show that the probability of the last person sitting in the right spot is $1/2$ 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).
Taking Seats on a Plane 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...
OED says:     doobie: a marijuana cigarette    Origin unknown. A relationship with dobby has been suggested.    dobby/dobbie: A silly old man, a dotard, a booby. Dialectal.      First citations:    1967   J. B. Williams Narcotics &amp; Hallucinogenics. Dubbe, Negro slang for a marijuana roach.    1982   A. Maupin Further Tales of City 95,  I smoke a doobie at lunch.   But I just chanced upon  in Yahoo Answers (don't laugh! :) where someone posted:     "It's obvious that Shaggy's a , I'm sure he's been feeding Scoob a bit of his hash and whatnot."    Scoob = , cartoon "pet" dog of constantly-hungry hippie/village idiot Shaggy (from 1969)   I'm not entirely convinced that 1967 dubbe was necessarily a significant factor in the appearance of  since around 1970 (the hippie era, some might say).  So I'd like to know whether it's credible to claim the usage arose (primarily?) through association with the TV show (specifically, the two characters therein closely associated with marijuana). Electromagnetic wave properties of light vs radio waves Find a formula for Taylor series of $\left(\frac{1}{1+z^2}\right)^n$ So the way I think I should approach this is by getting a result for $n=1,2,3...$  and then examine them. I could easily get the Taylor series expansion for $n=1$, but then I don't really know how to proceed, because I can't always break down a power of a polynomial. And a power of a summation doesn't seem as a legit answer either.   Can anyone help? Thanks "He is loved", is 'loved' an adjective or a verb? He is loved.   This is something that I've always kind of wondered.  In a sentence like this, is loved a verb or an adjective? Can it be considered either? What is the logic behind the timer of the Sojourner Badge? Does the 24 hours timer start:     when you hack a portal on a particular day     or     does it use the calendar days and check if you did hack a portal on a particular day   or     does it simply use a fixed international timezone to calculate the 24 hour cycle   or     does it use the agent's hacking cell's timezone to start it?   If it's the calendar days method than the explanation of the badge would be a little bit misguided because you could hack a portal on 00:01 on Saturday and then hack another one on 23:59 on Sunday and it would be practically two days between two hacks.  If it's based on the agent's last hack for a particular day then the explanation still would be problematic?  The third option would be a little bit unfair (or timezone racist) for the rest of the world.  The last option (the cell's timezone) would/could be both unfair for travellers and/or non-travellers.  Does anyone know any details about it? Degree of minimum polynomial at most $n$ without Cayley-Hamilton? Say a security has been trading at under $20.10, and there are limit orders selling at $20.10 and $20.21. Then Simon issues a limit order to buy at $20.21. According to , Simon would end up buying shares at $20.10.  Why does the exchange favor the buyer here?  I.e., why does the transaction execute at the seller's limit of $20.10 instead of the buyer's limit of $20.21?  (I.e., why don't Inara/River get to sell at $20.21 / above their limit, but Simon gets to buy at $20.10 / below his limit?)  Is the example flawed?  If it makes a difference, I'm mainly curious about NASDAQ and NYSE.  I tried Googling for specs (e.g. ), but couldn't find anything.  Only found that some  implementation executes the average of the buy/ask.  Just curious how the system works. Quote unquote (or end-quote), unseparated by the actual quotation In spoken English, people often say "quote-unquote" (or "quote-endquote") to indicate that part of what they are saying is a quotation (scare or otherwise). Sometimes the quoted material will go between quote and unquote, and occasionaly it will precede the quote-unquote. Often, though, "quote unquote" introduces the quoted material.  Why does quote unquote precede the quotation?  It would seem that quote opens the quoted text and that unquote closes it, but then why would we close the quotation before actually providing it. It would be the equivalent of writing your quotation marks together, "", rather than around the quoted text.  What are the origins of this construction?  