I was posed the following question recently:     Suppose a brand of cereal has four different toys included in their cereal boxes, each with an equal probability of being found: On average, how many boxes of cereal do you need to buy to get all four toys?   Since getting a toy in each box is an independent event and the probability stays constant, I guessed that you could model this using the binomial distribution:  \begin{align} X &amp;= \text{number of unique toys}\\ X &amp;\sim B(n, 0.25) \end{align}  Where $n$ is the number of trials. The probability function for getting four unique toys would therefore be:  $$P(X = 4) = {n \choose 4} \cdot 0.25^4 \cdot 0.75^{n-4}$$  So, to find the mean number of trials, I perform the infinite sum:  $$\sum \limits_{n=4}^\infty \left[n\cdot P(X=4)\right] = \sum \limits_{n=4}^\infty \left[n\cdot {n \choose 4} \cdot 0.25^4 \cdot 0.75^{n-4}\right]$$  Which, when I plug this into Wolfram Alpha returns $76$.   So, according to my calculation, on average you need to buy $76$ boxes of cereal to get all four toys. Is this right?
What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.  I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$  Can someone please explain this to me, possibly with a link to a general topic?
Expected number of siblings, given the expectation of children in a family The number of children in a family is a random variable $X$, where    $ \mathbb E(X) = 1.8 $ and $Var(X) = 0.36 $. If we randomly choose a child, which is the expected number of its siblings?      (it is given that the answer $&gt; 0.8$)   Let the number of a child's siblings be a random variable $Y$, so we are looking for $ \mathbb E(Y) $.  If found that,  $$ \mathbb E(Y) = \sum_{k=0}^{\infty}k \; \mathbb P(Y=k) = \sum_{k=0}^{\infty}k \; \mathbb P(X=k+1) = $$ $$ = \sum_{k=0}^{\infty}k \; \mathbb P(X=k)-\sum_{k=0}^{\infty}\mathbb P(X=k) +\mathbb P(X=0)= \mathbb E(X)-1+\mathbb P(X=0) $$ which is indeed greater than $0.8$, if  $\; \mathbb P(X=0)&gt;0$.  But, I don't know how to make use of the variance to find the final answer.  Thank you in advance Can a character use the Misty Step spell to escape from the Iron Bands of Bilarro? Can a character escape the  (DMG p. 177, listed in the SRD as the ) with the  spell? To clarify, I am not asking if the bands prevent teleportation. I am asking whether the bands are teleported with the character as he teleports using misty step. I have problems with getting an Asus PCE-N53 11n N600 PCI-E Adapter card to work on my desktop computer.  As far as I can tell no drivers are installed for the card. I know I can manually download the drivers directly from Asus, but I would rather not go that route.  If there are anyone that knows about any packages or other things I can do to make this work would be much appreciated.  Some systems details:  $ lsb_release -a No LSB modules are available. Distributor ID: Ubuntu Description:    Ubuntu 12.04.1 LTS Release:    12.04 Codename:   precise   $ sudo lshw -C network   *-network                       description: Ethernet interface        product: RTL8111/8168B PCI Express Gigabit Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: eth0        version: 06        serial: d4:3d:7e:03:b9:1d        size: 100Mbit/s        capacity: 1Gbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168e-3_0.0.4 03/27/12 ip=192.168.0.173 latency=0 link=yes multicast=yes port=MII speed=100Mbit/s        resources: irq:43 ioport:d000(size=256) memory:f2104000-f2104fff memory:f2100000-f2103fff  *-network UNCLAIMED        description: Network controller        product: Ralink corp.        vendor: Ralink corp.        physical id: 0        bus info: pci@0000:04:00.0        version: 00        width: 32 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list        configuration: latency=0        resources: memory:f7100000-f710ffff  $ lsmod Module                  Size  Used by nvidia              12319264  51  vesafb                 13844  1  snd_hda_codec_hdmi     32474  1  joydev                 17693  0  bnep                   18281  2  rfcomm                 47604  0  bluetooth             180104  10 bnep,rfcomm snd_hda_codec_realtek   224173  1  snd_seq_midi           13324  0  ppdev                  17113  0  snd_rawmidi            30748  1 snd_seq_midi usbhid                 47199  0  hid                    