A and B play a bit game. Unbiased bit generator is generating 0 or 1 repeatedly until one of the following happens.   The bit patterns to '00' (i.e., a 0 is immediately followed by a 0) for the first time. In this case A wins. The bit patterns to  '01' (i.e., a 0 is immediately followed by a 1) for the first time. In this case B wins.   Who has more probability of winning the game?  I am getting both probability to be the same. Can anyone help me out how to approach this question?
If A and B play a game with a fair coin such that A wins by getting consecutive TT for the first time, while B wins by getting a consecutive TH.  Who has higher probability of winning the game?   I tried the following method but not sure:  For A to win,     TT, HTT, HHTT ... that is, 2-k for each game length k   For B to win,     TH, HTH, HHTH ... that is, 2-k for each game length k   So ultimately I am getting same probability of each to win.
How to delete file with this name on linux: -]???????q Somehow a file named -]???????q got created on my terminal. I tried  rm -f "-]???????q"  (in double quotes), but it did not get deleted. (Error::  rm: invalid option -- ]).  How do I delete it? What Happened to The Latex Symbols In the Libreoffice-Texmaths Equation Editor? Why the galaxies form 2D planes (or spiral-like) instead of 3D balls (or spherical-like)? Question: As we know, (1) the macroscopic spatial dimension of our universe is 3 dimension, and (2) gravity attracts massive objects together and the gravitational force is isotropic without directional preferences. Why do we have the spiral 2D plane-like Galaxy(galaxies), instead of spherical or elliptic-like galaxies?   Input: Gravity is (at least, seems to be) isotropic from its force law (Newtonian gravity). It should show no directional preferences from the form of force vector $\vec{F}=\frac{GM(r_1)m(r_2)}{(\vec{r_1}-\vec{r_2})^2} \hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically.  If the gravity attracts massive objects together isotropically, and the macroscopic space dimension is 3-dimensional, it seems to be natural to have a spherical shape of massive objects gather together. Such as the globular clusters, or GC, are roughly spherical groupings , as shown in the Wiki picture:     However, my impression is that, even if we have observed some more spherical or more-ball-like , it is more common to find more-planar  such as our ? (Is this statement correct? Let me know if I am wrong.)  Also, such have a look at this more-planar Spiral "galaxy" as this NGC 4414 galaxy:    Is there some physics or math theory explains why the  turns out to be planar-like (or spiral-like) instead of spherical-like?  See also a   p.s. Other than the classical stability of a 2D plane perpendicular to a classical angular momentum, is there an interpretation in terms of the quantum theory of vortices in a macroscopic manner (just my personal speculation)?  Thank you for your comments/answers! Ever since I upgraded to 15.10, fdisk -l reports 16 ram disks (/dev/ram0 ... /dev/ram15). I'm a bit unsure what those are needed for. Is it safe to delete them? If not, how can I get rid of that fdisk output? Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer. Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer.   I'm not familiar to factorial and I don't have much idea, can someone show me how to prove this? Thank you. Young Adult book about kids being bred for genetic purity in a world where races no longer exist I'm looking for a book I read in Junior High (probably) about a world where everyone was basically the same race.  One genetically pure male and female teenager of each race still exists.  They are transporting a genetically pure boy to meet his genetically pure female counterpart and accidentally introduce him to a genetically pure female of a different race.  And, of course, chaos ensues.  Any thoughts? Why does Yoda speak the way he does? How does 3-wire RTD or resistance measurement works in bridge of wheatstone? I cannot understand how 3-wire measurement will improve the resistance measurement.    They say that IF lead resistance \$ R_L \$ is the same, it will compensate. How can this happen?  If bridge is balanced (without \$ R_L \$):  \$ \frac {R_1}{R_2} = \frac {R_g}{R_3} \$  If \$ R_L \$ is added: \$ \frac {R1}{R2}= \frac {R_g+R_L}{R_3+R_L} \$  \$ \frac {R_g}{R_3} \$ is not same as \$ \frac {R_g+R_L}{R_3+R_L}\$; is this true only if \$ R_G=R_3 \$ in the beginning? So this means that \$R_L\$ does not get compensated? What is the value of $p^2q + q^2r + r^2p$ for the given cubic equation? If p,q,r are the real roots of the equation $x^3-6x^2+3x+1=0$ then find the value of $p^2q + q^2r + r^2p$?  My Attempt: I tried $(p+q+r)(pq+qr+rp)$ but couldn't really figure out what to do with the extra terms. The roots are also not trivial to find. Any help would be appreciated. I tried to write , but I couldn't find any simple explanation page to link to. I ended up garbling a home-spun description of my own with a not-terribly-helpful image from 2012.  