"My computer tells me that $\\pi$ *is approximatively*"
"My computer tells me that $\\pi$ *is approximatively*"
]
]
},
},
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@@ -52,14 +43,8 @@
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@@ -52,14 +43,8 @@
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"## Buffon’s needle"
"## Buffon’s needle\n",
]
"Applying the method of [Buffon's needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get __approximation__"
},
{
"cell_type": "markdown",
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"source": [
"Applying the method of [Buffon's needle](https://en.wikipedia.org/wiki/Buffon%27s_needle_problem), we get **the approximation**"
]
]
},
},
{
{
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@@ -91,38 +76,72 @@
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@@ -91,38 +76,72 @@
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"## Using a surface fraction argument"
"## 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:"
},
{
"cell_type": "markdown",
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"source": [
"A method that is easier to understand and does not make use of the sin function is based on the\n",
"fact that if $$X ∼ \\mathcal{U}(0, 1)$$ and $$\\mathcal{Y} ∼ \\mathcal{U}(0, 1)$$, then $$\\mathcal{P}[X^2 + Y^2 ≤ 1] = π/4$$ (see \"[Monte Carlo method]\"(https://en.wikipedia.org/wiki/Monte_Carlo_method)\n",
"on Wikipedia). The following code uses this approach:"
"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:"
