Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible.  But when we look at the probability from a mathematics perspective, probability is defined as the frequency of occurrence over the how many times the experiment is performed, limit as the number of trials goes to infinity.   Doesn't this mean a probability of zero is an occurrence that is arbitrarily small but possible? What are some of the ways to make this line of argument more rigorous?
I read a comment under :     There are plenty of events that can occur that have zero probability.   This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering    if zero probability and impossibility mean the same? if an event with zero probability doesn't mean that the event is impossible to occur, how probability theory represents/describes impossibility?   Thanks and regards!
Can not make hibernate on UEFI (Secure boot) Composition of a harmonic function with a holomorphic function is still harmonic If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ is a conformal mapping of a domain $D_0$ onto $D$, is $f \circ g$ harmonic in $D_0$?   I noticed this question while reading several pdf of lecture notes, and I'm having trouble figuring it out. Can anyone help?  Thank you so much! How can I schedule timer to power on and off device? I want to know where the Metro applications (Mail, People, Messages, ...) executable file path is.  Also, how can I run Mail application when my PC starts? Find a polynomial f(x) of degree 5 such that 2 properties hold. I have been trying to find a polynomial $f(x)$ such that these $2$ properties hold:   $f(x)-1$ is divisible by $(x-1)^3$ $f(x)$ is divisible by $x^3$   To start, I set $f(x) =ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. This is divisible by $x^3$, so $d, e, f $  must be $0$. So polynomial $f(x) = ax^5 + bx^4 + cx^3$  This minus $1$ is divisible by $(x-1)^3$. So I used synthetic division and got that the remainder of $ax^5 + bx^4 + cx^3$ divided by $(x-1)$ is $a+b+c-1$. But it should divide out evenly, so $a+b+c-1 = 0$.  From here, I can't find any other equations involving the three variables. Could someone have any suggestions for how to continue? Thanks in advance. :) Lookup list dependant on the contents of another cell hope someone can help to crack this one?  I've got a table on my data entry tab (sheet 1) where each row includes two data validation drop down lists (source is on sheet2). I want the options in the second drop down list (k13) to be dependant on what is selected in the first drop down list (g13)...  ie. if you select 'Diary' in the first drop down box then the second drop down only has 'Cheese', 'Milk' and 'Yogurt' - likewise, if you select 'Fruit' in the first box - I want the only options to be available specific to that categories column in the lookup table on sheet 2 (which are 'apple', 'pear' and 'banana' in this case)?  Please help - i think this is a vlookup indirect formula?? I used the inbuilt dataset stackloss here. I used following R code for creating the multiple regression model.  &gt; head(stackloss)   Air.Flow Water.Temp Acid.Conc. stack.loss   1        80         27         89         42   2        80         27         88         37   3        75         25         90         37   4        62         24         87         28   5        62         22         87         18   6        62         23         87         18  &gt; fit &lt;- lm(stack.loss ~ .,data=stackloss) &gt; plot(fit)   And when I plot the model, the four following graphs are displayed.      How to interpret these graphs? What are the significance of these graphs? How to use these graphs? What is that component from the ADP5091 Schematic? I am looking for an explanation what that component does inside of that circuit.  It can be found on page 3 of the Datasheet of ADP5091:   It would help me out a lot. Thank you in advance! So I have a Palm Pre (original P100EWW) model that I enabled developer mode on, and installed a Debian Squeeze chroot.  Works great.  I have plans to use this for ANYTHING (bittorrent peer, web server) but a phone.  I noticed if I do a cat /dev/urandom &gt; /dev/fb0 it actually writes random pixels to the screen until a No space left on device error is generated.  Awesome, now I can use the display.  So what kind of utilites are there that will either A) let me use /dev/fb0 as a console I can output text to or B) render text on /dev/fb0 from the command line.  I don't know about recompiling the kernel for this yet (I'd love to eventually strip WebOS off entirely and turn this into a minimal ARM server) so userspace tools if they exist is what I'm asking about.  