From 272867e7f337e1d4712c518fe13273446ae83a3b Mon Sep 17 00:00:00 2001
From: 06295db534a9872198b1363b8b12f920
 <06295db534a9872198b1363b8b12f920@app-learninglab.inria.fr>
Date: Thu, 6 Mar 2025 14:41:45 +0000
Subject: [PATCH] test

---
 module2/exo1/toy_notebook_en.ipynb | 24 ++++++++++++++++++++++--
 1 file changed, 22 insertions(+), 2 deletions(-)

diff --git a/module2/exo1/toy_notebook_en.ipynb b/module2/exo1/toy_notebook_en.ipynb
index 0cd8d6c..c027cbf 100644
--- a/module2/exo1/toy_notebook_en.ipynb
+++ b/module2/exo1/toy_notebook_en.ipynb
@@ -4,9 +4,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# On the computation of &#960;\n",
+    "# 1 On the computation of &#960;\n",
     "\n",
-    "## Asking the maths library\n",
+    "## 1.1 Asking the maths library\n",
     "\n",
     "My computer tells me that &#960; is *approximatively*"
    ]
@@ -29,6 +29,15 @@
     "print(pi)\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 1.2 Buffon’s needle\n",
+    "\n",
+    "Applying the method of [Buffon’s needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the **approximation**"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 2,
@@ -54,6 +63,17 @@
     "2/(sum((x+np.sin(theta))>1)/N)\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 1.3 Using a surface fraction argument\n",
+    "\n",
+    "A method that is easier to understand and does not make use of the sin function is based on the\n",
+    "fact that if *X ∼ U(0, 1)* and *Y ∼ U(0, 1)* then P[X^2 + Y^2 ≤ 1] = &#960;/4 (see [\"Monte Carlo method\"\n",
+    "on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 3,
-- 
2.18.1