From e81bda087bb9ef4ec7ab71c0399ba7370b464e0b Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Wojciech=20=C5=81oboda?= <wojciech.loboda@swmansion.com>
Date: Wed, 15 Oct 2025 21:33:50 +0200
Subject: [PATCH] ex1

---
 module2/exo1/toy_document_en.Rmd | 50 +++++++++++++++++++++-----------
 1 file changed, 33 insertions(+), 17 deletions(-)

diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd
index 13b258d..4132751 100644
--- a/module2/exo1/toy_document_en.Rmd
+++ b/module2/exo1/toy_document_en.Rmd
@@ -1,33 +1,49 @@
 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
+title: "On the computation of pi"
+author: "Arnaud Legrand"
+date: "25 juin 2018"
 output: html_document
 ---
 
-
 ```{r setup, include=FALSE}
-knitr::opts_chunk$set(echo = TRUE)
+knitr::opts_chunk$set(echo = TRUE)	
 ```
 
-## Some explanations
-
-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>.
+## Asking the maths library
 
-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:
+My computer tells me that $\pi$ is *approximatively*
 
-```{r cars}
-summary(cars)
+```{r}
+pi
 ```
 
-It is also straightforward to include figures. For example:
+## Buffon’s needle
 
-```{r pressure, echo=FALSE}
-plot(pressure)
+Applying the method of [Buffon’s needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the **approximation**
+
+```{r}
+set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
 ```
 
-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.
+## Using a surface fraction argument
 
-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.
+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:
 
-Now it's your turn! You can delete all this information and replace it by your computational document.
+```{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()
+```
+
+It is then 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:
+
+```{r}
+4*mean(df$Accept)
+```
-- 
2.18.1