I've searched for an understandable answer to this exact question and have failed to find it.  How do you find the probability that exactly $k$ bins are empty, given $m$ balls and $n$ bins? (Each ball drop is independent).  The solution to  does not explain how to find the probability that exactly $k$ bins are empty. It mentions the solution in passing in the comments, but does not thoroughly explain how to find the answer.
A large number of variants of this question were already asked here, including these - , , which are close, but none seem to answer my question. Assume that $n$ balls are thrown randomly and independently into $k$ bins.  What is the probability of finding $x$ empty bins? What is the expectation of the number of empty bins?
Can I see which seats are available on a British Airways flight before I book? Python Implementation of Track-to/aim constraint Get exit status of process that's piped to another I have two processes foo and bar, connected with a pipe:  $ foo | bar   bar always exits 0; I'm interested in the exit code of foo. Is there any way to get at it? When asking a new question and typing in the title we get a list of similar questions named     Questions that may already have your answer     That suggestions contains posts across all tags. Like the example in the screenshot it returns similar questions for Python, jQuery, SQL, C#, Java, C++, ...  This makes the list less useless unlikely people find a question that already answers theirs. If people actually find a similar question in that list many duplicates could be avoided.  So can tags I entered below the question be considered to update the list? And can the tag textbox maybe put above the question title? pm-hibernate freezes and/or fails to resume on Dell XPS 13, 14.04, 3.19 kernel I'm running 14.04 on a new XPS 13. In order to support the newer hardware, I'm running a mainline 3.19 kernel.   Hibernate resume has been iffy with the entire 3.19 kernel line, but recently (3.19.5) pm-hibernate no longer works at all. The screen will either a) switch off momentarily, then back on with all inputs frozen or b) switch off entirely, and fail to resume upon power on (despite displaying the correct resume-from-uuid message).  I have enough swap space (and it has worked in the past), I have checked that the uuid is being correctly referenced, and nothing seems particularly amiss in /var/log/pm-suspend.log (last 200 lines here: )  Thanks for your help. Why do we have no enzyme to digest cellulose? As we know, cellulose is the most abundant polysaccharide in nature. Why don't we have an enzyme to digest cellulose? Prime dividing the binomial coefficients It is quite easy to show that for every prime $p$ and $0&lt;i&lt;p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the numerator is divisible by $p$ whereas the denominator is not (since it is a product of numbers smaller than $p$ and $p$ is prime).  My problem is with generalizing this argument for $q=p^n$. I'm looking for the most elegant and simple way to prove that $p$ still divides $\large q\choose i$. Enumerate behaves differently with beamer 3.42 The following is produced by beamer version 3.41 with enumerate:  \documentclass{beamer} \mode&lt;presentation&gt;     {\usetheme{Rochester}} \begin{document} \begin{frame}   \begin{enumerate}[(a)]     \item Foo     \item Bar   \end{enumerate}   \begin{enumerate}[1]     \item Foo     \item Bar   \end{enumerate} \end{frame} \end{document}     And with beamer version 3.42 the following:    The latter output is so, well, ugly, that it's probably a bug. While waiting for a bug fix (?), how can I temporarily fix this so that I get the output produced by beamer 3.41? I recently moved into an in-law unit of a large old house.  In the kitchen, there are two pipes coming out of the wall (one protruding out a little more than the other).  I had assumed they were gas lines to connect my stove.  But since the stove only connects to one pipe, I'm now wondering if they're actually water lines. Can anyone explain how I can tell the difference? On the other side of the wall is a large closet with the air conditioning system, water heater, and what I think is the gas shutoff valve. I came across following problem     Evaluate $$\int\frac{1}{1+x^6} \,dx$$   When I asked my teacher for hint he said first evaluate     $$\int\frac{1}{1+x^4} \,dx$$   I've tried to factorize $1+x^6$ as  $$1+x^6=(x^2 + 1)(x^4 - x^2 + 1)$$ and then writing  $$I=\int\frac{1}{1+x^6} \,dx=\int\frac{1}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx=\int\frac{1+x^2-x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$ $$I=\int\frac{1}{x^4 - x^2 + 1} \,dx-\int\frac{x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$  However $$x^4-x^2+1=\left(x^2-\frac12\right)^2+\frac{3}{4}$$ But I can't see how it helps  I've also tried to reverse engineer the   And I need to have terms similar to $$\frac{x^2-1}{x^4-x^2+1} \quad , \quad \frac{1}{1+x^2} \quad , \quad \frac{1}{(x+c)^2+1}\quad , \quad \frac{1}{(x+c)^2+1}$$ in integrand, How can I transform my cute looking integrand into these huge terms?  