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probability that there is at least one defective part I study economics and I am trying to solve this problem - without success, unfortunately.     A machine has a defect rate of 6%.       a) Chosen at random 4 pieces (with replacement) from the production flow, compute the probability that none is defective.   b) In the event that extractions are performed 60, compute the probability that there is at least one defective part.   I have no idea how to answer the first question. What logical processes can I use?  Regarding the second question, I thought of using the binomial random variable with $n = 60$ and $p = 0.06$. Then:  $$\begin{multline} P (X \geq 1) = 1 - P (X &lt; 1) = 1 - P (X = 0) = \\ = 1 - \binom{60}{0}  \cdot (0.06)^ 0 \cdot (0.94)^{60} = 1 - 0.02446 = 0.97558.\end{multline}$$  Can somebody please explain me how to solve this problem? Thank you very much. $X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)?  I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to $\frac{1}{n+1}$. Can anyone give a rigorous proof or some hints? Thanks! How to show that $f'(x)<2f(x)$ This is question B4 in 1999 Putnam Exam. I would appreciate if somebody could help me with this.      Let $f(x),f'(x),f''(x),f'''(x)&gt;0$ , $f'''(x)$ is a continuous function and $f'''(x)&lt;f(x)$ on $\mathbb{R}$. Then show that   $$f'(x)&lt;2f(x),~ \forall x\in \mathbb{R}.$$ Is  structured data (in Microdata, JSON-LD, or RDFa) relevant for SEO?   I know that structured data can provide benefits in other areas, but here I’m only asking about its significance for general web search engines. Jordan form, number of blocks. How to convert standard IP address format string to hex and long? Stop setInterval call in JavaScript Ratio of Boys and Girls In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same) In chapter 2, the section on bitwise operators (section 2.9), I'm having trouble understanding how one of the sample methods works.  Here's the method provided:  unsigned int getbits(unsigned int x, int p, int n) {     return (x &gt;&gt; (p + 1 - n)) &amp; ~(~0 &lt;&lt; n); }   The idea is that, for the given number x, it will return the n bits starting at position p, counting from the right (with the farthest right bit being position 0). Given the following main() method:  int main(void) {     int x = 0xF994, p = 4, n = 3;     int z = getbits(x, p, n);     printf("getbits(%u (%x), %d, %d) = %u (%X)\n", x, x, p, n, z, z);      return 0; }   The output is:     getbits(63892 (f994), 4, 3) = 5 (5)   I get portions of this, but am having trouble with the "big picture," mostly because of the bits (no pun intended) that I don't understand.  The part I'm specifically having issues with is the complements piece: ~(~0 &lt;&lt; n). I think I get the first part, dealing with x; it's this part (and then the mask) that I'm struggling with -- and how it all comes together to actually retrieve those bits. (Which I've verified it is doing, both with code and checking my results using calc.exe -- thank God it has a binary view!)  Any help? When a spaceship is experiencing a constant acceleration of $10m/s^2$, the astronauts will be moving at nearly the speed of light after about a year in the earth's reference frame. This means the spaceship's energy will start to diverge as a function of the speed $v$ so there will be a huge amount of energy necessary to increase the speed of the ship any further. This way, the speed of light can never be crossed.  All of this is clear to me, but all of this is also formulated in earth's reference frame. But from the astronaut's reference frame: the spaceship is simply accelerating at $10m/s^2$ and so the force on the spaceship is constant. Then why would we need huge amounts of energy to accelerate the spaceship?  For example, I read somewhere that the amount of energy that would be needed to accelerate a large spaceship to half the speed of light is more than 2000 times the current world annual energy consumption. How does this make sense in the astronaut's (non-inertial) frame? I know that for any nonzero $x,y\in\mathbf{R}$,  $$ax^2+by^2+cxy &gt; 0,$$  where $a,b,c\in\mathbf{R}$. What can I deduce about $a$, $b$, and $c$?    For example, letting $x=1$ and $y=0$, I know that $\boxed{a&gt;0}$. Letting $x=0$ and $y=1$, I know that $\boxed{b&gt;0}$. Letting $x=y=1$, I know that $\boxed{c&gt;-(a+b)}$.  What else can I deduce about $a$, $b$, and $c$? How will I know when it's time to stop looking? (Is that last question even answerable?) YA dystopia novel of battle school with unusual military control scheme Some sort of battle school (no, not Ender's Game), where teens are being trained as military commanders.   