EDITED  John is playing a game on $n$ days, each day being independent.  On each day $i$, his probability of success is $p_i$.   We have $\frac{1}{n} \sum_{i=1}^n p_i = p$, and typically, the standard deviation $\sigma$ of these $p_i$ is small. $\sigma$ is known.  So I have a succession of $Bernoulli(p_i)$. I want to model the probability of having k success after n trials, using $p$ and $\sigma$ for instance.  I can approximate this with a Binomial distribution, but even though it's close it still leads to biases. Any thoughts on how to improve this?
If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful?  Is there a better way of calculating these probabilities rather than simply summing together the combinations of success and failure probabilities?
How can I change the color of the lines (the frame) in a tabular environment? In the following MWE:  \begin{tabular}{c|c|c} a&amp;b&amp;c \\ \hline d&amp;e&amp;f \\ \hline g&amp;h&amp;i \end{tabular}   I would like the horizontal and vertical lines of the table to have, say a blue color. How to find solutions of linear Diophantine ax + by = c? I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan What does the notation like 8.6e-28 mean? What is the 'e' for? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 "He didn't know where New Jersey was" Should we put the references in slides/poster when we already have them in paper? After reading this question about  and asking myself two questions about  and , I realize that these activities are for showing what you found, make the audience be attracted. If they need further explanation, they should read your paper, which has successfully defensed and proved that your work is validated. One important thing when you attempt to attract people is concentrate on the important, which means unnecessary things need to be trimmed. I think the reference is one of those. So, should we put the references in slides/poster when we already have them in paper? Primes dividing $11, 111, 1111, ...$ Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct? Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?  The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.  This is the mentioned derivation:     $\sin \theta = \frac{\lambda}{\Delta y}$      assuming $\theta$ is small:      $\tan \theta = \frac{\lambda}{\Delta y}$      de Broglie equation:      $\lambda = \frac{h}{p_x}$      $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$      but also:      $\tan \theta = \frac{\Delta p_y}{p_x}$      equalize $\tan \theta$:      $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$      $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$      $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$      $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$   So: Which one is correct and why? Custom arrangement for fields with multiple values in Drupal 7 How to remove a filter that is an anonymous object? In my functions.php file I would like to remove the below filter, but I'm not sure how to do it since it's in a class. What should remove_filter() look like?  add_filter('comments_array',array( &amp;$this, 'FbComments' ));   It's on line 88 . Do's and Don't's of Undergraduate Research I was fortunate enough to get a position as a researcher for the Mayo Clinic's SURF Program this year. My PI's lab focus is on the the immune system's role in CNS axonal and neuronal injury, specifically through the lens of how innate and adaptive immune effectors interact w/ infected neurons.  Although I do research under a professor at my college and I volunteered for a state university lab during the previous summer, this is my first REU/SURF opportunity, and I REALLY want to make a good impression. Here are my questions:  What are the do's and don't's in terms of being a skilled and efficient researcher?  Since I am still an undergrad, I know that I will be a less useful asset to the lab than a grad student or post-doc, but what can I do as an undergrad to not burden my colleagues and PI?  Thank you all for your help! Wishing you all the very best! Short story about interstellar travel gone wrong Trying to find a story I read a couple of years ago in an anthology, not sure when the story dates from but likely to be recent - the last decade or so.   The story has passengers travelling between stars via a series of jumps. The calibration of the drive is affected by a special metal, any of which has to be stored in a shielded compartment (I think the jumps were measured in 'degrees'). One of the passengers unwittingly takes some of this metal on board, which throws off the calculations for the jumps, and they jump into an alternative universe where there are no signs of any humans ever existing. They start to try to reverse their steps to get back to the known universe, but the reverse calculation seems impossible.  That's as much as I read of the story before I lost the book. After many dreams (feverish nightmares!) re-living this story, I need to know how it ends. I know it's not much to go on. The story name, author, or the anthology it appears in would be fantastic. Pascal's triangle in tikz I would like to typeset the top part of Pascal's triangle. To get the triangle with the names of the binomial coefficients, i.e., {n \choose k}, I used the following code  \begin{tikzpicture} \foreach \n in {0,...,4} {   \foreach \k in {0,...,\n} {     \node at (\k-\n/2,-\n) {${\n \choose \k}$};   } } \end{tikzpicture}   The result is this   Now I want to be equally lazy and do something like this for the values of the binomial coefficients, i.e., replace {\n \choose \k} in the node label with \CalculateBinomialCoefficient{\n}{\k} where \CalculateBinomialCoefficient is a hypothetical macro that calculates the binomial coefficient. Has anyone done something like that?  The result should look like this: What is the best way to get security token of a API Only user? API only users cannot login and choose 'Reset my security token'. Right now, in my company, we select a System Administrator profile for users, log in for the first time, choose 'Reset my security token', then change the profile to a custom profile which has API only enabled.  The other way of doing it is, modifying the profile to not have API only user checked initially and then turn it on once the security token is received.  Is there a recommend way to do this? Is there an option to force the security token to be sent  when the user is created with a profile which has API user enabled? Why do C# multidimensional arrays not implement IEnumerable? indefinite article before proper names Kafka, discovered in the 1920s by the Expressionists, damned by the Nazis in the 1930s and by the Marxists at more or less the same time, was reinvented after World War II by the French Existentialists. They saw in him the father of Angst and the Angst for the father: an Albert Camus before his time.   Can you tell me what the purpose of using the indefinite article before the proper name is in the sentence above. What it be a mistake to write just "Albert Camus" without the article? Solve the equation $x^2+\frac{9x^2}{(x+3)^2}=27$ Truncate (not round) decimal places in SQL Server Third grade spiral in TikZ - dimension too large I'm trying to draw a smooth third grade spiral (r^3 = a^3\cdot \varphi) over $60\times 60$mm grid and I'm getting the "Dimension too large"-error if I set $a &gt; 320$ but I need to fill the grid area completely:  \begin{tikzpicture}[x=1mm, y=1mm]      \draw[step=1, line width=0.1mm, opacity=0.3] (-30, -30) grid (30, 30);      \draw [domain=0:50, variable=\t, samples=1000]          plot ({\t r}:{pow(320*\t, 1/3)}); \end{tikzpicture}     Is there any way to avoid this error? Entity Query field condition for Webform submissions I'm trying to query webform submissions based on a field value. The field is called "ticket_for". Somethings I tried and do not work:      -&gt;condition('ticket_for',$adnummer)     -&gt;condition('webform_submission_data:ticket_for',$adnummer,'CONTAINS') ...   What kind of condition should I be using in a query like this?  $query = \Drupal::entityQuery('webform_submission') -&gt;condition('webform_id', 'ticket') -&gt;condition('entity_id', $eventID) -&gt;condition('webform_submission_data:ticket_for',$adnummer,'CONTAINS'); Alt-C not working in Blender 2.8 I want to convert my text object to mesh. In 2.79 it was Alt-C, are the keys changed for 2.80?    I browsed it externally and I got some thing like this image as answer..   but any shortcut-keys will be easy to work! If you know one, you can comment below.