Two players, $A$ and $B$, alternately and independently flip a coin and   the first player to obtain a head wins. Player $A$ flips first. What is   the probability that $A$ wins?      Official answer: $2/3$, but I cannot arrive at it.   Thought process: Find the probability that a head comes on the $n$th trial. That's easy to do, it's just a Geometric random variable. Then find the probability that $n$th turn is player's A turn. Finally, multiply both probabilities.  When I came up with each probability, both of them depended on the amount of trials $n$, so my answer was a non-constant function of $n$.  However, what I find quite fantastic is that the answer is a constant, so the amount of tries until a head comes doesn't seem to matter.
Two players, $A$ and $B$, alternately and independently flip a coin and the first player to get a head wins. Assume player $A$ flips first.  If the coin is fair, what is the probability that $A$ wins?   So $A$ only flips on odd tosses. So the probability of winning would be $$ P =\frac{1}{2}+\left(\frac{1}{2} \right)^{2} \frac{1}{2} + \cdots+ \left(\frac{1}{2} \right)^{2n} \frac{1}{2}$$  Is that right? It seems that if $A$ only flips on odd tosses, this shouldn't matter. Either $A$ can win on his first toss, his second toss, ...., or his $n^{th}$ toss. So the third flip of the coin is actually $A$'s second toss. So shouldn't it be $$P = \frac{1}{2} + \left(\frac{1}{2} \right)^{2} + \left(\frac{1}{2} \right)^{3} + \cdots$$
Are punches with Gauntlets of Ogre Strength magical, improvised weapon attacks? My Gnoll Paladin just got the Gauntlets of Ogre Strength from Phandelver the other day, and it got me wondering:    Would it be possible/make sense for punches using them to count as an improvised weapon? (Since I'm not punching with empty fists, I'm wearing magical gauntlets) If so, would punches from a person using them count as magical weapon damage? How to combine multiple conditions to subset a data-frame using "OR"? Are positive real numbers $x,y$ allowed to be taken out during this proof? Prove $$\left(x^2 - y^2\right)\left(\frac1y - \frac1x\right) \ge 0$$ where $x$ and $y$ are positive real numbers.   Can I simplify $\frac{(x^2 - y^2)(x-y)}{xy} \ge 0$ and then $(x^2 - y^2)(x-y) \ge 0$ cancelling out the $xy$?  Is this valid because then  \begin{align} (x+y)(x-y)(x-y)&amp;\ge  0\cdot(x-y)^2\ge 0 \end{align}  which is always true and then prove backwards from here? Are these intended activities legal to do in the USA under the VWP? I'm a British citizen traveling to the USA next month with an ESTA to see my fiancee. I'm traveling there to plan our wedding, talk about a location to get married, and probably finalise a date and a booking with my fiancee (US citizen). This is the primary reason for my trip to the USA. Am I allowed to do this as a tourist? I would like to know for when the CBP officer asks me if this thing is allowed.  Am I allowed to enter the USA, on a separate trip (probably later this year), for the sole purpose of marrying my fiancee, provided I can persuade the CBP officer I intend to leave at the end of my trip? My fiancee and I are moving to the UK next year. How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$? How do you attach a debugger to a Web service? I am using a Web service. Can anybody tell me how to debug through it? Are global variables bad? I am a photography newbie and have only a 18 - 135mm canon kit lens at my disposal. I wanted to take macro shots but can't really afford a macro lens right now. One very optimistic goal of mine is to capture a single snow flake this winter.   Is there any cheap(er) way to take macro shots? Also, any specific advice from anyone one how to go about photographing a single snow flake. Reference request: Standard textbook for first-order predicate logic in english I am writing my bachelor thesis in english, but I have studied logic in my different native language. Since a rely on elementary results from first-order predicate logic (especially satisfaction in structures), I should reference the reader to a textbook for details.   Is there a standard textbook that starts with elementary propositional logic and goes to Incompleteness theorems? grub2 not able to find the new kernel This is very strange issue, i have CentOS7 running on HP server and everything was working till date until i did "yum install kernel" after reboot it drop me in grub&gt; shell that means it won't able to read or find grub.conf file.   in grub&gt; i found grub.conf has still old files there is no new entries of new kernel   I did this, and found there is no new kernel entries but if i boot with old kernel then i can see latest grub.conf (with new kernel)  grub&gt;cat /grub2/grub.conf   Question: where grub2 getting that old file from? why it's not able to find new kernel ? Should I use "who" or "which" when referring to a specific animal species? I'd like to know what option to use, specifically when referring to a species by its scientific name. I have some lines. I want to draw perpendicular lines at the start and end of the lines, on both sides of the lines.  Can somebody help me in Python?  I am using ArcGIS. What packages do i need to install on an ubuntu desktop for it to "become a server"? Alright, so due to problems connecting to a wifi network on the ubuntu server distribution, i want to install the neccesary packages for a lamp, mail and ssh server on an ubuntu desktop 14.04 lte distribution(i am able to connect to the internet with the desktop version).  what should i install from the command line?  Thanks in advance! $f(x)=x/1$ is not injective? I want the entries in the TOC for the indices to align with the Chapter entries. I can do this with \addcontents. \printindex also puts an entry in the TOC but it is aligned with the section entries.   So I want to:    change how printindex makes its entry into the TOC or  to eliminate printindex entry.   MWE:  \documentclass{book} \usepackage{multind}        % create the indices \makeindex{names}            \makeindex{subject} \newcommand{\Ndx}[1] { \index{names}{#1} }     % define the type of entry \newcommand{\Sdx}[1] { \index{subject}{#1} }   % to the indices \usepackage[headings]{fullpage}         % 8-1/2, 11 page with 1" margins \usepackage{fancyhdr} \pagestyle{fancy} \begin{document} \fancyhead{} \fancyfoot{} \renewcommand{\sectionmark}[1]{\markright{\thesection.