how would I be able to answer this question?  The first box contains 3 white and 7 black balls, and the second box contains 6 white and 3 black balls, A ball is chosen at random from the first box, and, without looking at its colour, put into the second box. Then a ball is chosen at random from the second box, and it is white. Is it more likely that the ball moved from the first box to the second was black?
Bag A has $3$ white and $2$ black marbles. Bag B has $4$ white and $3$ black marbles.  Suppose we draw a marble at random from Bag A and put it in Bag B. After doing this, we draw a marble at random from Bag B, which turns out to be white. Given this information, what is the probability that the marble we moved from Bag A to Bag B is white?  At first since the probability that white is $1/2$ is $2/5$, and the probability that white is $5/8$ is $3/5$, I thought a sufficient answer would be $$\begin{align}\tfrac 1 2 \cdot \tfrac 2 5 + \tfrac 5 8 \cdot \tfrac 3 5 &amp; = \tfrac 1 5 + \tfrac 3 8 \\ &amp; = \tfrac 8 {40} + \tfrac {15}{40} \\ &amp; = \tfrac {23}{40}\end{align}$$, but that was wrong.   Can someone provide me with an answer? Thanks
&quot;I hate wizards. If he so much as points a finger at me, you stick him, Hook.&quot; &quot;Remind me to find out the name of that flooring company so I can be sure never to buy so much as a carpet tile from it.&quot; &quot;I didn’t listen so much as survive the conversation.&quot; Coming from SketchUp one of the tools I miss most is the circle draw tool which lets you draw a circle on any face and then you could extrude it in/out. The only tool that I know of that comes close to this is the knife tool but this would be very impractical to cut out with the knife  Example:    Is there any tool in blender that can match this function? also how could you do this with squares? To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology:     Let X be an infinite set. Show that $\mathscr{T}_2=\{U \subseteq X : U = \emptyset $ or $ X\setminus U $ is countable$ \} $ is a topology.   My book calls this set the ""  Please let me know if my proof is valid.     corollary [A]: The union of countable sets are countable.    proof: I'm going to take as a given that that the union of finite set are finite, and just prove the case of . A set is countable infinite if there is a bijection between the set and the natural numbers. Let $A$ and $B$ be a countable infinite sets. Now let $f$ be the bijection between $A$ and $\mathbb{N}$ and $g$ be $B$'s bijection.  $f:\mathbb{N}  \rightarrow  A$  $g:  \mathbb{N}  \rightarrow B$  There can always exist a bijection $h$ that maps $A \cup B $ to $\mathbb{N}$. To account for non disjoint sets, let $B' = B \setminus A$; (note: $A \cup B = A \cup B'$)  $h:  \mathbb{N}   \rightarrow  A \cup B'$  $ h(x) = \left\{         \begin{array}{ll}             f(\frac{1}{2}x); &amp; \quad x \in 2\mathbb{N} \\             g(\frac{x+1}{2}); &amp; \quad x \in 2\mathbb{N}+1         \end{array}     \right. $   $h^{-1}(x)$ would map $A$ to the even numbers, and $B'$ to the odd numbers.  Therefore, The union of countable sets are countable. (QED)  Now to verify the definition.     (i) X and Ø are elements of $\mathscr{T}$.   Ø is given in the definition of $\mathscr{T}_2$. Also, the empty set is considered finite, with cardinality zero. Thus $X$ is in $\mathscr{T}_1$ because its complement is finite, and finite numbers are countable.      (ii) $\mathscr{T}$ is closed under finite intersections.   Let:$V_k \subseteq X$ be any countable set. (ie: $V_1, V_2...$ are all countable sets)  So Then, $U_k = X \setminus V_k$, would be the subsets of $\mathscr{T}_2$.  $U_k \cap U_j = X \setminus (V_k \cup V_j) $, due to De Morgan's law.   Thus, $U_k \cap U_j \in  \mathscr{T}_2$, since the Union of two countable sets is also countable[A]. And This reasoning can be extended to any finite amount of intersections.       (iii) $\mathscr{T}$ is closed under arbitrary unions.   Let $A=\bigcup_{i \in I} U_i$ be an arbitrary union of sets whose complements are are countable.Then by De Morgan's law, $A=X \setminus (\bigcap_{i \in I}V_i)$, for some countable sets $V_i$. By the definition of intersection, $\bigcap_{i \in I}V_i \subseteq V_i$ for all $i \in I$. Since $V_i$ is countable so is any of its subsets.  Hence the complement of $A$ is countable. Why does $\log(n!)$ and $\log(n^n)$ have the same big-O complexity? In an example that I found, it is said that $\log(n!)$ has the same big-O complexity as $\log(n^n)$. Please explain why this is the case. Linear Combination of Random Normal Variables In order to prove that the linear combination of two independent normal distributions(say Z=X+Y) is normal, i am using their MGFs to show that the linear combination also has a similar mgf. This works okay when Z = X+Y. But when Z=X-Y, i run into trouble :  E[$e^{tZ}$] = E[$e^{t(X-Y)}$] = E[$e^{tX}$]E[$e^{-tY}$]  Now i cannot put this as $M_{x}$(t) * $M_{y}$(t) because of the negative sign. Retrieve the position (X,Y) of an HTML element relative to the browser window I tried to write out the aura if statement but I am getting stuck if someone can explain it to me in detail. thank you  &lt;aura:if isTrue=&quot;{!lessthan(v.Tl.time, v.Tl.End_Date_Time1__c)}&quot;&gt;                            &lt;/aura:if&gt; Find the minimal polynomial for $\sqrt[3]{2} + \sqrt[3]{4}$ over $\mathbb{Q}$  I havent covered galois theory, this is an exercise from the chapter algebraic extensions in gallian.  I can see that $\sqrt[3]{2} + \sqrt[3]{4}$ $\in \mathbb{Q}(2^{1/3})$, after this I am clueless.  