A coin is tossed three times. The probability of zero heads is 1/8 and the probability of zero tails is 1/8. What is the probability that there is at least one head and at least one tail?  So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  We would use the complement rule and Multiplication Rule?   P(at least one head) = 1 - P(no heads) = 1 - 1/8= 7/8 P(at least one tail) = 1 - P(no tails) = 1 - 1/8= 7/8  would I multiply the two values?
A coin is tossed three times: The probability of zero heads is 1/8 and the probability of zero tails is 1/8.  And my question is: What is the probability that all three tosses result in the same outcome?   So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  If so, we could use the Addition Rule which is $P(A \cup B) = P(A) + P(B)$  where A and B are disjoint events, i.e. $A \cap B = \emptyset$, A is the event of tossing all heads and B is the event of tossing all tails.  I'm not sure how to continue after that... would the complement be used?
$(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit. Show $(z_n)_{n\in \mathbb{N}}$ converges to the same limit. $(z_n)_{n\in \mathbb{N}}$ is a complex valued sequence such that subsequences $(z_{2n})_{n\in \mathbb{N}}$ and $(z_{2n+1})_{n\in \mathbb{N}}$ converges to same limit.  How could I show that the sequence $(z_n)_{n\in \mathbb{N}}$ converges to the same limit using $\epsilon$, $N$ definition?    For every $\epsilon &gt;0$,  $|z_{2n}-l|&lt;\epsilon$ whenever $n&gt;N_1$, and   $|z_{2n+1}-l|&lt;\epsilon$ whenever $n&gt;N_2$.  We want to show that for every $\epsilon &gt;0$, there exists $N$ such that $|z_{n}-l|&lt;\epsilon$ whenever $n&gt;N$. I tried something like  $|z_{n}-l|\le|z_n-z_{2n}|+|z_{2n}-l|$ but wasn't sure what to do next How to inject keystrokes via a shell script? I'm trying to inject keystrokes into the input daemon so as to simulate typing from a Bash script. Is this possible, and if so, how can I make it happen? The first time I learnt LaTeX, I thought \newcommand properly fits for inline mode and \newenvironment for displayed mode.  Later I noticed it is not correct. So the question is   What is the rule of thumb when deciding whether we should use one over the other one? Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom Clicking on tags in global search results is broken What is a "double date"? remote ssh tcpdump command, through a gateway, to be opened, in realtime, in Wireshark, on local machine I (Host A - Windows based using plink) am trying to connect to a remote linux server (Host C), through a gateway linux server (Host B), and run this tcpdump on Host C to be piped all the way back to Host A (through Host B I assume), and opened, in real time, with Wireshark on Host A:  Below is what I saw as the command when not using a gateway ("direct remote fetch").  So, how would I make this work going through my gateway host B scenario above?  Host A$ plink.exe -ssh -pw abc123 root@Host C "tcpdump -s0 -U -n -w - -i eth0 not port 22" | "C:\Program Files\Wireshark\Wireshark.exe" -k -i - Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ Blender 2.78 file size grows and does never shrink I made a huge project, wanted to reduce file size from 1.3GB to below 1GB. Compressing did not help. I removed objects from scene. Also textures. I then realized, the file size even was the same when I delete ALL objects. Why?  Download of sample:   Before deleting all: 1.2GB, after: 1.1GB. When I'm looking down the front page on , I often wish I could tell at a glance what happened to the question to bump it there. Sometimes it is obvious from looking at the number of answers, other times you have to follow a link to find out if it has been edited or if an answer has been added or updated.  I'm guessing this is just as applicable on other SE sites - this is the kind of thing I'm suggesting:    A couple of suggestions from the comments that might be good:   To reduce clutter the word 'ago' could be removed at the same time. The new text could be visible only to established users - ie users who are likely to be reviewing/editing and would benefit from seeing at a glance what kind of change has occurred. Presumably after an update, google chrome has very serious problems with page rendering. I want to downgrade it. (Switching to vmlinuz.old and initrd.img.old did not help).  $ apt-cache policy google-chrome-stable google-chrome-stable:   Installed: 35.0.1916.153-1   Candidate: 35.0.1916.153-1   Version table:  *** 35.0.1916.153-1 0         500 http://dl.google.com/linux/chrome/deb/ stable/main i386 Packages         100 /var/lib/dpkg/status   on a computer where google chrome works ok:  $ apt-cache policy google-chrome-stable google-chrome-stable:   Installed: 34.0.1847.132-1   Candidate: 35.0.1916.153-1   Version table:      35.0.1916.153-1 0         500 http://dl.google.com/linux/chrome/deb/ stable/main amd64 Packages  *** 34.0.1847.132-1 0         100 /var/lib/dpkg/status   (If I had the .deb on the 2nd computer, everything would be easier. But there's no .deb. And no old .deb's on the google site.)  