Editing Monte Carlo method (section)

== Overview ==
Monte Carlo methods vary, but tend to follow a particular pattern:
# Define a domain of possible inputs.
# Generate inputs randomly from a [[probability distribution]] over the domain.
# Perform a [[Deterministic algorithm|deterministic]] computation of the outputs.
# Aggregate the results.

[[File:Pi monte carlo all.gif|thumb|upright=1.3| Monte Carlo method applied to approximating the value of {{pi}}]]
For example, consider a [[circular sector#Quadrant|quadrant (circular sector)]] inscribed in a [[unit square]]. Given that the ratio of their areas is {{sfrac|{{pi}}|4}}, the value of [[pi|{{pi}}]] can be approximated using the Monte Carlo method:{{sfn|Kalos|Whitlock|2008}}
# Draw a square, then [[inscribed figure|inscribe]] a quadrant within it.
# [[uniform distribution (continuous)|Uniformly]] scatter a given number of points over the square.
# Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1.
# The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, {{sfrac|{{pi}}|4}}. Multiply the result by 4 to estimate {{pi}}.
In this procedure, the domain of inputs is the square that circumscribes the quadrant. One can generate random inputs by scattering grains over the square, then performing a computation on each input to test whether it falls within the quadrant. Aggregating the results yields our final result, the approximation of {{pi}}.

There are two important considerations:
# If the points are not uniformly distributed, the approximation will be poor.
# The approximation improves as more points are randomly placed in the whole square.

Uses of Monte Carlo methods require large amounts of random numbers, and their use benefitted greatly from [[pseudorandom number generators]], which are far quicker to use than the tables of random numbers that had been previously employed.