"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__"
]
]
},
},
{
{
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@@ -71,13 +71,12 @@
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@@ -71,13 +71,12 @@
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"## Using a surface fraction argument\n",
"## 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 $[X^2 + Y^2 \\le 1] = \\pi/4$ (see [\"Monte Carlo method\"\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:"
"on Wikipedia](https://en.wikipedia.org/wiki/Monte_Carlo_method)). The following code uses this approach:"
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"cell_type": "code",
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"execution_count": 5,
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"outputs": [
{
{
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@@ -94,7 +93,6 @@
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@@ -94,7 +93,6 @@
}
}
],
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"source": [
"source": [
"%matplotlib inline\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib.pyplot as plt\n",
"\n",
"\n",
"np.random.seed(seed=42)\n",
"np.random.seed(seed=42)\n",
...
@@ -115,8 +113,7 @@
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@@ -115,8 +113,7 @@
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"It is then straightforward to obtain a (not really good) approximation to $\\pi$ by counting how\n",
"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:"
"many times, on average, $X^2 + Y^2$ is smaller than $1$:"