diff --git a/module2/exo1/toy_document_en.Rmd b/module2/exo1/toy_document_en.Rmd
index 13b258ddd0da29bc3bf08c64b6a1db742f6d5409..791440a2cbe554807dcb5164df65b5d5616fab26 100644
--- a/module2/exo1/toy_document_en.Rmd
+++ b/module2/exo1/toy_document_en.Rmd
@@ -1,7 +1,7 @@
 ---
-title: "Your title"
-author: "Your name"
-date: "Today's date"
+title: "On the Computation of Pi"
+author: "Jhouben Cuesta Ramirez"
+date: "20/05/2021"
 output: html_document
 ---
 
@@ -10,24 +10,38 @@ output: html_document
 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>.
-
-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:
-
-```{r cars}
-summary(cars)
-```
-
-It is also straightforward to include figures. For example:
-
-```{r pressure, echo=FALSE}
-plot(pressure)
-```
-
-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.
-
-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.
-
-Now it's your turn! You can delete all this information and replace it by your computational document.
+## Asking the maths library
+My computer tells me that $\pi$ is *approximatively*	
+
+```{r}	
+pi	
+```	
+## Buffon's needle	
+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))	
+```	
+
+
+## Using a surface fraction argument	
+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:	
+
+```{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)	
+
+```