n passengers board an airplane with exactly $n$ seats. The first passenger has lost the ticket and picks a seat randomly. But after that, the rest of passengers will: Take their own seat if it is still available, Pick other seats randomly when they find their seat occupied  What is the probability that the $n$-th person can get his own seat? The answer is  $1$, if $n = 1 $ $0.5$ if $n &gt; 1$  Following is my attempt For $n = 2$  let's have passengers $A$ and $B$  Sample space = {$AB,BA$}  Favorable = {$AB$}  Probability = #{$AB$}/#{$AB,BA$}= $\frac{1}{2} = 0.5$  For $n = 3$  we have passengers $A,B,C$  Sample Space = {$ABC,ACB,BAC,BCA,CBA,CAB$}  Favorable = {$ABC,BAC$}  Probability = $\frac{2}{6} = \frac{1}{3}$  Can anyone explain me in this way ?
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...
Short Story: Army investigates barrier and house I am trying to identify the author and title of a short story with the following plot.  A scientist narrates a story where s/he is called in to help the army investigate an unexplained phenomena.  The army has cordoned off an area where a prairie home has appeared. It is surrounded by a dome that allows things to enter but not to leave. Briefing the scientist, the Army explains that it appears that it is a family of five with mother, father, son, daughter, and grandfather. They look completely normal and live an idyllic life of the 40s or 50s.  They continue to send probes in to investigate the dome and its proprieties. Not sure if this a time anomaly or another dimension.  An Army lieutenant, so enamored by their life runs through the barrier to join the family. The Family faces split into an unnaturally wide smile full of sharp teeth runs towards him and tears him to pieces &amp; eats him. The grandfather then looks up acts as if he can see the forces gathered outside the barrier. He then retrieves a book and it looks as if he reads it out loud and the house and barrier disappear. The narrator worries that the aliens will return. Draw a vertical line over the entries of a column in an array I'd like to draw vertical/horizontal lines over the entries of a column/row in an array. I find the method to draw horizontal lines at . But when I used the same way to draw the vertical lines. It didn't look well. My codes are:  \usepackage{tikz} \usetikzlibrary{calc} \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};} \newcommand{\DrawLine}[3][]{% \begin{tikzpicture}[overlay,remember picture]     \draw [#1] ($(#2)+(0,0.6ex)$) -- ($(#3)+(0,0.6ex)$); \end{tikzpicture}% }%  \begin{displaymath} \left[ \begin{array}{cccc} \tikzmark{11}{15} &amp; 0 &amp; \tikzmark{13}{0} &amp;0\tikzmark{14} \\ 0 &amp; 50 &amp; 20 &amp; 25 \\ 35 &amp; 5 &amp; 0 &amp; 10\\ \tikzmark{41}{0} &amp; 65 &amp; \tikzmark{43}{50} &amp; 65 \end{array} \right]. \DrawLine[black,thick]{11}{14} \DrawLine[black,thick]{11}{41} \DrawLine[black,thick]{13}{43} \end{displaymath}   The result looks like     But the result I want is     Does anyone know how to fix this? Thank you. Why is the image of a smooth embedding $f: N \to M$ an embedded submanifold? I'm reading An Introduction to Manifolds (Tu) and got confused on p.123 Theorem 11.13. Let me briefly explain what was done before that.  The author defines an embedding between two manifolds $f: N\rightarrow M$ as an injective immersion (differential is injective everywhere) with the additional property that $f(N)$ as a subspace of $M$ is homeomorphic via $f$ to $N$. Then a canonical form theorem is proven, namely, that for any point $p \in N$, there are charts $(U,\phi = x^1, \ldots, x^n) \ni p$ and $(V,\psi = y^1, \ldots, y^m) \ni f(p)$, such that the transition map takes the following form:  $$\psi \circ f \circ \phi^{-1} : (r^1,\ldots,r^n) \mapsto (r^1,\ldots,r^n,0,\ldots,0).$$  (I guess it is implicitly assumed that $f(U) \subset V$).  Now in the process of proving that $f(N) \subset M$ is an , he states that $f(U) \subset V$ is defined by the vanishing of the coordinates $y^{n+1},\ldots,y^m$. I can prove that $$ f(U) \subset \{q \in V: y^{n+1}(q) = \cdots = y^m(q)\} $$ but not the reverse inclusion. Can anyone explain why this should be true? So I was thinking... If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?  When I thought about it more I figured out that wind usually carries some humidity, and since particles of liquid are moving same speed as the wind, they are basically static relative to each other, so no energy is transferred between them (wind and water particles). And if that water sticks to my skin and wind blows, it'll evaporate thus taking energy from my skin and make me feel cold.  