Since in exams I will neither have access to WA nor time to reverse engineer the solution moreover it does not seem intuitive,is there any way to solve this problem with some nice tricks or maybe substitutions? What is $\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$ for $c\in\mathbb{R}$ It is straightforward to show $$\int_{-\infty}^{\infty}\Phi(a+bx) \phi( x ) dx = \Phi\left(\frac{a}{\sqrt{1+b^2}}\right)$$ but what value does $$\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$$ have for $c&lt;\infty$, e.g., for $c=0$? Even a good approximation would really help me! Suppose I have 30 numbers that vary between 0 and 1.0 and which sum to 1.0.  The mean is obviously 0.033.  A client wants these scaled to lie between 0 and 1.0 but to have a mean of 0.5.  By the way, I'll probably have to do this for any set of numbers (positive, negative, outside the range 0 - 1) so that they lie in the range 0 - 1 and have a mean 0.5.  Any suggestions? Binomial Distribution Problem - Airline Overbooking What is tail call optimization? sys.dm_db_index_physical_stats is extremly slow I have a database that is around 4.5TB, since we have parallel inserts (to reduce daily load time) on one table (partitioned by month) the clustered index on this table tends to be heavily fragmented. When I do a select from sys.dm_db_index_physical_stats (Limited) on this table it takes ages (> 4-5 hours). Is there a faster and better way to check the fragmentation levels on the partitions of this table, the current time this takes is completely inacceptable. Test for finite variance? Is it possible to test for finiteness (or existence) of the variance of a random variable given a sample? As a null, either {the variance exists and is finite} or {the variance does not exist/is infinite} would be acceptable. Philosophically (and computationally), this seems very strange because there should be no difference between a population without finite variance, and one with a very very large variance (say > $10^{400}$), so I am not hopeful this problem can be solved.  One approach that had been suggested to me was via the Central Limit Theorem: assuming the samples are i.i.d., and the population has finite mean, one could check, somehow, whether the sample mean has the right standard error with increasing sample size. I'm not sure I believe this method would work, though. (In particular, I don't see how to make it into a proper test.) At least two questions (, ) on this site indicate that in more than a few isolated cases at least, letters of recommendation are in fact written by the applicant and only signed off (or perhaps rephrased somewhat) by the recommender, who would actually be responsible for writing the letter.  Is there any evidence about the prevalence of this practice that goes beyond anecdotes? I realize that cross-country (cultural) difference may play a role. Is the 9-speed Sora crankset compatible with 105 11-speed chain? I am trying to identify a sci-fi movie from the 80's or 90's, with the following parameters:  Setting:    Scenes of a red desert planet   Culture:    Either people roam fairly solo and try to acquire resources to survive, or you live in a weird society where if a stranger is captured he needs to play in a game of fight to the death.  Mobility:      One foil character has a giant battle-bot-looking tank as his form of transportation through the desert. Protagonist has a car that has a plastic half bubble on top to look out from.   Robot:      The protagonist runs around in the beginning of the film with a female robot/android whose face is melted off after it has acid touch its face.(This scene happens next to the protagonist's rover)  Love interest:      Somehow protagonist comes across female protagonist in desert. She smells really bad and is hungry. They sleep outside, and he tells her to sleep away from him. He uses a thermal blanket(the first I ever saw in my life). She wakes up next to him, smelling horrible. He bathes her in some water hole they find(as if water were not scarce and they could just waste water.)   Pupa monsters:      Scene where they are in a building and they are trying to get out. They come across a room that is white, in which multiple giant, white, snot, pupa are hanging securely from the ceiling. Through curiosity, they disturb them, and now they have to get out of that room fast.  Boss Antagonist:      Pervert bionic man, that is actually attached to a giant robotic arm, as only form of movement. He is in black and has a pale bald head as his most human looking part. Should "State" be capitalized on its own? Say we had the following:     Higher Education spending, clout, and influence in New York State is substantial. Within the State’s borders...   Should the latter instance of State be capitalized or not?