The control system for their army hinges around some sort of system that requires extremely fine motor control -- being able to "push" around digital representations of military units using a special interface that interprets minute finger movements.  The boy protagonist eventually meets a girl in the battle school, and is astonished to learn that most people do not live as well as he does -- for instance, he gets a fresh jumpsuit outfit every day and disposes of the old one; this is a luxury.  One scene I recall is them kissing for a moment in some sort of underground passageway; the boy protagonist notes the exact length of the kiss, as he has been trained to have heightened time perception.  I don't recall much about other commander characters; I think they have different special abilities, and one other major character might have physical deformities, possibly because of postwar-radiation-related birth defects.  They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. Mid 2013 MacBook Pro displaying multi color My MacBook Pro was displaying normally until recently. Now when I boot it up it  displays different colors, sometimes white on the screen, but when I hook it up to a TV screen it displays normally.  I need help as to what could be causing this problem and how to fix it?  So far I've tried resetting NVRAM and reseating the RAM, but this didn't help. I also haven't upgraded any software nor added any new items of hardware, so this change in behaviour cannot be attributed to any known actions on my behalf. Is starting a group with 5 newbies (DM included) possible? Some friends and I want to start playing D&amp;D (5e), but none of us, DM included (not me) have played before to any great extent. The DM and I have some cursory experience with 4e, but have never been able to play to any great extent. Is this advisable, or should we play with existing groups first? We live in a relatively remote area, making it hard to find experienced players. What is the difference between dpkg and aptitude/apt-get? When installing a software via dpkg -i packageA.deb, will the dependencies required by packageA be downloaded and installed automatically? How is this different from using apt-get or aptitude? Can I transfer trophies and data to a new account? Okay I have created a sub account on my ps3 as a new user. I wanted to change my psn online id. So I have copied all game data from the old account to my new account, but when I go to play the game it still says the trophies and data belong to the old account. I don't want to delete the old one and lose all my saved games and trophies, is there a way to either unlock and allow my new user to own the data and trophies or transfer the trophies and data so that my new account may own it? There has got to be the establishment of a universal auxiliary language.   Does the verb got add more to the context or is it there just for emphasis? Is it even grammatical?  Using Google-Ngrams I found a few similar versions as follow: . For example;   For many from the North, however, it has seemed difficult and there has seemed to be a problem. There has come to be a convolution between this become enmeshed concept of marriage as it exists in the sacred and how it exists in the secular. Most of the time since I have been here, there has appeared to be considerable feeling in the minds of a few. There has had to be some negotiation, obviously, but I think we provided what our soldiers need and what our country needs.   In general, I wanna know if 'have/has'+past-participle+'to be' is grammatical. Lie Algebra of U(N) and SO(N) U(N) and SO(N) are quite important groups in physics. I thought I would find this with an easy google search. Apparently NOT! What is the Lie algebra and Lie bracket of the two groups? I am stuck while calculating the peak current in the boost converter through the inductor.  Can you please check if I am right?  Ipeak=(Max Voltage/ R2) = 18/1000=0.018. So, my peak current through the inductor is just 18mA? Am I doing it correct. If this is correct, how will I account for the Vce drop inside the pin1 and pin2 of the IC. The darlington transistor inside the IC is just like a normal transistor. I have added two transistor outside the IC instead of using the darlington switch in the IC for thermal purposes.   Can you please help me with the calculation of Vswce and Ipeak. The calculation of Vswce is not mentioned in the datasheet. So, please help. And the calculation of Peak current is mentioned in the datasheet. But is my approach of calculating the peak current through the inductor right? Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer. Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer.   I'm not familiar to factorial and I don't have much idea, can someone show me how to prove this? Thank you.