Suppose you are a prisoner. The king gives you $100$ marbles, $50$ black and $50$ white. You have to put all the marbles into $2$ urns such that none of the urns is empty. The king will then choose an urn at random and from that a marble at random. If the marble is white, you will be set free, otherwise you will be sentenced to death. So you have to distribute the marbles in such a way so as to maximize the chance of survival. Basically you have to maximize the probability that the king chooses white marble. My attempt : Let there be $b$ black and $w$ white marbles in the 1st urn . Therefore, there will be $50-b$ black and $50-w$ white marbles in the 2nd urn. Therefore P(Survival) = P(Choosing white marble) = $\frac{1}{2}\cdot\frac{w}{w+b} +\frac{1}{2}\cdot\frac{50-w}{100-w-b}$ . Now I need to maximize the above equation but I don't know how. Any ideas?
I've tried searching for this question but couldn't find it on stackexchange. This is a common type of interview question; I ran into it doing brain teasers on a probability puzzles app, and if you fine people agree with my logic, I will inform the app developer that his/her answers are incorrect.   The problem is essentially this:  You are sentenced to death for thievery. The King is magnanimous and decides to put your fate in the hands of chance. You are given $100$ white marbles and 100 black marbles, and $2$ urns. The king will choose an urn at random and pull out a single marble at random; if the marble is white, you live, if its black, you die. If you place the marbles in the best way possible, what is your probability of survival?  I started with the base case: $100$ white marbles in one urn, $100$ black marbles in the other. This comes down to a $50$-$50$ chance of survival. I then worked my way to deciding that placing $1$ white marble in one urn and $99$ white marbles + $100$ black marbles in the other urn would be the "best way possible", which yields the following:  $$P(\text{Survival}) = \frac{1}{2}(1+\frac{99}{199}) \approx .749$$  Selecting $1$ of $2$ urns at random gives $\frac{1}{2}$, the urn containing $1$ marble gives $1$, and the other that contains $99$ white marbles and $100$ black marbles gives $\frac{99}{199}$ because there are $99$ possible white marbles to select out of $199$ total marbles.  The app claims that the correct answer is $\frac{1}{2}(1+\frac{99}{200}) \approx .748$   I see where the $200$ comes from, but I do not think it is right to say that there are $200$ marbles in the other urn. Who is correct?
How many solutions are there to $x_1 + x_2 + ... + x_5 = 21$? How many solutions are there to the equation      $x_1 + x_2 + x_3 + x_4 + x_5 = 21$    where $x_1, i = 1,2,3,4,5$ is a nonnegative integer such that $0 ≤ x_1 ≤ 3, 1 ≤ x_2 &lt; 4$, and $x_3 ≥ 15$?  I have correctly completed the previous parts of the question but am having trouble with this particular restriction.  The total number of solutions to the equation is $C(25,21) = 12650$.  I have tried to do each restriction individually:  For $0 ≤ x_1 ≤ 3$, I solved this by subtracting the number of solutions when $x_1 ≥ 4$ from the total number of solutions. This is $C(25,21) - C(21,17) = 5985$.  For $1 ≤ x_2 &lt; 4$, I have $C(24,20) - C(21,17) = 5985$.  For $x_3 ≥ 15$, I have $C(10,6) = 210$.  However, the restriction is when these are all together. I am not sure how to proceed. The answer is $106$. How to get object size in memory? 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I want to match a portion of a string using a  and then access that parenthesized substring:  var myString = "something format_abc"; // I want "abc"  var arr = /(?:^|\s)format_(.*?)(?:\s|$)/.exec(myString);  console.log(arr);     // Prints: [" format_abc", "abc"] .. so far so good. console.log(arr[1]);  // Prints: undefined  (???) console.log(arr[0]);  // Prints: format_undefined (!!!)   What am I doing wrong?    I've discovered that there was nothing wrong with the regular expression code above: the actual string which I was testing against was this:  "date format_%A"   Reporting that "%A" is undefined seems a very strange behaviour, but it is not directly related to this question, so I've opened a new one, .    The issue was that console.log takes its parameters like a printf statement, and since the string I was logging ("%A") had a special value, it was trying to find the value of the next parameter. How to catch creation of DOM elements and manipulate them with jQuery implementing a shell in C Reading data from a relational database into a LaTeX table I'm using Tex Live 2009 on Debian squeeze. I can upgrade if I really, really  have to, but I'd rather not. I was wondering if it would be possible to do the following.  Given some data in a (relational) database, can I extract it and automatically format it in a LaTeX table, so one would not need to manually recreate the LaTeX table if the data changes? Could one do such a thing with Lua? One would normally use a suitable language database adaptor for this, and I see Lua has . I'd like to use PostgreSQL. I'm currently using version 8.4.  