final changes

module2/exo1/toy_notebook_en.ipynb

@@ -4,7 +4,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# On the computation of $\\pi$\n",
+    "# On the computation of $\\pi$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "## Asking the maths library\n",
     "My computer tells me that $\\pi$ is *approximatively*"
    ]
@@ -32,7 +38,7 @@
    "metadata": {},
    "source": [
     "## Buffon's needle\n",
-    "Applying the method of [Buffon's needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the **approximation**"
+    "Applying the method of [Buffon's needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get the _approximation_"
    ]
   },
   {
@@ -65,7 +71,7 @@
    "metadata": {},
    "source": [
     "## Using a surface fraction argument\n",
-    "A method that is easier to understand and does not make use of thesin function is based on thefact that if $X \\sim U(0,1)$ and $Y \\sim U(0,1)$, then $P[X^2 + Y^2 \\leq 1] = \\frac{\\pi}{4}$ (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 $P[X^2+Y^2\\leq 1] = \\pi/4$ (see [\"Monte Carlo method\" on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:"
    ]
   },
   {
@@ -87,17 +93,18 @@
     }
    ],
    "source": [
-    "%matplotlib inline\n",
+    "%matplotlib inline  \n",
     "import matplotlib.pyplot as plt\n",
     "\n",
     "np.random.seed(seed=42)\n",
-    "N=1000\n",
-    "x=np.random.uniform(size=N, low=0, high=1)\n",
-    "y=np.random.uniform(size=N, low=0, high=1)\n",
-    "accept=(x*x+y*y)<=1\n",
-    "reject=np.logical_not(accept)\n",
+    "N = 1000\n",
+    "x = np.random.uniform(size=N, low=0, high=1)\n",
+    "y = np.random.uniform(size=N, low=0, high=1)\n",
+    "\n",
+    "accept = (x*x+y*y) <= 1\n",
+    "reject = np.logical_not(accept)\n",
     "\n",
-    "fig, ax=plt.subplots(1)\n",
+    "fig, ax = plt.subplots(1)\n",
     "ax.scatter(x[accept], y[accept], c='b', alpha=0.2, edgecolor=None)\n",
     "ax.scatter(x[reject], y[reject], c='r', alpha=0.2, edgecolor=None)\n",
     "ax.set_aspect('equal')"