There are  n  people in line, indexed by  i=1,…,n , to enter a theater with  n  seats one by one. However, the first person ( i=1 ) in the line is drunk. This person has lost her ticket and decides to take a random seat instead of her assigned seat. That is, the drunk person decides to take any one of the seats  1  to  n  with equal probability. Every other person  i=2,…,n  that enters afterwards is sober and will take his assigned seat (seat  i ) unless his seat  i  is already taken, in which case he will take a random seat chosen uniformly from the remaining seats.  Suppose that  n=3 . What is the probability that person 2 takes seat 2?  (Enter a fraction or a decimal accurate to at least 3 decimal places.)
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...
Split polygons into equal areas within each region I've a set of regions within a city. I'd like to split each region into five subregions and then get the centroid of each one of them.  I know how to make a vector grid, but the problem I have is that the generated points are not taking into account the regions, they're only related to the general map.  I have this:    And I want to split each region in five equal size areas. Something like this:    I'd like to build a visualization similar to this one: . In order to accomplish that, I need to split each region, because the original ones are pretty big. Prove that a finite union of closed sets is also closed Let $X$ be a metric space.  If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed.  I'm looking for a direct proof of this theorem.  (I already know a proof which first shows that a finite intersection of open sets is also open, and then applies De Morgan's law and the theorem "the complement of an open set is closed.")  Note that the theorem is not necessarily true for an infinite collection of closed $\{F_\alpha\}$.  Here are the definitions I'm using:  Let $X$ be a metric space with distance function $d(p, q)$.  For any $p \in X$, the neighborhood $N_r(p)$ is the set $\{x \in X \,|\, d(p, x) &lt; r\}$.  Any $p \in X$ is a limit point of $E$ if $\forall r &gt; 0$, $N_r(p) \cap E \neq \{p\}$ and $\neq \emptyset$.  Any subset $E$ of $X$ is closed if it contains all of its limit points. filling an occluded plane with the smallest number of rectangles I've got a specific problem which I'll try to describe as clearly as possible.   I have a defined rectangular region on a cartesian plane, and within that region there are other given rectangular sub-regions that are described in terms of their 4 vertices, ie {(x1, y1), (x1, y2), (x2, y1), (x2, y2)}, so that these regions form 'occlusions' on the plane. These regions don't overlap, but they can form more complex polygons when different-sized rectangles adjacent to each other appear joined.  here's an illustration:   I am interested in the space between these shapes, and how to define the space in the same way the occlusions are defined, that is as a set of rectangles. In particular I want the definition optimised so that the space is described using the smallest possible number of rectangles. For example, an incomplete rendering might look like this:   Can anyone suggest a way forward with this? How can I have the original set of vertices (describing the black rectangles) generate the 'complementary' set of vertices (red rectangles) such that the number of red rectangles is minimal?  I suspect it's a variation of a 'packing problem', but I have a feeling it might be fairly simple... Bridged interfaces and VLAN tags DBRefs (Mongo Document references) not fetched eagerly How do I get the number of days between two dates in JavaScript? Is there an SE for Cell-phones? Where would I ask a question pertaining to an ordinary phone? Or just cell phones in general? I saw Windows Phone while browsing the SE database, nothing else really popped out at me. Google didn't bring up anything either. Is that just too small a community to cover with SE? I want to copy a live production database into my local development database.  Is there a way to do this without locking the production database?  I'm currently using:  mysqldump -u root --password=xxx -h xxx my_db1 | mysql -u root --password=xxx -h localhost my_db1   But it's locking each table as it runs. Multi armed bandit for general reward distribution Omission of "the" in "elected him president" and "made captain" Is there a dollar amount that, when adding Massachusetts Sales Tax, precisely equals $200? Product Snippets with JSON-LD + Schema.org - Are there any live examples in Google index? ** EDIT Feb 2016 - Finally JSON-LD seems to be working now in SERPS, with extremely fast changes-to-index time, assuming you have everything with Google merchant center feeds syncd and linked. Make sure the price, stock, and identifier are the same on your JSON-LD as it is on the Google shopping feed. New organization-&gt;localbusiness types are live, organization reviews are live... however JSON-LD feed alt= endpoints still do not exist, the homepage schema defining custom site name is not active, and G support for schema itemList/dateModified is still questionable. You can still get pagination results and date modified snippets by using literal text strings in