Thing is, I don't think that's really the case but even if it is, if I somehow dry out the wind, will it burn me if it's strong enough? And winds can reach some pretty high velocities (though I must admit I'm not sure if they are comparable to movement of atoms in warm bodies etc...).  So. Bottom line. Can I be burned by wind in some perfect scenario? Windows 7: Disable Lock Workstation After 10 Minutes Using Group Policy On my Windows 7 PC the system becomes locked after 10 minutes of inactivity. Usually I would find this setting next to the screen-saver configuration. The setting is grayed out, however.  I think this is because of corporate group policy. As I am an administrator on this computer I should be able to reconfigure the group policy setting using gpedit.msc.  What is the group policy setting that I need to change to prevent automatic locking of my workstation?  Edit: I don't have configured a screen-saver. I also want to continue to be able to lock the workstation manually. Can awakened mind be used with suggestion? Can the warlock class feature "awakened mind" be used in conjunction with the suggestion spell, allowing the warlock to make the suggestion telepathically?  Awakened Mind, Player's Handbook p. 110:     You can telepathically speak to any creature you can see within 30 feet of you. You don't need to share a language with the creature for it to understand your telepathic utterances.   Suggestion, Player's Handbook p. 279:     You suggest a course of activity (limited to a sentence or two) and magically influence a creature you can see within range that can hear and understand you.   If it works with suggestion, would it also work with mass suggestion?  To be clear, I am not asking about the verbal component of the spell, only the method the caster "suggests a course of activity". Normal subgroup of prime order is characteristic I'm stuck on this exercise from Herstein's topics in algebra:  Suppose that $G$ is a group and $|G| = pm$, where $p \nmid ~ m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.  How to prove that? What we know:   $H$ and $\varphi(H)$ have prime order $p$, so they must be cyclic and abelian They are both normal subgroups of $G$   how to show that, in fact, we have $H = \varphi(H)$ for every automorphism $\varphi$ ? how to host multiple pages from diferent domains all together? Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$ As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$   When trying to prove that P(n+1) is true if P(n) is, then I get that:  $$\sum_{i=1}^{2^{n+1}} \frac{1}{i} = (\sum_{i=1}^{2^n} \frac{1}{i} ) +(\sum_{i=1}^{2^n} \frac{1}{2^n+i} ) \ge 1+\frac{n}{2} + (\sum_{i=1}^{2^n} \frac{1}{2^n+i} )$$ using the inductive hypothesis. And for P(n+1) to be true it would be sufficient to prove that  $$1+\frac{n}{2} + (\sum_{i=1}^{2^n} \frac{1}{2^n+i} ) \ge 1+\frac{n+1}{2}$$  or rewriting $$\sum_{i=1}^{2^n} \frac{1}{2^n+i} \ge \frac{1}{2} $$ and here I'm a bit stuck  Thank you for your help! Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$.   Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of $\frak{g}$ is also a linear transformation. Now i have several questions:   How do i prove that $Aut(\frak{g})$ is actually a lie subgroup of $GL(\frak{g})$? I've tried playing with actions to realize $Aut(\frak g)$ as an orbit or an isotropy subgroup but nothing worked out... Say $H \subset G$ is an abstract subgroup (not necessarily lie). Are the following statements equivalent?   $$\{T_e H \text{  is closed under the lie bracket of } \frak{g} \}$$  $$ \{H \text{is a lie subgroup of } G\}$$   How does the lie algebra of $Aut(\frak g)$ look like? When does $Aut(\frak g)$ $= G$? what does $Aut(\frak g)$ tell me about $G$? Seeking open source solution to access rasters in File Geodatabase? I'm looking for a way to access raster layers in an Esri File-GeoDatabase without using ArcObjects or ArcPy, specifically I would like to access the raster layers with GDAL. I understand that the GDAL FileGDB and OpenFileGDB give access to vector datasets within File-GeoDatabases, and that ESRI's own FileGDB C++ API does not support raster access.   Is anyone out there aware of any open source solutions? Uploading video to youtube in right format No custom off topic message when flagging on SO I have less than 3000 rep on StackOverflow. Now that the new closing changes are live, I cannot supply a custom off topic message:    I can enter a custom off topic flag message without issue here on meta, though it only has three canned off topic reasons, rather than five.  