I need some help in solving the following problem. We roll a fair die repeatedly until see the number $6$. Let $X$ be a random variable is a number of times we roll the die. Let $A$ be the event "The results of all rolls are even numbers". What is the value of $\mathbb{E}(X|A)$?
So earlier today I came across , and I am having some trouble understanding the solution. The question is as follows:     You throw a fair six-sided die until you get 6. What is the expected   number of throws (including the throw giving 6) conditioned on the event   that all throws gave even numbers?   Quoted from Jimmy Jin in "Elchanan Mossel’s dice problem"  In the paper it goes on to state why a common wrong answer is $3$. Then afterwards explains that this problem has the same answer to,   "What is the expected number of times you can roll only $2$’s or $4$’s until you roll any other number?"  I don't understand why this is the case. If the original problem is asking for specifically a $6$, shouldn't that limit many of the possible sequences?  I also attempted to solve the problem using another method, but got an answer different from both $3$ and the correct answer of $1.5$.  I saw that possible sequences could have been something like:  $$\{6\}$$ $$\{2,6\}, \{4,6\}$$ $$\{2,2,6\}, \{2,4,6\}, \{4,2,6\}, \{4,4,6\}$$ $$\vdots$$  To which I set up the following summation and solved using :  $$\text{Expected Value} =\sum_{n=1}^\infty n\left( {\frac{1}{6}} \right)^n 2^{n-1} = 0.375$$ Obviously this is different and probably incorrect, but I can't figure out where the error in the thought process is.  Any help on understanding this would be greatly appreciated.  A blog post discussing the problem can be found
What is the difference between object keys with quotes and without quotes? Counter with 4 flip flops jk synchronous from 3 to 13! I have to create a counter with 4 flip flops jk synchronous. The counter shall count in this order: 3 to 13 and then to 3 to 13 and so on..  I have to use logisim simulator.   Ok. So i think its solved. I realized that the circuit needs to start at 0 ou 1 and stabilize after the 13 and then it goes to 3. Thanks for your help.  Hope this will help for somebody else too.  Thank you all for your help! On , this is my tag blacklist:        Note well that I have them set to hidden, because I would prefer not to see questions on this topic. Bridge players, don't take offence &mdash; I don't have problem with the game itself. I don't understand the game's mechanics, though, so I find questions about it incomprehensible, and I'd rather stick to just seeing questions I can understand.  I still see [bridge] questions galore on the Android app though. Could it be updated to pay attention to whether I have certain tags ignored and set to be hidden? Question is :  Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that  $Y\cup A$ and $Y\cup B$ are connected.  What i have tried is :   Suppose $Y\cup A$ has  separation say $C\cup D$ then all i could see is that   Either $Y\subset C$ or $Y\subset D$ as $Y$ is connected and $ C\cup D$ is separation for a subset that contains $Y$.  Without loss of generality we could assume $Y\subset C$  As $Y\cup A=C\cup D$ and $Y\subset C$  i can say $D\subset A$ (I do not know how does this help)  I have $X-Y=A\cup B$ with $A\cap B=\emptyset$  I have not used connectedness of $X$ till now, So i thought of using that and end up with following :   $X-Y=A\cup B\Rightarrow X=Y\cup A\cup B=(Y\cup A)\cup (Y\cup B)$  I do not know what to conclude from this...   I would be thankful if some one can help me by giving some "hints"  Thank you I had this "almost bonus" question on the final in Group Theory recently: prove that there is no such group $G$ which would satisfy Aut$(G)\cong \mathbb{Z}_n$, where $n$ is an odd integer. I don't have much certainty if this proof is OK, and one's opinion would be appreciated.  Here's my attempt:     Suppose that there exists such a group $G$ that satisfies the above condition. Since $\mathbb{Z}_n$ is cyclic, Aut$(G)$ is also cyclic and $G$ is abelian, which implies that Inn$(G)\cong \{e\}$. Thus Aut$(G)=$Out$(G)$. (I don't think this fact is important here though). Now, since an automorphism sends a generator to a generator, and since each automorphism is completely determined by such mapping, |Aut($G$)| must be of factorial order. But an integer factorial is always even. However, $|\mathbb{Z}_n|$ is odd, which is a contradiction. Therefore, no such $G$ exists.   I'm afraid that my proof is not very nice, but at least a genuine attempt was made. Limit $\frac{x^2y}{x^4+y^2}$ is found using polar coordinates but it is not supposed to exist. Resolving ambiguous overload on function pointer and std::function for a lambda using + How do you create a dropdownlist from an enum in ASP.NET MVC? I'm trying to use the Html.DropDownList extension method but can't figure out how to use it with an enumeration.  