I am having a very hard time understanding the , from an intuitive level. I've had a lot of people trying to explain this to me, including my professor and my brother, who is a graduate statistics student. However, I can only understand up to the part that the host's door choice is not random, but cannot wrap my head around the fact that this non-randomness makes the probability to win, if the player switches doors, jumps from 1/3 to 2/3. Note: I am a maths graduate student who doesn't have much experience with everyday life's statistics situations. Learning statistics/probability to me has always been challenging, not because of the maths, but because of the (intuitive) meaning of the equations. This Monty Hall problem is an example of my inability to comprehend a real-life statistics situation. While answering the question, please keep in mind that I hope to relate this Monty Hall problem to something intuitive that I can experience in real life.
From Wikipedia :     Suppose you're on a game show, and   you're given the choice of three   doors: Behind one door is a car;   behind the others, goats. You pick a   door, say No. 1, and the host, who   knows what's behind the doors, opens   another door, say No. 3, which has a   goat. He then says to you, "Do you   want to pick door No. 2?" Is it to   your advantage to switch your choice?   The answer is, of course, yes - but it's incredibly un-inituitive. What misunderstanding do most people have about probability that leads to us scratching our heads -- or better put; what general rule can we take away from this puzzle to better train our intuition in the future?
I would like to look at my referral traffic over time on a per domain-basis in Google Analytics. This is what I have done:   Set New visits as advanced segment  Navigated to traffic-sources -> Referrals Checked a coulple of domains Clicked plot rows   The individual domains now show up in the graph, but the problem is that "all traffic" is also visible. All traffic is much bigger than the graphs for individual domains. This makes it very hard to see any changes as they look all values are Zero.  Is there any way to hide "Visists" from the graph in order to zoom in on the vertical axis? How to turn off "unread messages" feature on Windows XP login screen? In Windows XP, if you have logged onto Windows Live Messenger at some point, then it'll start displaying how many unread messages on the Windows login screen. How can I turn it off? I have a COM dll written in vb6. When I try to create a new object of a class module from this dll I get a Run timer error 430: Class does not support automation or does not support expected interface. The interesting thing is that this happens only from outside the IDE, when I am debugging from within the IDE there is no error thrown and the new object of the class is created successfully. What can be the cause?  In general I occasionally get these kind of errors in COM dlls. What is the best way to debug COM issues? How can I know the path of the dll that is being used when a program is running? How to edit a file in vi editor (arrow keys prints letters)? I am open a file in vi:  vi index.php   Here the problem is: when I press the down arrow and it prints 'B'. Why it prints 'B'? Is abstract algebra (mostly?) restricted to $2$-ary operators? This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic objects consisting of a set together with one or more binary operators defined on that set, perhaps over some other algebraic structure. Not counting categories here, just "low-level"-ish stuff.  I'm an undergraduate so my knowledge is of course very limited but I can't help to wonder why I never stumble upon algebraic structures with $n$-ary operators? Is there a good reason for this, something along the lines of $n$-ary operators behave "badly" when $n&gt;2$ or is it just my ignorance because I'm a beginner in the field? Name for these kinds of "sayings"? Why is $\sqrt{-x}*\sqrt{-x}=-x?$ Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$  Q2 - I was thinking it would be:  $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?)  