Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

@@ -13,14 +13,16 @@ knitr::opts_chunk$set(echo = TRUE)
 
 My computer tells me that $\pi$ is _approximatively_
 
-```pi
+```{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__
 
-```set.seed(42)
+```{r}
+set.seed(42)
 N = 100000
 x = runif(N)
 theta = pi/2*runif(N)
@@ -29,7 +31,7 @@ theta = pi/2*runif(N)
 
 ## 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 thath if $\X ~ U (0,1)$ and $\Y ~ U (0,1)$, then $\P[X^2 + Y^2 <=1] = $\pi$/4$ (see ["Monte Carlo method" on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:
+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 $\le$ 1] = $\pi$/4$ (see ["Monte Carlo method" on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:
 
 ```set.seed(42)
 N = 1000
@@ -38,7 +40,7 @@ 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 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 :
+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 :
 
 ```4*mean(df$Accept)
 ```