From f619a48dd94fc82b9b571f7740174b875d14f8e9 Mon Sep 17 00:00:00 2001
From: 87599ea463bc80856e7cb0d6ce6fa3a1
 <87599ea463bc80856e7cb0d6ce6fa3a1@app-learninglab.inria.fr>
Date: Wed, 25 Oct 2023 22:21:37 +0000
Subject: [PATCH] Update toy_document_en.Rmd

---
 module2/exo1/toy_document_en.Rmd | 43 +++++++++++++++++++++-----------
 1 file changed, 29 insertions(+), 14 deletions(-)

diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd
index 13b258d..c33e052 100644
--- a/module2/exo1/toy_document_en.Rmd
+++ b/module2/exo1/toy_document_en.Rmd
@@ -1,7 +1,7 @@
 ---
-title: "Your title"
+title: "À propos de pi"
 author: "Your name"
-date: "Today's date"
+date: "25 Oct 2023"
 output: html_document
 ---
 
@@ -10,24 +10,39 @@ output: html_document
 knitr::opts_chunk$set(echo = TRUE)
 ```
 
-## Some explanations
+## Asking the maths library
+My computer tells me that $\pi$ is approximatively
 
-This is an R Markdown document that you can easily export to HTML, PDF, and MS Word formats. For more information on R Markdown, see <http://rmarkdown.rstudio.com>.
+```{r}
+pi
+```
 
-When you click on the button **Knit**, the document will be compiled in order to re-execute the R code and to include the results into the final document. As we have shown in the video, R code is inserted as follows:
+## Buffon’s needle
+Applying the method of [Buffon’s needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the approximation
 
-```{r cars}
-summary(cars)
+```{r}
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
 ```
 
-It is also straightforward to include figures. For example:
+## Using a surface fraction argument
 
-```{r pressure, echo=FALSE}
-plot(pressure)
-```
+A method that is easier to understand and does not make use of the $\sin$ function is based on the fact that if $X\sim U(0,1)$  and $Y\sim U(0,1)$, then $P[X^2+Y^2\leq 1]=\pi /4$ (see [“Monte Carlo method” on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:
 
-Note the parameter `echo = FALSE` that indicates that the code will not appear in the final version of the document. We recommend not to use this parameter in the context of this MOOC, because we want your data analyses to be perfectly transparent and reproducible.
+```{r}
+set.seed(42)
+N = 1000
+df = data.frame(X = runif(N), Y = runif(N))
+df$Accept = (df$X**2 + df$Y**2 <=1)
+library(ggplot2)
+ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
+```
 
-Since the results are not stored in Rmd files, you should generate an HTML or PDF version of your exercises and commit them. Otherwise reading and checking your analysis will be difficult for anyone else but you.
+It is therefore straightforward to obtain a (not really good) approximation to $\pi$ by counting how many times, on average, $X^2+Y^2$ is smaller than 1 :
 
-Now it's your turn! You can delete all this information and replace it by your computational document.
+```{r}
+4*mean(df$Accept)
+```
\ No newline at end of file
-- 
2.18.1