"My computer tells me that $\\pi$ is *approximately*"
]
},
...
...
@@ -64,8 +64,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# 1.3 Using a surface fraction argument\n",
"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:"
"# Using a surface fraction argument\n",
"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:"
]
},
{
...
...
@@ -95,7 +95,7 @@
"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",
"accept = (x*x+y*y) <= 1\n",
"reject = np.logical_not(accept)\n",
"\n",
"fig, ax = plt.subplots(1)\n",
...
...
@@ -108,7 +108,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"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:"
"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:"
