I would like to ask a simple question but I'm unable to understand the logic behind it.     From an urn containing $M$ balls $n$ balls are drawn with replacement. What is the probability that at least one ball is drawn more than once?   Thank you in advance.
In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding probability is $\left[1- \left(\frac{1}{1-n}\right)^r\right]$.  Please help me show how this is proved.
Let $D \subset \mathbb{C}$ be an open, connected set and let $\{ u_n \}$ be a sequence of harmonic functions with $u_n: D \longrightarrow (0, \infty)$. Show that if $u_n(z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.  If you could offer a hint or a helpful question to get me started I'd appreciate it.   Progress: I now see why showing it for the unit disk suffices (any compact subset in $D$ can be covered by finitely many disks each which is contained in $D$). If you shine a light away from the center of a Schwarzchild black hole, will it stay still? $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? wp_get_nav_menu_items order doesn't work How are "Little Planet" photos created? How do you make mini-planet photos like this one?   By Guido Radig (Own work) [CC BY 3.0 (, via Wikimedia Commons  Here are some examples of what I mean on Flickr:  - though I would be open to a better source of examples. Baby Rudin Theorem 1.20 (b) Proof I have a question about Rudin's proof of Theorem 1.20 (b) in his book Principles of Mathematical Analysis.  Theorem 1.20 is stated as follows:     (a) If $x\in R, y\in R$, and $x&gt;0$, then there is a positive integer $n$ such that $$nx&gt;y.$$   (b) If $x\in R, y\in R$, and $x&lt;y$, then there exists a $p\in Q$ such that $x&lt;p&lt;y$.     I understand Rudin's proof of (a).  The beginning of Rudin's proof of (b) is given below:     Since $x&lt;y$, we have $y-x&gt;0$, and (a) furnishes a positive integer $n$ such that $$n(y-x)&gt;1.$$  Apply (a) again, to obtain positive integers $m_1$ and $m_2$ such that $m_1&gt;nx$, $m_2&gt;-nx$.  Then $$-m_2&lt;nx&lt;m_1.$$  Hence there is an integer $m$ (with $-m_2\leq m\leq m_1$) such that $$m-1\leq nx&lt;m.$$   I don't understand the justification for this last sentence beginning "Hence...."  How is $m$ found, and why are $m_1$ and $m_2$ needed to find $m$? What does this residuals versus fitted plot mean about my model? This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$ let $0&lt;a\le b\le c\le d$, and such $abcd=1$,show that   $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$   it seems harder than Studying the equation $\ln(x)- 1+ x^2=0$ What do I call someone who marries or has a sexual relationship with someone much younger than themselves? Their partner is someone who is at least 18 years old.    The term paedophile is not the correct word, because it refers to minors. Add days to JavaScript Date How can I embed a View inside a content of a node? How can I embed a view inside a content of a node in Drupal 8?   I found something similar, for example in the question about "" or the  module, but it is all for Drupal 7.  I tried with an Entity reference field pointing to a View but it display only a label and from the Manage Display tab I can choose only from Label and Entity ID. I need something like the "Rendered entity" option that is present on Entity reference fields pointing to other nodes. Not enough free disk space when upgrading Prove the equality of circle's areas Through a random point inside the circle, we draw $4$ lines with $45$ degrees between each other. Prove, that the total area of odd pieces, equals total area of even pieces.    I suppose there is a solution using integrals, but maybe there is geometric solution as well. Mouse cursor doesn't change to pointer when hovering "full site" on mobile When (accidentally) clicking "mobile" in the footer, one gets the mobile site in a regular browser too. To switch back, one can click "full site" in that same footer:    However: when hovering it, the mouse is changed as to select text, rather than to indicate one can click (even though it's clickable).  All other links do change the mouse into a pointer.  (Not a big deal, except when somehow  getting into the mobile mode, I guess.) Creating your own Tinyurl style uid Correlation with a constant Suddenly my Ubuntu stopped playing video. When I open a video, the player suggests that I install  gstreamer1.0-libav. However, it fails to install.  I tried installing it manually, here is the error message:  $ sudo apt-get install gstreamer1.0-libav Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  gstreamer1.0-libav : Depends: libavcodec54 (&gt;= 6:9.1-1) but it is not going to be installed or                                libavcodec-extra-54 (&gt;= 6:9.13) but it is not going to be installed                       Depends: libavformat54 (&gt;= 6:9.1-1) but it is not going to be installed E: Unable to correct problems, you have held broken packages. How to copy files via terminal? I have read about copying files with terminal but these examples will help me a lot. So here is what I want to do:  Examples:   I have a file in /home/levan/kdenlive untitelds.mpg and I want to copy this file to /media/sda3/SkyDrive and do not want to delete any thing in SkyDrive directory. I have a file in /media/sda3/SkyDrive untitelds.mpg and I want to copy this file to /home/levan/kdenlive  and do not want to delete any thing in kdenlive directory  I want to copy a folder from home directory to sda3 and do not want to delete any thing on sda3 directory and opposite I want to cut a folder/file and copy to other place without deleting files in that directory I cut it into. Use of 'prototype' vs. 'this' in JavaScript? What's the difference between  var A = function () {     this.x = function () {         //do something     }; };   and  var A = function () { }; A.prototype.x = function () {     //do something }; I have heard of some methods, but none of them have stuck. Personally I try to avoid complex types in C and try to break them into component typedef.  I'm now faced with maintaining some legacy code from a so called 'three star programmer', and I'm having a hard time reading some of the ***code[][].  How do you read complex C declarations?