\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
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\(\pi\)
e
\(\frac{a}{b}\)
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\(\sqrt{\,}\)
\(a^{b}\)
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\(\div\)
log
ln
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\(\times\)
cos
cos⁻¹
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sin
sin⁻¹
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tan
tan⁻¹
We conduct a computer simulation to estimate the value of \(\pi\).
Imagine a square with side length \(2\) (area \(= 4\)) enclosing a circle with radius \(1\) (area \(= \pi\)).We generate \(n\) random points uniformly inside the square. Let \(X_i\) be a random variable such that \(X_i = 1\) if the point falls inside the circle, and \(X_i = 0\) otherwise.
Calculate the theoretical probability \(p\) that a single random point falls inside the circle.
Let \(\overline{X}_n\) be the proportion of points inside the circle after \(n\) trials. According to the Law of Large Numbers, what value does \(\overline{X}_n\) approach as \(n \to \infty\)?
Propose a formula to estimate \(\pi\) using the experimental result \(\overline{X}_n\).
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