\ #1}} \fancyhead[LE,RO]{\rightmark} \fancyhead[RE,LO]{\thepage} \tableofcontents        % put table of contents here  \chapter{\bf Logic, Plausibility, and Probability} \label{Ch:Logic} \section{Logical Arguments and Reasoning } It is obvious that logical arguments are essential to the judicial system. The logic must be able to treat syllogisms\Sdx{syllogism} that range from deductive\Sdx{syllogism!deductive} to abductive\Sdx{syllogism!abductive}, that is from strong to weak syllogisms \Sdx{syllogism!weak}\Sdx{syllogism!strong} as illustrated in the table taken from Author \Ndx{Author} \Ndx{Cohen} \Sdx{Cohen on probability} \clearpage \addcontentsline{toc}{chapter}{Subject Index} \printindex{subject}{Subject Index} \addcontentsline{toc}{chapter}{Name Index} \printindex{names}{Name Index} \end{document} What is the $n$ times iterate of $f(x)=\frac{x}{\sqrt{1+x^2}}$? Flight cancelled but doesn't show up on status I booked a SpiceJet flight from Dubai to Ahmedabad on Dec 16 (SG-16) scheduled to depart at 7:25 PM. At noon I received an SMS from SpiceJet to advise me that my flight has been delayed by 4h 50m. Sure enough, when I checked flight status on SpiceJet's website, that was the case. But on other websites like flightstats.com and dubaiairports.ae, the status was that flight was "delayed by 2h 25m".    At 5 PM, I got an SMS that flight had been cancelled. I double checked on Spicejet's website and that too showed the flight to be cancelled. But both flightstats.com and dubaiairports.ae still showed the flight as delayed by 2h 25m.    I went to the airport and met SpiceJet officials. Sure enough, the flight was cancelled. But somehow, the flight cancellation status never made it to the system. I have a Tripit Pro account, and even that never sent me an alert for that. I contacted the travel agent through which I booked my flight (makemytrip) for a refund, and they said that their system doesn't show that flight was cancelled. Now, even the Spicejet website has removed the cancellation status from that flight.    Apart from the SMS, I now have no proof that my flight was cancelled.    Can someone tell me, how can I proceed with a refund claim? How can we unlock an iPhone backup file if we don't know the password? Starting with the episode "Crossroads Part II" the      final five Cylons begin to hear Bob Dylan's "All Along the Watchtower" in their heads.   Have the creators/writers ever stated why they picked that song in particular and what it might mean to the story? I can't open google-related sites I'm  getting timeout errors on google-related sites. If I try to open google.com, gmail.com, etc, it will keep displaying the "waiting reply" on the browser bottom, and the page won't open. If I am browsing another site that includes, say, www.google-analytics.com, ajax.googleapis.com, google fonts, etc, then it start loading, displaying a "connecting to [google-related site]" on the browser bottom, and stall at that point for a long time.  I can browse other sites fine and they load quickly. I can ping the domains with 0% loss and &lt;100ms times. I can ping 8.8.8.8, I am not behind a proxy, there is nothing on my hosts file, I am not at work or in a networked environment: it is a home network behind a router. I connect using wifi (I am too distant to use wired).   Other users on the same network don't have the issue. I can both ping and browse the sites. I have pinged ajax.googleapis.com from my android and my other laptop, and they return the same IP than on this laptop. They also share the same external IP address, according to   The issue happens on ubuntu 12.04, and as it is cyclic but not frequent enough to make sure, I don't honestly remember if it happens on Windows 7 as well (it is a dual-boot machine). Rebooting the router won't help. Rebooting the computer fixes the problem, until it happens again.  It's been happening for a few months now. The last time it happened (earlier today) I performed a ping before and after the reboot. Here's the results:  before  $ping ajax.googleapis.com PING googleapis.l.google.com (74.125.21.95) 56(84) bytes of data. 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=1 ttl=47 time=157 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=2 ttl=47 time=163 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=3 ttl=47 time=156 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=4 ttl=47 time=165 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=5 ttl=47 time=157 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=6 ttl=47 time=155 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=7 ttl=47 time=159 ms 64 bytes from yv-in-f95.1e100.net (74.125.21.95): icmp_req=8 ttl=47 time=157 ms   after  $ping ajax.googleapis.com PING googleapis.l.google.com (74.125.134.95) 56(84) bytes of data. 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=1 ttl=47 time=207 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=2 ttl=47 time=154 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=3 ttl=47 time=154 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=4 ttl=47 time=152 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=5 ttl=47 time=157 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=6 ttl=47 time=160 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=7 ttl=47 time=154 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=8 ttl=47 time=157 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=10 ttl=47 time=154 ms 64 bytes from gg-in-f95.1e100.net (74.125.134.95): icmp_req=11 ttl=47 time=152 ms   The issue happens on different browsers, I installed epiphany and it shows the same behaviour.  I have inspected the network tab on firefox and it shows a single, unreplied GET request:   I installed wireshark and listened for HTTP GET requests; on other sites it pops up on the report, but on ajax.googleapis it shows nothing. I listened then for DNS requests, there are some differences:     If connect via telnet to the port 80 of a random site and perform a GET it works, but I can't telnet to ajax.googleapis.com (or it's IP), the connection never completes.