Let's say I have an enumeration like this:  public enum ItemTypes {     Movie = 1,     Game = 2,     Book = 3 }   How do I go about creating a dropdown with these values using the Html.DropDownList extension method?  Or is my best bet to simply create a for loop and create the Html elements manually? Can I see which seats are available on a British Airways flight before I book? How do I remap a key in macOS Sierra, e.g., Right Alt to Right Control? Defining macros with arguments With newcommand I can define customized command like \ABC. Is there any way to define a command with arguments. For example I want \brat{A}{B} be the same as [\![A,B]\!]. a / an - adjective - noun The article a becomes to an when the following noun starts with a vowel.     Not a adventure but an adventure.   But how is it, when there's an adjective before the noun?     Not a thrilling adventure or not an thrilling Adventure?   And how is it, when the adjective starts with a vowel but the noun does not?     Not a exciting tournament or not an exciting tournament? Empty drives show 2 GB used, Explorer shows nothing I have 4 drives in Windows: C:, D:, E: and F:. Except C:, all of them are empty.    Suddenly it shows up to 2 GB space used – even when there are no files.     [  When I format the drives, it's fine for a while until it shows used space again.  I use Windows 8.1 Pro 64 Bit. How does Princess Leia know Obi-Wan Kenobi is "Ben" Kenobi? When Luke rescues Leia from her cell on the Death Star:  Leia: Aren't you a little short to be a stormtrooper? Luke: Huh? Oh, the uniform. I'm Luke Skywalker. I'm here to rescue you. Leia: You're who? Luke: I'm here to rescue you. I've got your R2 unit, I'm here with Ben Kenobi! Leia: Ben Kenobi, where is he?! Luke: Come on!  How does Princess Leia know Obi-Wan Kenobi is &quot;Ben&quot; Kenobi? Shouldn't her response have been: &quot;Ben Kenobi? You mean the General??&quot; How to measure the component value on a PCB? Word for dismissing someone's opinions as racist, liberal, etc, instead of debating back I'm looking for either a single word or phrase that would describe either someone or the action of dismissing someone's opinions as something "socially unpopular", without giving any reasoning why, either just to spite them, because they have a poor counter-argument, have been backed into a corner, or something similar.  For example, someone who supports something politically incorrect (like, say, stronger gun control or help for male domestic abuse) might have their views dismissed as racist or sexist or liberal without actually acknowledging and debating against them, thus labeling the former as something they might not be.  I was talking to a friend earlier about politics, and she complained about Germany's immigration problem. She said one issue was that many politicians that try to support stronger immigration laws would immediately be shot down as racist or a nazi and forced to resign, instead of being presented with counter-arguments. Regardless of whether you agree with them or not, what would describe that?  EDIT: Forgot to specify -- the word or phrase I'm looking for would be used in an informal context. And to address further questions, to be used when the person's opinion has legitimate worth and isn't just "being racist" etc, i.e. the dismisser is incorrect, and is only making these accusations because they can't argue further. (Added emphasis to the last sentence of the first paragraph) wrong positioning of figure with tabular in xelatex I recently updated MikTex and it seems Xelatex is not able to place figures correctly anymore. Before updating all figures were shown correctly. I work with Texworks and Miktex 2.9  Here it is the code that I run:  \documentclass[10pt,a4paper]{report} \usepackage{graphicx} \graphicspath{{method1/}{method2/}{dpfig/}}  \begin{document}  \begin{figure}[!htbp] \centering \begin{tabular}{cc} \includegraphics[trim=6cm 2.5cm 6cm 2.5cm,clip=true,width=0.33\textwidth]{figure1.eps}&amp; \includegraphics[trim=6cm 2.5cm 6cm 2.5cm,clip=true,width=0.33\textwidth]{figure2.eps}\\ \end{tabular} \caption{Two figures} \end{figure} \end{document} \end{figure}   With this code, figures are correctly trimmed and the relative position between figure 1 and figure 2 is maintained. However, the position in the document is wrong: figure are placed upper and more right than expected (as the bottom-left corner of the table was placed in the middle of the page). The unusual thing is that this does not happen in draft mode and picture are shown centered as it would be.  I usually compile with Xelatex, Makeindex and Bibtex.  I hope to have been clear.. bye!  EDIT: Sorry, I forgot to add the images I prepared from the file that run that script: Why are hand-built wheels considered to be better than machine-built wheels? Why are hand-built wheels considered to be better than machine-built wheels? I would think that a machine-built wheel would be better and cheaper, since machines generally do a more precise job than humans. Live refresh shows "modified" on tagged questions even when unmodified Evaluation order of Button's action