Also would prefer to render directly to /dev/fb0 and not use X. Why does the following code sometimes causes an Exception with the contents "CLIPBRD_E_CANT_OPEN":  Clipboard.SetText(str);   This usually occurs the first time the Clipboard is used in the application and not after that. What is the difference between Θ(n) and O(n)? Sometimes I see Θ(n) with the strange Θ symbol with something in the middle of it, and sometimes just O(n). Is it just laziness of typing because nobody knows how to type this symbol, or does it mean something different? In Java, what is the best way to determine the size of an object? Meta Stack Exchange should not penalize downvotes How do I get lightning:datatable to do some undocumented thing X? Canonical Question  I want to do &lt;some really cool thing&gt; with a lightning:datatable, but I can't seem to figure out how to do it. I checked the , but I did not see a way to implement &lt;this really cool feature I need&gt;.  Perhaps I want to provide tooltips, or a pick list, or alter the behavior of the menus, or display images, or show a custom component I created, or define a new row-level click behavior, or alter the keyboard event handlers, etc.  Can I somehow do this? How to cleanup deadlinks I was reading some answers about ruby and there are a lot of links to _why's sites.  and took all his sites with him.  Almost every introduction question about ruby will include at least one or two links to one of his sites. Examples:          With a bit more effort I could probably find more questions with a lot of deadlinks.  My suggestion for discussion was, what would be a good way to solve deadlinks in the answers/comments? Not every site _why created has resurfaced somewhere else, but some could be replaced by new links.  But I wanted the discussion to go beyond this specific problem of deadlinks in the ruby answers, an automatic solution would be to let the links be scanned in a batch and when found dead be replaced by a google cache or webarchive version of the page, and perhaps mark it so that the community can replace the link with an alternative? Does Process Builder run with System permissions or User permissions? Regarding field-level security, does Lightning Process Builder run as system-level permissions or user?  That is:   Can I trigger an update to a field that the current user cannot edit? Can I filter based on a field the current user cannot read? How do I add myself back as a sudo user? I clean-installed Ubuntu 11.10 today, and then installed VirtualBox.  This required me to add myself to the vboxusers group, and since 11.10 seems to no longer have a graphical app to add users to a group, I ran the following command:  sudo usermod -G vboxusers stephane   This is a problem.  I now see what I should have run instead is:  sudo usermod -aG vboxusers stephane   The end result is I'm no longer in the groups I should be in.  Including whatever group is required to run "sudo".  When I run any command as sudo now, I get the following:  $ sudo ls [sudo] password for stephane:  stephane is not in the sudoers file.  This incident will be reported.   Is there a way to fix this, or do I need to re-install from scratch again? How can I get the page numbers only of a pattern in a pdf file, regardless if the pattern is multiline? How does gravity escape a black hole? Distribution of $Z_n=n \log\left(\frac{\max\left(X_{(n)},Y_{(n)}\right)}{\min\left(X_{(n)},Y_{(n)}\right)}\right)$ The random variables $X_1,X_2,...X_n,Y_1,Y_2,...Y_n $ are independent and $U(0,a)$-distributed. Determine the distribution of $$Z_n=n\cdot \log\left(\frac{\max\left(X_{(n)},Y_{(n)}\right)}{\min\left(X_{(n)},Y_{(n)}\right)}\right)$$ where $X_{(n)},Y_{(n)}$ denotes $n$th order variable.  This problem is giving me a headache. The answer is that $Z_n$ is $\text{Exp}(1)$-distributed. But i can't derive that conclusion. I'd like to show you my derivation, but its so long and contains so many integrals and fractions that it would take a while to write it here. But basically my approach was to derive the distribution of $M=\max(X_{(n)},Y_{(n)})$ and $L=\min(X_{(n)},Y_{(n)})$. I get the following cdf's: $F_M=(F_X\cdot F_Y)^n=(F_X)^{2n}$ and $F_L=1-(1-(F_X)^n)\cdot (1-(F_Y)^n)=1-(1-(F_X)^n)^2$ ,  correct? $F_X $ stands for the cdf of $X_i$  Then i define the random variable $T=\frac{M}{L}$ and compute its cdf to finally be able to derive the distribution of $Z_n=n\cdot \log(T)$. But thats where i think everything goes wrong. Any help appreciated, thanks.