This  puts it better than I can:     As a new user in 2015 can there please be more information about what is WinterBash. The link took me to a bunch of stick figures running around making (I think) numbers (I think its a countdown clock). It doesn't explain the event or even what a hat is. – Ian Miller Dec 4 at 2:58   All I found on Meta was dead links. For example,  was downvoted to -2 and answered with a link to an FAQ page that no longer exists. How to "Hide a Collection in viewport" but not "Disable in Viewport" via script in Python Hi Blender coders and artists.  I am updating one of my add-ons for Blender 2.8 and have run into some  issues finding the appropriate way to hide a collection that is added to my scene.  I want to deselect the eyeball on the toolbar with a python command in a script but I don't want to disable the collection in the view-port entirely as it is used as a collection in a particle system.   I just want to deselect the eyeball circled in green below.  Anyone know the python API for this? Since upgrading to OS X El Capitan, I've noticed a change in the Terminal: Executed lines show an opening bracket on the left, and a closing bracket on the right, as shown in this screenshot:    Similar questions have been asked here:          Some of the questions are referring to these brackets as marks, and the Terminal's Edit menu has a couple of entries related to marks, for example:   Edit > Marks > Mark as Prompt and Send Return Edit > Clear to Previous Mark Edit > Navigate > Jump to Previous Mark     Is there a summary of what these marks can be used for? I have not seen any reference to this new feature in any of the El Capitan feature overviews. How do wizards cast spells which appear on their spell list when it has a divine focus component? What does a wizard do to cast a spell which contains a DF component? Take  for example,  Components V, S, DF    A divine focus component is an item of spiritual significance. The divine focus for a cleric or a paladin is a holy symbol appropriate to the character's faith. The divine focus for a druid or a ranger is a sprig of holly, or some other sacred plant. If the Components line includes F/DF or M/DF, the arcane version of the spell has a focus component or a material component (the abbreviation before the slash) and the divine version has a divine focus component (the abbreviation after the slash).  So do wizards have holy/unholy symbols in their spell components pouch or what? Why multiple PTR records in DNS is not recommended? I   that using multiple PTR records in a DNS configuration is not recommended.   However, the reasons are often vague, or not so obvious, naming:    "it can cause problems",  "can trigger bugs in programs expecting a single answer" : it's the software's problem then, isn't it?! "can make DNS answer packet too large" : isn't this fixed with ?   Are these good reasons? Do you know of any other (good) reasons? All this kinda looks like a "legacy fear"... I want to georeference jpg map without any digitizing data Can I mix different RAM modules in PC? I have 4 RAM slots on the motherboard P5K. At the moment I have two 1GB RAM modules and two 2GB RAM modules. Everything is working fine.   Can I buy one 4GB module and put it instead of the 1GB module?  Another question, can I mix MHz values, like that I have two 667MHz modules and two 1066MHz modules? Is this dangerous for the comp? Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}[\frac{1}{X};X>y]=0$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $X$ be a nonnegative random defined on this space. For any $A\in \mathcal{F}$, let $\mathbb{E}[X;A]:=\mathbb{E}[X\mathbb{1}_{A}]$, where $\mathbb{1}_{A}$ denotes the indicator random variable on $A$. Without assuming $\mathbb{E}\left[\frac{1}{X}\right]&lt;\infty$, show that  \begin{eqnarray} \lim\limits_{y\downarrow 0} ~~y~\mathbb{E}\left[\frac{1}{X};X&gt;y\right]=0. \end{eqnarray} Any initial ideas will be greatly appreciated. How many research interests should be included in a statement of purpose/objective I am wondering how many research interest should be included in a statement of purpose. Is it better to include just one core interest or to include up to three as in my case? In the article class, it seems that using the abstract environment has an effect to the rest of the document. More precisely, it seems that it redefines the \linewidth, which equals \textwidth without abstract, is smaller with the abstract. My feeling is that \linewidth is redefined for the abstract environment, but the value is not reseted at the end of the abstract.  Here is a minimal example:  \documentclass{article} \begin{document} \abstract{This is an abstract. This is an abstract.}\bigskip  This is text. This is text. This is text. This is text. This is text. This is text.\\  \rule{\textwidth}{3em} \rule{\linewidth}{3em} \end{document}   The output is not the same if you comment the \abstract line or not.  Do you have the same problem as I do when you compile (if not, I will upload some PDF files to show)? Is this a (known) bug in LaTeX2e? How can we circumvent the problem?  Note: I compile with the pdflatex command, and I use pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian). How to install ns2 on Ubuntu