Update toy_document_orgmode_R_en.org

module2/exo1/toy_document_orgmode_R_en.org

@@ -20,8 +20,10 @@ pi
 
 #+RESULTS:
 : [1] 3.141593
+
 * Buffon's needle
 Applying the method of [[https://en.wikipedia.org/wiki/Buffon%2527s_needle_problem][Buffon's needle]], we get the *approximation*
+
 #+begin_src R :results output :session *R* :exports both
 set.seed(42)
 N = 100000
@@ -29,10 +31,13 @@ x = runif(N)
 theta = pi/2*runif(N)
 2/(mean(x+sin(theta)>1))
 #+end_src
+
 #+RESULTS:
 : [1] 3.14327
+
 * 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 [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method" on Wikipedia]]). The following code uses this approach:
+
 #+begin_src R :results output graphics :file figure_pi_mc1.png :exports both :width 600 :height 400 :session *R* 
 set.seed(42)
 N = 1000
@@ -41,11 +46,15 @@ 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()
 #+end_src
+
 #+RESULTS:
 [[file:figure_pi_mc1.png]]
+
 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:
+
 #+begin_src R :results output :session *R* :exports both
 4*mean(df$Accept)
 #+end_src
+
 #+RESULTS:
-: [1] 3.156
\ No newline at end of file
+: [1] 3.156