'm just curious as to what niche the beta "computer science" site is meant to fill, as I see many topics in both forums that overlap. How to repeat a string a variable number of times in C++? I want to insert 'n' spaces (or any string) at the beginning of a string in C++. Is there any direct way to do this using either std::strings or char* strings?  E.g. in Python you could simply do  &gt;&gt;&gt; "." * 5 + "lolcat" '.....lolcat' how does LEA and several instruction work? Since I first learned English, I have been holding this understanding that "and", as a conj. but unlike "but",  can only connect two clauses, not two sentences ended with periods.  But recently, I have seen so many prints, either in entertainment or in academia, where "And" is popularly used in the beginning of a sentence. It seems like the author is trying to connect the sentence just right before and the sentence following "And" in some intended meaning which I don't quite get.  I was wondering if I have been wrong all the time, or if there is a new trend that I fail to understand and accept? How to understand such usage of "and" in rigorous English grammar? If I am right, why it is used differently from "but" in terms of what I mentioned at the beginning of this post? Why does Kylo Ren's lightsaber look different from normal lightsabers? I can already scroll left and right with two-finger swiping, however, in Windows I can use the same gestures to go forward and back in browsers and such. Is this possible in Ubuntu? If $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid$ then vectors $a_1,\dots ,a_n \in \mathbb{R}^n$ are linearly indendent. Let $a_1,\dots ,a_n \in \mathbb{R}^n$ with $a_i = (a_{i1}, \dots , a_{in})$.  Show that when $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid \; \forall \; j\in \{1,\dots ,n\}$, then $a_1,\dots ,a_n$ are linearly independent.  I want to show that the matrix \begin{matrix} a_{11} &amp; a_{21} &amp; \cdots &amp; a_{n1} \\ a_{12} &amp; a_{22} &amp; \cdots &amp; a_{n2} \\ \vdots &amp; &amp; \ddots &amp; \\ a_{1n} &amp; a_{2n} &amp; \cdots &amp; a_{nn} \end{matrix} can be transformed into an upper triangular matrix with all entries in the main diagonal $\neq 0$.  For the cases $n=1$ and $n=2$ this is easy. However, I fail to show this for the general case; the problem being that the tranformations mess up the coefficients such that I can't use the assumption on the $a_{jj}$'s anymore.  I'm pretty sure that this has already been solved somewhere on here, but I don't know how to search for it. In many practical applications, one can consider the Earth-Centered Inertial () reference frame approximately as an inertial reference system, though strictly speaking, it is non-inertial.   Is there any quotable reference, where this claim is supported by a detailed estimation how small the effects are that one neglects if one considers the Earth's frame as inertial? Javascript to sort contents of select element is there a quick way to sort the items of a select element? Or I have to resort to writing javascript?  Please any ideas.  &lt;select size="4" name="lstALL" multiple="multiple" id="lstALL" tabindex="12" style="font-size:XX-Small;height:95%;width:100%;"&gt; &lt;option value="0"&gt; XXX&lt;/option&gt; &lt;option value="1203"&gt;ABC&lt;/option&gt; &lt;option value="1013"&gt;MMM&lt;/option&gt; &lt;/select&gt; Why didn't the Alliance send for Yoda instead of Obi-Wan? Can a self-complementary graph on 100 vertices have precisely one vertex of degree 50? Host name changed remotely by wifi? Am am currently visiting TU Wien and today I connected my Debian Linux laptop to their eduroam wlan using wpa_supplicant and the credentials of my home institute - as always when I am visiting another scientific institution.  When I opened a terminal I noticed that my command promt was showing a different host name, and in fact, excecuting hostname gave me e244-082.eduroam.tuwien.ac.at instead of the usual host name of my machine x301.  I am very puzzled by this. How on earth can it be possible that connecting to a wlan changes my host name without my consent? "Dare" with and without "to" To my surprise, there's a missing question about this particularly interesting verb, dare. All I know about it is the fact it can be in two forms, as an auxiliary (without to: "I dare not mention their names") and main (with to: "Did anyone dare to admit it?") verb and the difference has something to do with agreeing with subject. Can somebody explain in a greater detail? For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$ For a polynomial $p(z)=a_0+\ldots+a_nz^n$, prove there exist $R&gt;0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$.  I get $|p(z)|\geq |a_n|R^n$, which makes that factor of $1/2$ useless. So I must have done something wrong.  $$|\frac{\partial^n{f}}{\partial z^n}(P)|\leq \frac{\sup_{z\in \overline{D}(P,r)} |f(z)|k!}{r^k}$$. This is the Cauchy estimate.   $$|\frac{\partial^n{p}}{\partial z^n}(z)| = |a_n n!|$$ $$|\frac{\partial^n{p}}{\partial z^n}(0)|\leq \sup_{\zeta\in \overline{D}(0,|z|)} |p(\zeta)|n!/R^n$$ $$|a_n n!|\leq \sup_{\zeta\in \overline{D}(0,|z|)} |p(\zeta)|n!/R^n$$ $$R^n |a_n|\leq \sup_{\zeta\in \overline{D}(0,|z|)} |p(\zeta)|$$  There exist a $z_0$, such that $|z_0|&gt;0$ and $$\sup_{\zeta\in \overline{D}(0,|z_0|)} |p(\zeta)| = |p(z_0)|$$.  This is saying there exist a disk centered at the origin, such that $|p|$ is maximum on some point on the boundary.   Let $R=|z_0|$, we have $R^n |a_n|\leq |p(z_0)|$  Edit: I know what I got wrong. $z_0$ is just one point on the circle $|z| = R$. One can chose another point $z_1$ on the circle, such that $p(z_1)&lt;p(z_0)$. Validity of conditional statement when the premise is false.