your view/template Showing 1-28 of 8977 Results (321 Pages)and Last updated on February 3rd 2016. To proceed to a deeper exploration of JSON-LD support, see these:       Original Post Below: Google seems to be flip flopping JSON-LD -&gt; Schema.org product [or other types of] snippets. The snippets verify without error in both the testing tool and long term (10+ months) across 5 properties in GWT search console. The snippets have been applied to index, removed, applied, and removed again, seemingly not being applied again in regards to various facets. Since they worked in the past via microdata, and were applied long term, I assume it's due to the switch to JSON-LD. I have not found any official docs nor examples that say &quot;Google will support and render products snippets that use JSON-LD&quot;. Same thing with breadcrumb snippets and review looping. All I see is a handful of Google examples for events, movies, recipes, very basic [incomplete] organization etc. This seems strange to me considering the sheer amount of ecom pages using product snippets trumps the amount of movies, recipes, and events combined. So the question is, does Google officially support, index, and render product markup using JSON-LD? If so, can anyone point me to a live example to verify this? ** This question is not intended to be &quot;opinion/recommendation based&quot;....im looking for a specific link and subsequent SERP result [evidence] showing that JSON-LD products/etc are indeed rendered within Google.com realm as of late August 2015. ** EDIT Nov 16 2015 - Another annoying caveat has been found. Google merchant center validates certain shopping feed fields such as condition, stock, and price against live microdata. However, it seems to not recognize JSON-LD.... although, as stated above, it's recommended to use in all the G examples, and it validates the snippet test. So due to this, your merchant center account could be suspended because of &quot;differences in presentation data&quot; I have multiple characters linked to a blender file, but i moved the original character files and now blender is sating they're missing.  Is it possible to get blender to find the missing blender files like it finds missing image files so that i can preserve the previously made animation?  On a side note is there a way to also view a list of all the missing files? Are there any official specifications regarding the torque for camera mounts? I've been wondering why I've never seen any official specifications of the maximum allowed torque that can be exerted upon lens mounts. It seems to me like a very useful specification, but I can't seem to find it from any manufacturer.  indicates that there might be no specifications for Nikon mounts. Are there really no manufacturer that publish these specification of their mounts? What books are recommended for learning calculus on my own? I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take one math class, introduction to calculus (and it was hardly calculus, it was more of a math class for designers)  Since graduating I have pursued a career in web development. I'm considering going back for a MS in computer science or MS in web technologies.   I feel my math has suffered dramatically as I have always been behind most others on my math studies (middle school through college). I want to learn calculus as I feel it's an integral part of any CS degree.   Does anyone have any recommendations on any great books that I can read through and teach my self a bit of calculus or other advanced math topics?  Thanks for any advice Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls. I know that every open set in $\mathbb{R}$ can be decomposed as the union of countably many disjoint open intervals and I'm trying to show that this property no longer holds in higher dimensions. For this I'm trying to show that there is no countable, pairwise disjoint open balls whose union is the open square $(0, 1)^2 ⊂ \mathbb{R}^2$. I know that it is because of the corners, but I don't know how to write this in mathematical terms. Thanks for any help. Show that $\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$ In Ruby on Rails, how do I format a date with the "th" suffix, as in, "Sun Oct 5th"? I want to display dates in the format: short day of week, short month, day of month without leading zero but including "th", "st", "nd", or "rd" suffix.  For example, the day this question was asked would display "Thu Oct 2nd".  I'm using Ruby 1.8.7, and  just doesn't seem to do this.  I'd prefer a standard library if one exists. Can Blender render pngs with the background transparent? I would like to render an animation as a series of .png images with transparent backgrounds for use in a video. Is this possible? Remove the shortcut of an application that was installed using Wine from the Unity Dash How do I remove an application shortcut from the Unity Dash (not from the Unity Launcher)?  My specific issue is I tried to install a game using Wine a while ago. I then uninstalled the game, then uninstalled Wine. Today I found that shortcuts to the game (and other related content that came with it) remain when I push Super and type the game name. (Wine is not visible though.)