Editing Pi (section)

=== Monte Carlo methods ===
{{multiple image
| direction         = horizontal
| image1            = Buffon needle.svg
| caption1          = [[Buffon's needle]]. Needles ''a'' and ''b'' are dropped randomly.
| alt1              = Needles of length ''ℓ'' scattered on stripes with width ''t''
| image2            = Pi 30K.gif
| caption2          = Random dots are placed on a square and a circle inscribed inside.
| alt2              = Thousands of dots randomly covering a square and a circle inscribed in the square.
| align             = left
| total_width       = 225
}}
[[Monte Carlo methods]], which evaluate the results of multiple random trials, can be used to create approximations of {{pi}}.{{sfn|Arndt|Haenel|2006|p=39}} [[Buffon's needle]] is one such technique: If a needle of length {{math|''ℓ''}} is dropped {{math|''n''}} times on a surface on which parallel lines are drawn {{math|''t''}} units apart, and if {{math|''x''}} of those times it comes to rest crossing a line ({{math|''x''}}&nbsp;>&nbsp;0), then one may approximate {{pi}} based on the counts:<ref name="bn">{{cite journal |last=Ramaley |first=J. F. |title=Buffon's Needle Problem |jstor=2317945 |journal=The American Mathematical Monthly |volume=76 |issue=8 |date=October 1969 |pages=916–918 |doi=10.2307/2317945}}</ref>

<math display=block>\pi \approx \frac{2n\ell}{xt}.</math>

Another Monte Carlo method for computing {{pi}} is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal {{math|π/4}}.<ref>{{harvnb|Arndt|Haenel|2006|pp=39–40}}.{{br}} {{harvnb|Posamentier|Lehmann|2004|p=105}}.</ref>

[[File:Five random walks.png|thumb|Five random walks with 200 steps.  The sample mean of {{math|{{abs|''W''<sub>200</sub>}}}} is {{math|''μ'' {{=}} 56/5}}, and so {{math|2(200)''μ''<sup>−2</sup> ≈ 3.19}} is within {{math|0.05}} of&nbsp;{{pi}}.|left]]
Another way to calculate {{pi}} using probability is to start with a [[random walk]], generated by a sequence of (fair) coin tosses: independent [[random variable]]s {{math|''X<sub>k</sub>''}} such that {{math|''X<sub>k</sub>'' ∈ {{mset|−1,1}}}} with equal probabilities.  The associated random walk is

<math display=block>W_n = \sum_{k=1}^n X_k</math>

so that, for each {{mvar|n}}, {{math|''W<sub>n</sub>''}} is drawn from a shifted and scaled [[binomial distribution]].  As {{mvar|n}} varies, {{math|''W<sub>n</sub>''}} defines a (discrete) [[stochastic process]].  Then {{pi}} can be calculated by<ref>{{cite journal |last=Grünbaum |first=B. |author-link=Branko Grünbaum |title=Projection Constants |journal=[[Transactions of the American Mathematical Society]] |volume=95 |issue=3 |pages=451–465 |year=1960 |doi=10.1090/s0002-9947-1960-0114110-9 |doi-access=free}}</ref>

<math display=block>\pi = \lim_{n\to\infty} \frac{2n}{E[|W_n|]^2}.</math>

This Monte Carlo method is independent of any relation to circles, and is a consequence of the [[central limit theorem]], discussed [[#Gaussian integrals|below]].

These Monte Carlo methods for approximating {{pi}} are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximate {{pi}} when speed or accuracy is desired.<ref>{{harvnb|Arndt|Haenel|2006|p=43}}.{{br}}{{harvnb|Posamentier|Lehmann|2004|pp=105–108}}.</ref>