Q3 - What are the formal algebra rules to use? Can I calculate this without using i such as in: $\sqrt{-x}*\sqrt{-x}=i\sqrt{x}*i\sqrt{x}=-\sqrt{x^2}=-x$. When will data.SE get updated data? promised that data.SE would &quot;soon&quot; get live data:     I realize it's a long weekend, but given that data.SE is now over 2 months out of date, it would be nice if this was finally rolled out. Looking for intro to Conformal Bootstrap Tourist Visa (30 days) Dubai valid until date Today I was informed by VFS that my tourist visa for UAE got approved. I'm travelling to UAE on 16th December 2014 and returning on 7th Jan 2015. But when I checked the "valid until" date on the visa it states 30 December 2014.    Does this date mean I should return before December 30th? Should I get it corrected from VFS? Why do we have more male infants born? Direct product of Sylow subgroups Proposition II.7.5 of Hungerford's Algebra goes     Proposition 7.5. A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.   Let $G = (\mathbb{Z}_6, +)$. $G$ is abelian, so it is nilpotent and thus by the proposition is the direct product of its Sylow subgroups.  The Sylow subgroups of $G$ are $H = (\mathbb{Z}_3, +)$ and $K = (\mathbb{Z}_2, +)$, so the proposition says $G = H \times K$. In particular, $0 \in H$ and $0 \in K$, so $(0,0) \in H \times K = G$... but that's not true, since $G = \{0,1,2,3,4,5\}$.  Where is/are my misunderstanding(s)? How to get the last argument to a /bin/sh function What's a better way to implement print_last_arg?  #!/bin/sh  print_last_arg () {     eval "echo \${$#}"  # this hurts }  print_last_arg foo bar baz # baz   (If this were, say, #!/usr/bin/zsh instead of #!/bin/sh I'd know what to do.  My problem is finding a reasonable way to implement this for #!/bin/sh.)  EDIT: The above is only a silly example.  My goal is not to print the last argument, but rather to have a way to refer to the last argument within a shell function.    EDIT2: I apologize for such an unclearly worded question.  I hope to get it right this time.  If this were /bin/zsh instead of /bin/sh, I could write something like this  #!/bin/zsh  print_last_arg () {     local last_arg=$argv[$#]     echo $last_arg }   The expression $argv[$#] is an example of what I described in my first EDIT as a way to refer to the last argument within a shell function.  Therefore, I really should have written my original example like this:  print_last_arg () {     local last_arg=$(eval "echo \${$#}")   # but this hurts even more     echo $last_arg }   ...to make it clear that what I'm after is a less awful thing to put to the right of the assignment.  Note, however, that in all the examples, the last argument is accessed non-destructively.  IOW, the accessing of the last argument leaves the positional arguments as a whole unaffected. Please add TeX rendering on the Android app MathJax isn't currently working on the Android app, which makes it totally useless for the 'technical' sites like  or .  For example, in the Android app:    Same post in a (mobile) browser:     Please add this functionality! I needed to create a list of lists in Python, so I typed the following: my_list = [[1] * 4] * 3  The list looked like this: [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]    Then I changed one of the innermost values: my_list[0][0] = 5  Now my list looks like this: [[5, 1, 1, 1], [5, 1, 1, 1], [5, 1, 1, 1]]    which is not what I wanted or expected. Can someone please explain what's going on, and how to get around it? Macro with lstlisting inside I have been following  about using \NewDocuentCommand with lstlisting and the accepted answer is nice except it is too big and it doesn't quite work for me.   I came up with this example to test xparse:    \documentclass{article} \usepackage{xparse} \usepackage{listings}  \ExplSyntaxOn \NewDocumentCommand{\code}{m +v}{     \begin{lstlisting}[style=#1,numbers=none,xleftmargin=0.35cm,framesep=0mm]         #2     \end{lstlisting} } \ExplSyntaxOff  \begin{document}  \code{C}{ int x = 200; unsigned int y = 0; }  \end{document}   Like stated in the  I first used m as a normal parameter and then +v as a multiline verbatim parameter, but it fails. It looks like I am still missing something... How to revert uncommitted changes including files and folders? If I have an i.i.d. random variables $X_i$ from a uniform distribution on $[0,1]$.   How would I find scaling sequences $a_n$, $b_n$ such that $a_n(M_n − b_n)$ converges in distribution to a non-trivial limit function $G$ for  $ Y_i = X_i,$ and $M_n = \max(Y_1,...,Y_n) $ Cleaning up factor levels (collapsing multiple levels/labels) Modifying C string constants?