Solutions not involving Lua are also fine. Just to be clear, I don't know anything about Lua and my knowledge of LaTeX programming is also extremely skimpy. I'm looking for a recipe which I can adapt if necessary. While this is not really important for my current use case, it seems like a useful general technique if possible. I am confused about the usage of the words like and unlike in sentences. Like and unlike make me confused because I see them being used everywhere almost interchangeably, and to make matters worse I even saw commas making difference in meaning.  Which of the following sentences is most correct, and least ambiguous?      Unlike Xbox, Playstation can't play Halo.     Playstation can't play Halo unlike Xbox can.     Playstation can't play Halo, unlike Xbox can.     Like Xbox, Playstation can't play Halo.     Playstation can't play Halo like Xbox can.     Playstation can't play Halo, like Xbox can.     Playstation can't play Halo unlike Xbox.     Playstation can't play Halo, unlike Xbox.     Playstation can't play Halo like Xbox.     Playstation can't play Halo, like Xbox.    In case you don't know what I actually wanted to say in those sentences: I'm trying to say that Playstation can't run the Halo video game and Xbox can run that game. I probably should have tried sentences with no brands but I can't imagine any meaningful sentences at this moment which contains like/unlike. How to ask "If you have any problem then ask me" in polite manner Can a hyphen be used without anything on the right side? Solving an Op-amp circuit The image of the circuit is shown below and it required to find \$V_0\$,  My first attempt at solving this problem is by changing the current source into a voltage source with 1-V and 2k\$\Omega\$ resistance. The fact that the inverting and non-inverting terminals aren't grounded make this problem look difficult.To the point which, I don't know how to proceed with this question or where to start. I would appreciate any help. Following some thought and another schematic from a hint suggested by Alfred, I produced a schematic representing our work.  And  my solution for the problem is below, Using node equation at nodes A and B we have, $$\frac{V_A-V_B}{1k}=-0.5 \text{mA}$$ $$\frac{V_B-(2+V_A)}{1k}=-0.5-x$$ where x is the current that is sent in the output of the op-amp. Using KCL, at the bottom node near the current source we see that the same current that goes through the op-amp also goes through the \$2\text{k}\Omega\$ resistor. Hence, we have, $$\frac{-V_A}{2k}=x$$ Replacing this in the second equation, $$\frac{V_B-(2+V_A)}{1k}=-0.5+\frac{V_A}{2k}$$ And and solving the equations yields \$V_A=-2 \text{-V}\$ and \$V_B=-1.5 \text{-V}\$ Is there a safe way to run a diff on two zip compressed files? Seems this would not be a deterministic thing, or is there a way to do this reliably? getting error when I compile the Java code using package in commandline? 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The New York Times article specifically mentions a 2006 NIST standard, where two Microsoft engineers later discovered a serious problem in the scheme.  The New York Times article explains how the standard was written by the NSA, who pushed the scheme hard:     Simultaneously, the N.S.A. has been deliberately weakening the international encryption standards adopted by developers. One goal in the agency’s 2013 budget request was to “influence policies, standards and specifications for commercial public key technologies,” the most common encryption method.      Cryptographers have long suspected that the agency planted vulnerabilities in a standard adopted in 2006 by the National Institute of Standards and Technology, the United States’ encryption standards body, and later by the International Organization for Standardization, which has 163 countries as members.      Classified N.S.A. memos appear to confirm that the fatal weakness, discovered by two Microsoft cryptographers in 2007, was engineered by the agency. The N.S.A. wrote the standard and aggressively pushed it on the international group, privately calling the effort “a challenge in finesse.”    After some sleuthing, I'm pretty certain this is a reference to the  pseudorandom number generator scheme described in NIST SP 800-90.  The weakness is that Dual_EC_DRBG appears to contain a backdoor, and anyone who knows the backdoor can totally break the PRNG.  The weakness was first described in  and was subsequently .  As a consequence, there are now good reasons to suspect that Dual_EC_DRBG contains a NSA backdoor, and the NSA might be able to spy on anyone who is using Dual_EC_DRBG.  My questions:   Who uses Dual_EC_DRBG? Do any commercial products use Dual_EC_DRBG? Does anyone use it?