I'm not sure if this is the same for users with enough rep to vote to close directly, but the lack of a custom message eliminates the ability to use the correct flag reason in some cases.  For example, I wanted to flag , which is simply off topic on SO. However, I can't use a custom message, and none of the canned ones are appropriate.    appears to be the most up to date official information on custom reasons, and indicates that they are remaining in play for now.  supports that.  If this is by design, how should I flag such questions? This is not a case worth bothering the mods over, so I don't want to use a custom diamond attention flag. Windows 10 high memory usage (unknown reason) A few days ago I updated my PC to Windows 10. However, after some use my PC started to slow down until it was impossible to use – it was due to high memory usage. After a restart, everything came back to normal (total usage around 25% of 8GB of RAM). However, during a few hours of usage the memory builds up again to 70%, and if not restarted it further goes to 100%, and later on even freezes. Task manager does not help very much as it does not show all the processes (added screenshots below). Also tried RAMMap but it gives an error: "error refreshing database". I tried Googling this question, yet without much success.  I do not know much about PCs, but maybe some of you know this issue, or could help to find out what is using my RAM. What are the advantages of stacking multiple LSTMs? What are the advantages, why would one use multiple LSTMs, stacked one side-by-side, in a deep-network? I am using a LSTM to represent a sequence of inputs as a single input. So once I have that single representation— why would I pass it through again?  I am asking this because I saw this in a natural-language generation program. Hide parameterless constructor on struct How do we prove $\cos(\pi/5) - \cos(2\pi/5) = 0.5$ ?. What to do about a 2-year-old who won't sleep without someone else in the room? My 2-year-old son won't fall asleep without having someone in his room with him. He doesn't want to be touched or held. He just wants you to sit in the chair. And after he does go to sleep he wakes up about 4 to 5 hours later, crying and screaming and is out of his bed and by the door. I go to him and give him a hug and put him right back in his bed. If I walk out he screams and won't go back to sleep for the rest of the night. If I stay he goes right back to sleep. But he wakes right back up when I walk out of his room.   I'm at a loss as to what to do. I've tried lots of stuff and nothing seems to work. It's almost as if he is having separation anxiety. I dont know what to do. Please help. Why doesn't my ~/.bash_profile work? I'm using Linux Mint. My login shell (cat /etc/passwd | grep myUserName) is bash.  After I start my graphical desktop environment and run a terminal emulator from it, I can see that .bash_profile is not sourced (environment vars that are exported in it are unset). But if I log in from a text console (ctrl+alt+F1) or manually run bash -l from the terminal emulator, .bash_profile works fine.  Am I wrong when I think that .bash_profile should be sourced when X starts and all export'ed vars should be available in the terminal, running from X?  P.S. Placing everything in .bashrc and sourcing it from .bash_profile is not good idea (): environment stuff should be sourced only once. Digitally selectable filter I have asked a  but I did not get an answer, as it was too specific. So I will build on it but make it a bit more broad spectrum.  I would like to make a digitally selectable BPF using transistors to select a component in the filter; resulting in the center frequency being changed.  I have used FilterPro to generate quick second order filters and get a rough filter response.  For all the designs, a Multiple-feedback was selected:  In the first picture, a center frequency at 108KHz with a bandwidth @ -3dB of 103K to 113KHz is given.    For the second, a center frequency at 117.5KHz with a bandwidth @ -3dB of 112.8K to 122.2KHz is given.    The only difference between these two circuits is R3.  If I were to implement a circuit as shown below, would it work and what transistor would be recommended? I have selected the  as the Opamp in the circuit design as that is what was used for the .    For the transistors, I was thinking of using the MOSFETs (2N7002) used in the ST7590 design as well.