I went through some books and found this problem. I am unable, and am really curious to see the proof.  There are 3 different colored balls in an urn: Red, Blue and Green.  There are 6 reds, 7 blues and 8 green.  Balls are drawn until one color is completely drawn.  Find with proof the probability that all the blue balls are drawn first.
Suppose that a box contains 10 red balls, 20 green balls, and 30 blue balls.   Suppose also that balls are drawn from the box one at a time at random.  What is the probability that all the red balls are drawn before the blue or green balls are themselves exhausted. What is the probability that as the last red ball is drawn, there remains at least one blue and one green left in the box.  The answer I was given is $\dfrac{7}{12}$ and a general equation is: $$ \dfrac{b g}{1-b}+\dfrac{b g}{1-g} $$ where $$ g=\dfrac{20}{60},b=\dfrac{30}{60} $$ but why?
I'm new to Linux but not to app development. I come from a RealBasic on a Mac background.   I've accepted that I need to learn a new language to develop for Linux (since RB won't run on 64 bit distros and is a cloed proprietary language) but I want to make sure I pick the right one.   I've been looking at Python and was all set to go until I read that PyGTK is becoming obsolete with the introduction of GTK 3. Is this true? The Ubuntu developer website stills seems to recommend it with Glade for the UI but I want to make sure that I don't learn something that's on the way out! Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I have some questions regarding suspended users:   Why do suspended users have a reputation of one? According to the default reputation of a new user, he/she will have reputation of one and he/she has the privilege of posting questions. When comparing this, this is getting violated when compared to suspended users' reputation. Even though they have one reputation point, they cannot post questions. As suspended users do not have privilege of posting questions, there is no use of making "Ask Question" visible. This is an unnecessary server hit. Am I right? As suspended users do not have the privilege of answering questions, there is no use of making "Answer Question" visible. How to burn or mount an ISO file? What is aperture, and how does it affect my photographs? Word that means "outdated name" Is there a word that means "outdated name"? For example:   Record, although very little music is on vinyl Film, although most movies are digital Horsepower, although no one uses horses as a metric anymore Phone, although it's mostly used to access the web   and so on. Problems with Groups in Traminer I'm having a problem using the group function in TraMineR. I have a data set that contains SPELL data, so multiple rows per case. I also have demographic data per case, at one row per case. I merge these together and end up with data that has a demographic covariate per row, so multiple rows per case. An example of this data would be:  id startmin stopmin activity educ4 4      1      20       work    HS 4      20     40       play    HS 8      1      15       sleep   College 8      15     40       work    College   I can make sequence data from this, but when I try to run a plot using the group command        seqiplot(atus.seq, group = atus.seqdata$educ4, border=NA,                 withlegend="right",  cex.plot=.55)   it tells me:       &gt;"Error: group must contain one value for each row in the sequence object".   I have gotten this to work with the example of the mvad data in the training manual, but I can't seem to get it to work with the groups, whether I link to the original demographic data, the merged data, or try to pass the covariates by seqformat and seqdef. Ideas? Why would a spinning space station create a centrifugal force on an astronaut rather than simply spinning around him/her? We often see films with spinning space station that create  by having the astronauts pulled outwards by centrifugal force.  I'd like to know if this would really happen, and if so, why is the following scenario not true:   Take an astronaut in open space. He doesn't move. Put a big open spinning cylinder around him - surely he still doesn't move. Close the cylinder. I still see no reason for him to be pulled outwards. A positional number system for enumerating fixed-size subsets? Many combinatorial objects have some associated positional number system.  For example, the subsets of a (finite) set S can be listed off by observing the bits of all $|S|$-bit numbers, treating the 1s and 0s as indicators for whether to pick a particular element or not.  Permutations of a finite set S can be found by examining the  representations of the first $|S|!$ natural numbers.  Is there an analogous positional number system for encoding k-element subsets of an n-element set?  Thanks! How to effectively close a page? Is it alright to pay real money for useful help? Convert a preprocessor token to a string I'm looking for a way to convert a preprocessor token to a string.  