From c8d941c5e988d9b03201fdf864cc6fc616904c62 Mon Sep 17 00:00:00 2001
From: Alexandre Jesus <adbjesus@gmail.com>
Date: Sat, 2 May 2020 15:15:43 +0100
Subject: [PATCH] Minor fixes

---
 module2/exo1/toy_document_orgmode_R_en.org | 27 +++++++++++-----------
 1 file changed, 13 insertions(+), 14 deletions(-)

diff --git a/module2/exo1/toy_document_orgmode_R_en.org b/module2/exo1/toy_document_orgmode_R_en.org
index e7f9cbe..d001869 100644
--- a/module2/exo1/toy_document_orgmode_R_en.org
+++ b/module2/exo1/toy_document_orgmode_R_en.org
@@ -5,37 +5,36 @@
 
 My computer tells me that $\pi$ is approximately
 
-#+BEGIN_SRC R :results output :exports both
+#+begin_src R :results output :exports both
 pi
-#+END_SRC
+#+end_src
 
 #+RESULTS:
 : [1] 3.141593
 
 * Buffon's needle
 
-Applying the method of Buffon's needle, we get the *approximation*
+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 :exports both
+#+begin_src R :results output :exports both
 set.seed(42)
 N = 100000
 x = runif(N)
 theta = pi/2*runif(N)
 2/(mean(x+sin(theta)>1))
-#+END_SRC
+#+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 $P[X^2 + Y^2 \le 1] = \pi/4$ (see [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo method"
-on Wikipedia]]). 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 [[https://en.wikipedia.org/wiki/Monte_Carlo_method]["Monte Carlo
+method" on Wikipedia]]). The following code uses this approach
 
-
-#+BEGIN_SRC R :session *R* :results output graphics :file "./pi.png" :exports both :width 600 :height 400
+#+begin_src R :session *R* :results output graphics :file "./pi.png" :exports both :width 600 :height 400
 set.seed(42)
 N = 1000
 df = data.frame(X = runif(N), Y = runif(N))
@@ -45,7 +44,7 @@ ggplot(df, aes(x=X,y=Y,color=Accept)) +
   geom_point(alpha=.2) +
   coord_fixed() +
   theme_bw()
-#+END_SRC
+#+end_src
 
 #+RESULTS:
 [[file:./pi.png]]
@@ -54,9 +53,9 @@ 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 :session *R* :results output :exports both
+#+begin_src R :session *R* :results output :exports both
 4*mean(df$Accept)
-#+END_SRC
+#+end_src
 
 #+RESULTS:
 : [1] 3.156
-- 
2.18.1