Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

 ---
-title: "On the Computation of Pi"
+title: "On the computation of Pi"
 author: "Jhouben Cuesta Ramirez"
 date: "20/05/2021"
 output: html_document
@@ -11,37 +11,37 @@ knitr::opts_chunk$set(echo = TRUE)
 ```
 
 ## Asking the maths library
-My computer tells me that $\pi$ is *approximatively*	
+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))	
+```{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:	
+## 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}	
+```{r}
 set.seed(42)
-N = 1000	
-df = data.frame(X = runif(N), Y = runif(N))	
-df$Accept = (df$X**2 + df$Y**2 <=1)	
-library(ggplot2)	
-ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()	
+N = 1000
+df = data.frame(X = runif(N), Y = runif(N))
+df$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)	
+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)
 
-```	
+```