99559  1 usbhid nouveau               774641  0  parport_pc             32866  1  snd_hda_intel          33773  5  ttm                    76949  1 nouveau snd_hda_codec         127706  3 snd_hda_codec_hdmi,snd_hda_codec_realtek,snd_hda_intel drm_kms_helper         46978  1 nouveau drm                   242038  3 nouveau,ttm,drm_kms_helper snd_seq_midi_event     14899  1 snd_seq_midi snd_hwdep              13668  1 snd_hda_codec snd_seq                61896  2 snd_seq_midi,snd_seq_midi_event i2c_algo_bit           13423  1 nouveau mxm_wmi                12979  1 nouveau wmi                    19256  1 mxm_wmi mac_hid                13253  0  snd_pcm                97188  3 snd_hda_codec_hdmi,snd_hda_intel,snd_hda_codec psmouse                97362  0  video                  19596  1 nouveau snd_timer              29990  2 snd_seq,snd_pcm snd_seq_device         14540  3 snd_seq_midi,snd_rawmidi,snd_seq snd                    78855  20 snd_hda_codec_hdmi,snd_hda_codec_realtek,snd_rawmidi,snd_hda_intel,snd_hda_codec,snd_hwdep,snd_seq,snd_pcm,snd_timer,snd_seq_device serio_raw              13211  0  soundcore              15091  1 snd snd_page_alloc         18529  2 snd_hda_intel,snd_pcm mei                    41616  0  lp                     17799  0  parport                46562  3 ppdev,parport_pc,lp r8169                  62099  0 What to do about rep inflation? Overall reputation scores are headed nowhere but up.  Pretty soon there will be thousands of mods.  What to do?  Thoughts on possibilities:   Thousands of mods is fine.  Carry on. Adjust the threshold values for high-level SO privileges. Do a straight-up inflation adjustment.  You had x rep, then at the turn of the year we applied the Consumer Rep Index to all scores, now you have (x / inflation) rep. Implement a rep decay mechanism, so that inactivity sheds some of the overall rep that's built up. My roommate put a hunk of beef roast in the slow-cooker for a good 6 hours.  It's a fairly large sized slow-cooker, at least 8 quarts.  The meat was cooked through but very chewy, and I never remember my mom having that issue with a slow-cooker.  My roommate tends to put only enough water to get halfway up the side of the roast, and always uses a slow-cooker 'bag' (another thing my mom never did).  What advice can I offer them to get a better outcome? How to write a function (piecewise) with bracket outside? How to install mods for the new Minecraft launcher I've recently changed to the new Minecraft launcher, and I have noticed a change in the .minecraft folder layout. I can no longer install mods just by copying it over into the versions/1.6.1 jar file, as this file refreshes every time it is launched.  For those who don't know what I am talking about, here is the new layout:    I am specifically trying to install Optifine, which does work with 1.6.1. Where do I drag in the mod files now? I can't seem to find a jar file that is suited to it, except in the version folder, but as I stated above, this refreshes every time.  After copying a recent version and renaming it in the version folder, I get this error in the development console:   Unexpected exception refreshing version list java.lang.IllegalArgumentException: Version '1.6.1' is already tracked     at net.minecraft.launcher.updater.VersionList.addVersion(VersionList.java:91)     at net.minecraft.launcher.updater.LocalVersionList.refreshVersions(LocalVersionList.java:44)     at net.minecraft.launcher.updater.VersionManager.refreshVersions(VersionManager.java:47)     at net.minecraft.launcher.Launcher$2.run(Launcher.java:164)     at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source)     at java.util.concurrent.FutureTask$Sync.innerRun(Unknown Source)     at java.util.concurrent.FutureTask.run(Unknown Source)     at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source)     at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source)     at java.lang.Thread.run(Unknown Source) Why is gradient the direction of steepest ascent? $$f(x_1,x_2,...x_n):\mathbb{R}^n \rightarrow \mathbb{R}$$ The definition of the gradient is  $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\  ... +\frac{\partial f}{\partial x_n}\hat{e}_n$$  which is a vector.  Reading this definition makes me consider that each component of the gradient corresponds to the rate of change with respect to my objective function if I go along with the direction $\hat{e}_i$.  