Update toy_document_en.Rmd

module2/exo1/toy_document_en.Rmd

@@ -9,4 +9,28 @@ My computer tells me that π is *approximatively*
 
 ```{r}
 pi
-```
\ No newline at end of file
+```
+
+##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*∼*U*(0,1)** and ***Y*∼*U*(0,1)**, then **P[X^2+Y^2≤1]=π/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$Y**2 <=1)
+library(ggplot2)
+ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
+```
+It is therefore straightforward to obtain a (not really good) approximation to π by counting how many times, on average, ***X*^2+*Y*^2** is smaller than 1 :
+```{r}
+4*mean(df$Accept)
+```