Giving the following question:  A and B flips a cion, alternately. A starts. The one who gets H first, wins. Let X be a random variable denotes the number of flips at the end of the game. Calculate: $P(X is even)  I found two approached that yields two difference solutions:   Let A flips a coin , and Let B flip a coin. (Assume A didn't win)  I'll call it a round. $round_i$ is independent. By a Tree, $P(A)_{head} = \frac{1}{2}$ and $P(B)_{head} = \frac{1}{4}$ When $X$ is even, it's $B$'s round and therefore answer is $\frac{1}{4}$.  Second approch is simply:  $\sum_{i=1}^\infty (\frac{1}{2})^{2i} \approx \frac{1}{3}$ For each 2 flips, or for each round, we have a probability of $\frac{1}{2}$ to get X even.  What is the right approach? If none of the above, that is the right approach? Thanks in advance.
A fair coin is tossed until a head comes up for the first time. The probability of this happening on an odd number toss is?  How do i approach this problem ?
Compute $e^{e^z}$, where z is a complex number I am self-studying the MAT 67 linear algebra course and I got stuck at a problem in Chapter 2:     Compute the real and imaginary parts of $e^{e^z}$ for $z$ $\in$ $\Bbb C$.   I tried to expand it using Euler's formula but it seemed endless. Any help? How to make a pattern follow a stroke, path or line? Here is my MS Paint job on a real bad attempt at what I'm trying to achieve:    I have access to Photoshop and Illustrator CS6, and I want to be able to draw a thick line (that is not straight) and have a pattern within the line show arrowheads within the line to indicate the direction of the line. File locked after sending it as attachment Get OS-level system information I'm currently building a Java app that could end up being run on many different platforms, but primarily variants of Solaris, Linux and Windows.  Has anyone been able to successfully extract information such as the current disk space used, CPU utilisation and memory used in the underlying OS? What about just what the Java app itself is consuming?  Preferrably I'd like to get this information without using JNI. I know that IList is the interface and List is the concrete type but I still don't know when to use each one. What I'm doing now is if I don't need the Sort or FindAll methods I use the interface. Am I right? Is there a better way to decide when to use the interface or the concrete type? What's the best way to store class variables in PHP? I currently have my PHP class variables set up like this:  class someThing {      private $cat;     private $dog;     private $mouse;     private $hamster;     private $zebra;     private $lion;      //getters, setters and other methods }   But I've also seen people using a single array to store all the variables:  class someThing {      private $data = array();      //getters, setters and other methods }   Which do you use, and why? What are the advantages and disadvantages of each? I've tried almost any method to root my ASUS ZenFone Live, but no one seems to work. Also I only need root, no unlocked bootloader (don't want to wipe data) or custom recovery (not essential ATM). How do I get a list of installed files from a package? How to generate a chirped signal What causes "java.lang.IllegalStateException: Neither BindingResult nor plain target object for bean name 'command' available as request attribute"? Java: GUI - JButton opening new JPanel and reading the JTextFields Re-mounting an un-mounted USB stick How can I add a new value to a picklist using APEX? I have queries pulling all of the information correctly but what I can't seem to find was a way to add new options to a picklist using APEX.  I currently have   Package__c.Products__c.add(new Schema.PicklistEntry(products[j].Name));   Package__c is my custom object, and Products__c is the picklist. products[j] is pulling a list of products pulled from our products page.  And I've tried adding it as a select option, or even casting the variable type.  And this information has to go to the actual picklist within a custom object, and not within a visual force page or anything like that. How to select more layers with GIMP? In Adobe Photoshop I am able to select multiple layers at once with Shift+Click.  How can I do that in GIMP? Chatroom deleted? I created a chatroom about the Qt Framework some time ago but it seems it's no longer available. The  just says &quot;Page Not Found&quot; and the possible reasons for removal are &quot;spam, off topic, abuse&quot; but I don't think any of this concerns the chatroom (though I was rarely checking it). Any idea what could be the issue? What do \makeatletter and \makeatother do? Many LaTeX “hacks” begin with \makeatletter and end with \makeatother. What do these commands do? Prefix or adjective meaning "one and a half" Is there a prefix or adjective that means "one and a half", as "tri-" or "triple" is for "three"?  The exact usage I have is to describe "18" in terms of a dozen. Where I live they've started making 18-egg cartons, and the local grocer was wondering what name to give them in his computerised cash register, hoping for a single word.  Even slang, informal or invented term would do if there isn't such a term already. Gluing Lemma for Closed Sets: Infinite Cover Counter-Example OED says:     doobie: a marijuana cigarette    Origin unknown. A relationship with dobby has been suggested.    dobby/dobbie: A silly old man, a dotard, a booby. Dialectal.      First citations:    1967   J. B. Williams Narcotics &amp; Hallucinogenics. Dubbe, Negro slang for a marijuana roach.    1982   A. Maupin Further Tales of City 95,  I smoke a doobie at lunch.   But I just chanced upon  in Yahoo Answers (don't laugh! :) where someone posted:     "It's obvious that Shaggy's a , I'm sure he's been feeding Scoob a bit of his hash and whatnot."    Scoob = , cartoon "pet" dog of constantly-hungry hippie/village idiot Shaggy (from 1969)   I'm not entirely convinced that 1967 dubbe was necessarily a significant factor in the appearance of  since around 1970 (the hippie era, some might say).  So I'd like to know whether it's credible to claim the usage arose (primarily?) through association with the TV show (specifically, the two characters therein closely associated with marijuana). I want to boot Ubuntu into recovery mode but there is no grub menu entry allowing me to do so. I pressed e and now I am able to edit boot options. It looks like that:  setparams 'Ubuntu , with Linux 3.6.0-26-generic'  recordfail gfxmode $linux_gfx_mode insmod gzio insmod part_msdos insmod ext2 set root='(hd0,msdos5)' search --no-floppy --fs-uuid --set=root &lt;uuid&gt; linux /boot/vmlinuz-3.5.0-26-generic root=&lt;uuid&gt; ro ipv6.disable=1 quirt splash $vt_handoff initrd /boot/initrd.img-3.5.0-26-generic   The  documentation says:     Boot to the recovery mode even if it is not listed on the menu:         Remove existing kernel options from the linux line and add single      But this sentence doesn't make any sense, maybe the wiki page is broken somehow. How to modify the boot menu to boot directly into recovery mode?  Note: I don't have access to grub config or anything else, only the boot menu edit page is available.