Three years ago ChrisR asked  on whether it is correct to use quote, unquote. The answers did not deal with origins, but one commenter suggested that quote-unquote is a corruption of quote-on-quote, but he provided no evidence, nor any real explanation. Extract information from a text file After a table of addresses is geocoded, ArGIS provides information about each geocoded address, among those the "match score" of the candidate to which the address was matched, which ranges from 0 to 100. According to their documentation "The match score is based on how well the locations found in the reference data match with the address data being searched."  It seems intuitive that 100 means an address with the exact name was found in the Address locator and 0 means no such address was found. However, I could not find any information about how exactly this score is calculated, particularly if values are somewhere between the extremes. I this known?  I found the pointer to  in the answer to , but I could not find any information in that paper that would answer the question. Remote start up a machine that is in the shut down state? Possible Duplicate:          May I know how can one remote start up a machine that is in a shut down state? If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$ I'm having a problem rendering the environment map. I downloaded a sample project that has the same settings as my project but it renders without a problem. I tried using both a .hdr and .png image. Attached are screenshots with node settings. First is the incorrect render and 2nd image is the project the renders correctly. How do I go about fixing this? What is the monomorphism restriction? Something special about energy eigenstates when it comes to time evolution? A particle is subject to an infinite square well potential with  $$V(x)= \begin{cases}  0  &amp;   −a \lt x \lt a\\   \infty &amp;  \,\,\,\,\text{otherwise} \end{cases}$$  At a time $t=0$ its wavefunction is given by $$\psi(x, t=0)=\frac{1}{\sqrt{5a}}\cos\left(\frac{\pi x}{2a}\right)+\frac{2}{\sqrt{5a}}\sin\left(\frac{\pi x}{a}\right)$$       (a) What are the possible results of an energy measurement at $t = 0$, and with what probabilities?     Re-writing the terms in $\psi(x,t=0)$ as $$u_1(x)=a^{-1/2}\cos\left(\frac{\pi x}{2a}\right)\qquad \text{&amp;} \qquad u_2(x)=a^{-1/2}\sin\left(\frac{\pi x}{a}\right)$$ such that $$\psi(x,t=0)= \frac{1}{\sqrt{5}}u_1(x)+\frac{2}{\sqrt{5}}u_2(x)$$ with energies $$E_1=\frac{\hbar^2 \pi^2}{8ma^2}\qquad \text{&amp;} \qquad E_2=\frac{\hbar^2 \pi^2}{2ma^2}$$ and probabilities $$P(E_1)=\frac15 \qquad \text{&amp;} \qquad P(E_2)=\frac45$$       (b) If no measurement is performed, what is the wavefunction $\psi(x, t)$ at all times $t$? What are the possible energies and their probabilities if a measurement is first performed at time $t$?     At a time $t$ the wavefunction is given by $$\psi(x, t=0)=\frac{1}{\sqrt{5a}}\cos\left(\frac{\pi x}{2a}\right)\exp\left(\frac{-i E_1 t}{\hbar}\right)+\frac{2}{\sqrt{5a}}\sin\left(\frac{\pi x}{a}\right)\exp\left(\frac{-i E_2 t}{\hbar}\right)$$    The problem is that I cannot answer the second part of the question (b).    The answer states that:      Because the two terms are energy eigenstates, the relative probabilities do not change and are as in the previous part.     I have some questions about the above statement: why does an "energy eigenstate" mean that the probablities don't change with time? Or, put in another way, does a "momentum eigenstate" (for example) have constant relative probabilities when measured at any time?   The answer seems to be implying there is something special about energy eigenstates when it comes to time evolution. The eigenvalues of energy measurements don't seem to change regardless of how many times you measure them, and no matter how long you wait until you measure them. I'm not sure if this is the case, but could someone please explain why all quantum operators (except Hamiltonian) do not exhibit this behaviour. I have downloaded an ISO file recently.  How do I burn it to a CD or DVD or mount it? How do I allow HTTPS for Apache on localhost? How to Start Web Mapping? I want to start working on a web map at work to map some of our statistics by city.    I have an idea, but I'm not exactly sure where to start.   Are there any definitive resources that provide step by step guidance on how to do this? Performing multiple layer intersection in QGIS? I am travelling form Delhi (DEL) to San Francisco (SFO), I have to make a connection flight in Vienna that would take me to Frankfurt (FRA). My flight to SFO is from FRA, I have one ticket for entire journey and the flights are operated by Lufthansa or its subsidiaries.  Do I need a transit visa? My flight arrives at Vienna international airport and leaves from the same, and for second leg it arrives at Frankfurt international airport.  I asked this question as an addendum to my  but was advised to make a new post.  I hold Indian passport and valid US visa.