Specifically, I've somewhere got:  #define MAX_LEN 16   and I want to use it to prevent buffer overrun:  char val[MAX_LEN+1]; // room for \0 sscanf(buf, "%"MAX_LEN"s", val);   I'm open to other ways to accomplish the same thing, but standard library only. I am aware    but I ask this regarding SE-2.0 sites, as the example I will take gaming:  Both answers on the question  make perfect sense to me, i.e. on the one hand this might overuse the tagging system but on the other hand it would help a lot for filtering.  Therefore I suggest a system which, in this case, would internally store the platforms to which a tag is relevant and   if a question is tagged with one of these platforms, nothing special will happen and the question is to be considered platform specific (e.g. a bug which only appears on the PS3 version) if the question is not tagged with a platform, the only result will be that the tag filter mechanism works a bit different compared to what would happen if the question was (invisibly) tagged with all platform tags:   If the invisible tagging is definite on interesting or ignore (i.e. there is not at least one ignored and one interesting tag) the usual mechanism works. if the tagging is ambiguous (i.e. contains both interesting and ignored tags), no filtering happens at all    The actually attributed tags must of course override this behaviour. That way questions about a game one likes but which is available for a platform one doesn't care about will still be highlighted unless specifically tagged with that platform, but no additional tags have to be used to filter a game which is only available on ignored platforms.  This can of course easily be extended to other sites, e.g. the  may be interesting both general and for c++ but totally uninteresting if the question is only about a JavaScript implementation.  I am only unsure whether search should include those hidden tags, too. Clarification for non-answers One of my flags for "non-answer" was declined and I was wondering if I misunderstood what "non-answer" means. I saw someone post an answer that was just a block of code, with nothing in it that was even the slightest related to the question, it was just a random block of code with no explanation. Should I have not used the "not an answer" tag even though the description explicitly says that it should be used when answers "do not attempt to answer the question"? Relation between AB and BA Let $A$ and B be $n\times n$ matrices over a field $F$. Then prove or disprove  $(i)\, AB$ and $BA$ have same characteristic values.  $(ii)\, AB$ and $BA $ have same characteristic polynomial.  $(iii)\, AB$ and $BA $ have same minimal polynomial.  Proof of $(i)$   Let $c$ be an eigen value of AB and v be corresponding eigen vector. Then    AB(v)=cv,    B(ABv)=B(cv),    BA(Bv)=c(Bv),   this implies $Bv$ is an eigenvector of $BA$ corr. to eigen value $c$.  But this statement holds if $Bv$ is nonzero.  What if $Bv=0$?  $(ii)$ If $A$ is invertible then ($A$ inverse)$(AB)A= BA$, i.e. $AB $ and $BA$ are similar and hence have same charateristic polynomial.   Similar proof when $B$ is invertible.  But what if both $A$ and $B$ are not invertible?  I have no idea about $(iii)$.  Please help me to complete these proofs. I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ is a positive term as the equation above.  Thanks I'd like to download automatically tons of papers listed on .bib file so I'm trying to find a way to facilitate the process by, e.g. I import or input to a tool a certain .bib file which contains a list of literature, the output is the literature's .pdf file. Any idea to do so? Currently the migration path options from SO are:   Meta.SO Serverfault SuperUser Webmasters Programmers   If we are going to be limited to 5 options, shouldn't we periodically re-evaluate which sites are most appropriate?   (10k only) indicate that the top sites for the last 90 days are:   Superuser - 564 - 5% rejected ServerFault - 278 - 10% rejected Tex - 157 - 13% rejected DBA - 144 - 2% rejected Sharepoint - 127 - 1% rejected Programmers - 126 - 26% rejected CodeReview - 100 - 3% rejected Meta.SO - 81 - 18% rejected Wordpress - 75 - 14% rejected Webmasters - 58 - 8% rejected   All these migrations for sites out of the list required a flag and a moderator to process it.  This is just extra work for the mod team!  If we had the top 5 destinations by volume as the 5 options, we would have had a net savings of 160 flags to be processed.  I suspect we will actually see these numbers change quite a bit if it's easier for users to put questions in the appropriate spot without moderator intervention.  Can we have a review of the migration path list based on migration volume?  Think of the flags! How do I edit a file in /proc that can be viewed with cat, but is not editable by mousepad and abiword? Convert line voltage heating system to use low voltage thermostats I currently have two line voltage ceiling radiant heating systems in my condo each with their own DPST thermostat. I'd like to use low voltage thermostats such as the nest. I found , but it only has 3 wires. Will this work with a system designed to use DPST thermostats? If not, is there something else out there that will allow me to use low voltage thermostats with line voltage heaters?