Understanding Conditional Probability Basics In many online sources I've read a statement similar to:     Probability of B happening given A is equal to the probability of A and B both happening divided by B happening   or  $p(A | B) = p(A \cap B) / p(B)$  But what is the difference between "B happening given A" and "A and B happening at the same time" ? If B happens given A, didn't A and B happen at the same time ? Why is that chance smaller than the intersection ?
Intuition behind the Definition of Conditional Probability (for 2 Events) What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a definition, but no true development of that definition. Hopefully that's not too much to ask.
How to make grass appear on top of ground not in middle? When I take the pth root, the bottom of the p comes very close to the \sqrt symbol.  In the New Century Schoolbook font (which I am using), the p actually intersects the \sqrt symbol, which looks even more terrible.  Is there a clean way to fix this?  \documentclass{article} \usepackage{fouriernc} % use the New Century Schoolbook font \begin{document} When I write $a^{(1/p)}$ as $\sqrt[p]{a}$, the bottom of the $p$ touches the top of the root symbol, which looks ugly. Even when the root symbol is larger, such as with $\sqrt[p]{\frac{1}{n}}$, it still looks bad (since the tail of the p almost hits that line). \end{document} Why would one suppress the intercept in linear regression? In a number of statistical packages including SAS, SPSS and maybe more, there is an option to "suppress the intercept". Why would you want to do that? Class of sets of a given infinite cardinality This question was inspired by the question on .  From the discussion in the comments there, it seems there does not exist a set of all sets of a given cardinality.  To me, it seems easy to see that this is true for any nonzero finite cardinality $n$. Given any set $x$, you can always make a set of size $n$ by adding $n-1$ other sets to $x$ to make a set of size $n$, which exists by repeated use of the pairing axiom.   But how would you do this for infinite cardinalities? For suppose $\kappa$ is some given infinite cardinality. To show that the set of all sets of cardinality $\kappa$ does not exist, it seems you would have to show that for any set $x$, there exists a set of that size with $x$ as an element, and then you could take the union of that set to find the set of all sets, and thus a contradiction. But it doesn't seem reasonable to simply say, for a given set of size $\kappa$ if $x$ is in the set, we have no problem. If not, just take an element out of the set and put $x$ in. Something about that seems like it would not be allowed.  So how would you do this for infinite cardinalities? Why would spacetime curvature cause gravity? How to map stock ticker symbols to ISIN (International Securities Identification Number)? How can I can convert ticker symbols (which include information on the exchange) to an International Securities Identification Number (ISIN) (which contains no information on the exchange)? Also I am interested in the other way round — given an ISIN and a market identification code (MIC) how can I determine the ticker symbol? Any databases around? $(document) vs. $("document") How do I tell if ACR will work with my camera and how do I get the correct version for Photoshop? Proving a direct product of groups is a group I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.    Let $n \geq 1$ be an integer. Suppose $\{G_1,...,G_n\}$ is a finite set of groups. Define the set $G = G_1 \times G_2 \times ... \times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i \in G_i$. Show that the operation $\bullet$: $G \times G \rightarrow G$ given by $(g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.    Proof of associativity: $((g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...g'_n)) \bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) \bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) \bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) \bullet ((g'_1,g'_2,...,g'_n) \bullet (g''_1,g''_2,...,g''_n))$  \  Proof of the existence of an identity For each $G_i \in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) \bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.  \  Proof of the existence of an inverse: Since each $G_i \in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) \bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$. Why does this CSS margin-top style not work? Dealing with questions of nefarious intent 'Which', 'whose' or something else? Android and setting width and height programmatically in dp units Multiple returns from a function mkdir error: no such file or directory on a linux machine running CENTOS v 7.8.2003 I am in a directory Neutrinos I now do  mkdir /pionloop/ this works. I then go to into this new directory pionloop and do mkdir /E_0.3 and this works as well. I thus have as a result a directory Neutrinos/pionloop/E_0.3.  Now, starting in Neutrinos I want to do this in one command and therefore do mkdir /pionloop/E_0.3 and get: `mkdir: cannot create directory ‘/pionloop/E_3.0’: No such file or directory  What is going wrong here?? In some papers or documents, "linear times series" seems to refer to a time series that can be modeled as a linear AR (auto-regressive) model.   In other places, "linear time series" seems to refer to a time series with a linear trend.   When someone just drops "linear time series" out of the blue without any context, what do they mean usually? Are human bodies programmed to die? Linear regression best polynomial (or better approach to use)? Any ideas on other polynomials I could successfully use for applying regression? My goal is a solution that has fit error strictly based on the noise. Is this possible since it is a bell-like curve? The tails can be long, short or non-existent. Am I looking for the impossible? Please note it has been a while since I have plunged this far into linear regression.   Anyway, my data points do not fit the $ax+bx^2+c$ polynomial well enough. I would like a replacement for this polynomial to apply regression (or different approach). In the following example, each point misses the fit curve by $20\%$ of the $Y$ range on average (of the data points). I looked at splines, but I do not know how to apply the regression since they look piece-wise.   Example $Y$ values (evenly spaced in $X$). It looks kind of like a bell curve. I would be applying 5 to 50 data points to determine the polynomial coefficients. Ultimately, I am interested in the best $X$ location for the peak based on the data points.     X      Y -40    -21142.1111111111 -30    -21330.1111111111 -20    -12036.1111111111 -10      7255.3888888889   0     32474.8888888889  10     32474.8888888889  20      9060.8888888889  30    -11628.1111111111  40    -15129.6111111111 Prove that for any positive integer $n$,       $$n^{1/n}  &lt; 1 + \sqrt{\frac{2}{n}}.$$   This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29.   Hint:      Use the binomial theorem. How to change the language for the front-end only?