But I can't see why this vector (defined by the definition of the gradient) has anything to do with the steepest descent.  Why do I get maximal value again if I move along with the direction of gradient? I am getting a warning when trying to include the .net 3.5 sp1 prerequisite for my setup project.  The warning states Prerequisite could not found for bootstrapping.  Any suggestions?  Thanks Combo box in Java - nullpointer I am looking for an English translation of Markov's 1958 paper, On insolubility of the homeorphy problem, which I remember coming across on a website for a computational topology course (taught by Afra Zomorodian). I am  with the Google login for Pokemon Go. Is there any way, be it contacting support or through Trainer Club, to either link my Google account to a new Trainer Club account or transfer my account to a Trainer Club account?  It seems like a better idea to have a Trainer Club account, so if there are Google login problems it won't matter.  Does anyone know if something like this is possible? If a stock doesn't pay dividends, then why is the stock worth anything? I read somewhere that companies are not required to pay dividends to shareholders (this is correct, yes?).   If so, then if company A never pays dividends to its shareholders, then what is the point of owning company A's stock? Surely the right to one vote for company A's Board can't be that valuable.   What is it that I'm missing? Below is my code to create the component dynamically - The component is "ui:inputselect"  $A.createComponent(             RecoveryLineItem.Attributes[i].name,{                 "class" : "slds-input ",                 "aura:id" : 'ExampleId',                 "label" : RecoveryLineItem.Attributes[i].label,                 "multiple" : RecoveryLineItem.Attributes[i].multiple             }, ....   ..  console.log(component.find('ExampleId').get("v.value"))    Or console.log(component.find('ExampleId'))  is Giving as 'Undefined"  How to access the aura:id of the dynamically created components? USB-C Dongles killing Wifi on 2016/2017 MacBooks – Who’s to blame? I recently switched from my beloved MacBook Pro Retina (Late 2013) i5 2.6/16/512 to MacBook Pro (2017) Non-TB i5 2.3/16/512.  I purchased what from the build-quality looked like a good USB-C-to-DisplayPort cable with 4k/60Hz support, just to run into the very problem the Logicboard repair guru Louis Rossmann had described in : As soon as you plug in the USB-C dongle, the (2.4 GHz) Wifi stops working despite full antenna signal. Unplug the USB-C connector and you are immediately back in the game.  The big question I'm asking myself now: Who is to blame? Is this a design flaw in the current generation MacBooks with improper shielding or is it a design flaw in the various dongles?  I mean the fact that even Apple’s very own USB-C Digital AV Multiport Adapter is causing these WiFi issues (as you can learn from the many horrific ; but AFAIK have been fixed through a ), show that there’s something going horribly wrong. How do I increase my code coverage, or why can't I cover these lines? What are the specifics of 18.04's ten year support? When is a cellular automaton "bidirectional"? I wonder about the foundations of set theory and my question can be stated in some related forms:   If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not allowed to contain the notion of sets? The axioms of Zermelo–Fraenkel set theory seem to already expect the notion of a set to be defined. Is there are pre-definition of what we are dealing with? And where? In set theory, if a function is defined as a set using tuples, why or how does first order logic and the axioms of Zermelo–Fraenkel set theory contain parameter dependend properties $\psi(u_1,u_2,q,...)$, which basically are functions? Most of the wand-carriers that come from a non-Muggle family seem fairly uneducated when it comes to interacting with Muggles.  This implies that they don't go to any type of Muggle school before entering into their education at Hogwarts (or whichever Wizarding school they go to), or else they would at least understand some Muggle culture.  What do children from magical families do before going to Hogwarts if they don't go to a traditional school?  Is there any evidence that they're homeschooled?