So I try:  $ sudo aptitude install google-chrome-stable=34.0.1847.132-1 Unable to find a version "34.0.1847.132-1" for the package "google-chrome-stable" Unable to find a version "34.0.1847.132-1" for the package "google-chrome-stable" No packages will be installed, upgraded, or removed. 0 packages upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B of archives. After unpacking 0 B will be used.   You see, it does not work. The Synaptic manager also shows no version that I could set.  So, how do I downgrade google-chrome-stable?  Probably someone knows an archive of old google-chrome .deb's? Flag apron shows superfluous closing anchor tag I awoke this morning to a spam flag on a site I moderate, and noticed this when I went to read the post.    Notice answered&lt;/a&gt;, which I imagine isn't meant to have the &lt;/a&gt; there.  I also confirmed that this displays asked&lt;/a&gt; for other types of flags (namely, closed without comment).   I looked around for a duplicate of this but can't seem to find it, so maybe it's a recent change? What are the prerequisites to studying general relativity? This question recently appeared :     Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade.   It seems like something that would be a good addition to this site: I think it's specific enough to be answerable but still generally useful. The textbook aspect is covered pretty well by , but beyond that: What college-level subjects in physics and math are prerequisites to studying general relativity in mathematical detail? How should ip2long converted IP's be stored in MySQL? I read that it makes a difference whether you calculate $\frac{dE(\lambda) }{d \lambda}=0$ or $\frac{dE(\omega)}{d \omega}=0$ in the sense that the maximum energy density with respect to the wavelength does not coincide with the frequency maximum that one would assume to be at $\omega_{max} = \frac{c}{\lambda_{max}}$. Actually, I do not understand why this is so ( Now, I am only interested in a pure physical explanation, mathematically this is clear and due to the chain rule). Does anybody know how to explain this odd thing?  Somehow I feel that the core of the question has not been completely answered. Although I can easily look up wikipedia, where the need for the chain rule is explained, I am rather interested in understanding where the following argument breaks down, which seems to be why this question has caused some confusion in the past(as you can see by googling this question):  So we have $E(\omega)$ the energy radiated at a given frequency. Now this function has a maximum somewhere, so there is a frequency where a maximal amount of energy is emitted. In other words: If you add the sum of the photon's energies at each frequency that are emitted, you will notice that the maximum is reached at this frquency. Now $E(\lambda)$ tells you basically the same for the wavelength, but again: We know where at which frequency the maximal amount of energy is radiated, so we know the corresponding wavelength.   I think there is a need to explain this.   If anything is unclear, please let me know. I'm trying to fill empty space in my mesh, and I looked at the knife tool page on the Blender Wiki, and I'm not sure if there's a way to make it so that a new edge is made using the knife tool. I took a screenshot, and hopefully it'll explain better than I can of what I'm trying to do. Oh and I'm using Blender version 2.71, by the way. Isometries of the sphere $\mathbb{S}^{n}$ Got this as homework and I don't know how to tackle this. Help please!  Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to $\mathbb{S}^{n}$ of the linear orthogonal transformations. How to execute several commands after each other with one request to the terminal (without using a file)? "This is considered..." vs. "This is considered as..." Does this "as" change the meaning or correctness of this sentence?     This is considered as socially desired.   This is considered socially desired.   Are both of them grammatically correct? Why or why not? Can we have some tools to handle link rot? There are many links posted in answers, and some in questions, that are either stale or broken.  An example is the change to  on-line documentation this year to do away with separate 'Optional' and 'Core' tasks areas in the documentation - the old links 404 now.  I've fixed a bunch of 'optionaltasks' broken links, but searching for 'coretasks' in Stack&nbsp;Overflow questions returns over 300 matches...  Are there any tools for us to review and fix these - with an emphasis on the fixing - without sifting through completely manually?  Is it time for some refactoring tools for Stack&nbsp;Overflow content, in particular, for links?  Also, I'd be interested to know if there is an analysis of links (as found in the Stack&nbsp;Overflow data dumps) out there. Answer: there is a  of a sample of links and images, but it's getting a bit out-of-date.