Need hints will finish the proof. Show reference label when it is undefined When a reference is undefined, LaTeX shows simply [??] which doesn't give much information. I would like it to show the label instead. This is useful when more collaborators work on the same preprint and the references are yet to be done.  I'm using neither biblatex nor natbib. I do use bibtex but it is irrelevant for this question.  MWE:  \documentclass{article}  \begin{document}  This: \cite{undef} should look like: [{\bfseries ?? undef}]  \end{document} Solving equation involving Euler's totient function Given $m\in\mathbb{N}$, is there a general way to solve the equation \begin{equation} \varphi(n)=m \end{equation} where $\varphi$ is the Euler's totient function? For example, I want to solve for all the $n$'s such that $\varphi(n)=4$. Is there any way to solve it that are also applicable to many (if not general) cases? I would like to install Ubuntu 14.04.02 LTS on my machine which currently has Mageia 4.1.   So, I have the following partitions: One for /, one for swap and one for /home.   I have some important data on /home which I wouldn't like to lose (I have backed them up, but I would prefer to keep them) after a clean install. I plan to reformat partitions one and two as above and keep the /home one.   Will I be able to tell Ubuntu to do so during an installation? By this I mean: Will it be able to recognize the /home partition and keep all its data intact while formatting the rest? And one more question: Will it be able to keep the users as they are our will I be forced to change them too? (I have the bad feeling that it won't be able to keep the users as they are...) Flexible table with calculating column width fails starting from division by 10 Finding the value of $\int_0^{\pi/2} \frac{dt}{1+(\tan(x))^{\sqrt{2}}}$ The problem: Find the value of $\displaystyle\int_0^{\pi/2} \frac{dx}{1+(\tan(x))^{\sqrt{2}}}$   I tried a few different substitutions and the closest I got to an okay looking integral is $$\int_0^\infty \frac{t^2 \, dt}{t^{1/\sqrt{2}}(1+\sqrt{t})},$$ which still looks scary. How do I approach this integral? Internet through wireless hotspot, intranet through wired network - can I use both at the same time? I have been tring this for a while now, without success. I have read online this can be done, either by changing the default gateway (DG) or messing with the DG's metric  The short story is: I need to connect to my company's ERP system (port 80 for some web services and port 22 for sftp and bash) with using my notebook computer's LAN card - but I can't connect to the Internet on this network. So I have my mobile phone's 3g hotspot when I need to go online. But I want to use both at the same time without so much as opening another web browser process.  At the moment, I am using the ERP system regularly - and whenever I need to go online to check a mail or research something related to work, I need to unplug the network cable so the wifi can kick in... which is starting to become annoying especially if I am connected to the ERP system using FTP (Filezilla) or Shell (putty)  Can someone help me with the configurations?  Below are some of the details I can provide:  Wired network - office LAN - for connection to the ERP system: DHCP -    Yes IPv4      10.64.200.245 Subnet    255.255.240.0 DG        10.64.193.254 DHCP Serv 10.64.17.191 DNS       10.68.72.200           10.68.20.200 WINS      10.68.77.200           10.64.20.200   Below are the details for my WiFi internet connection - over my mobile hotspot  DHCP      Yes IPv4      192.168.43.39 &lt;- I'm guessing this changes every time I restart the wifi subnet    255.255.255.0 DG        192.168.43.1 DHCP      192.168.43.1 DNS       192.168.43.1   Finally, here is the tracert results (btw, I'm on Windows 7, 64bit and the remote system should be on Linux) - the target ERP ip address is 10.76.142.39  Tracing route to 10.76.142.39 over a maximum of ...    1    &lt;1 ms    &lt;1 ms    &lt;1 ms  10.64.193.253   2    &lt;1 ms    &lt;1 ms    &lt;1 ms  10.64.4.13   3    &lt;1 ms    &lt;1 ms    &lt;1 ms  10.76.23.124   4     1 ms    &lt;1 ms    &lt;1 ms  10.76.23.93   5     1 ms    &lt;1 ms    &lt;1 ms  2xx-xx-xx-9.adsl...   6     1 ms     1 ms     1 ms  2xx-xx-xx-9.adsl...   7     1 ms     1 ms     1 ms  2xx-xx-xx-10.adsl..   8     2 ms     1 ms     2 ms  10.76.135.125   9     1 ms     1 ms     1 ms  10.76.142.39   sorry, I had to mask out some details above... wasn't sure if they would have any confidentiality issues.  Any help is much appreciated. "Similar questions" overlaps footer and has no background When I can scroll down far enough to have the hot network questions overlap the footer (because my question is short) there is no background under the text:    Could they either be given a white background: (just disable the transparency option)    Or stop at the footer:      This also happens with the help text on some sites, as per this bug report:     Ubuntu 14.04.1 (Latest LTS Version), . Same problem with  and .  Windows 10 Pro,  and .  (Images linked to screenshot of browser)  My question is different from  because my issue is the lack of background on the footer.   I think there was a site redesign? Convert line voltage heating system to use low voltage thermostats I currently have two line voltage ceiling radiant heating systems in my condo each with their own DPST thermostat. I'd like to use low voltage thermostats such as the nest. I found , but it only has 3 wires. Will this work with a system designed to use DPST thermostats? If not, is there something else out there that will allow me to use low voltage thermostats with line voltage heaters? Sort a Map by values In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible. How to turn cheats on or off