Mathematics Pi Estimator with Random Sampling Method
?Community
00
Estimate pi by sampling random points in the unit square, counting those inside the quarter circle, and using 4x the inside fraction as pi.
NOTEMethod
Each iteration samples one point (x,y) uniformly in [0,1]x[0,1]. The point is inside the quarter circle if x^2 + y^2 <= 1. The probability of being inside equals the quarter-circle area, pi/4, so pi ≈ 4 * fraction_inside.
NOTEAccuracy
Increase iterations for a tighter estimate. Sampling error shrinks about like 1/sqrt(N).
VARIABLERandom X Coordinate
uniform(0, 1)
Captures uncertainty in x location for a uniformly random point in the unit square.
x
VARIABLERandom Y Coordinate
uniform(0, 1)
Captures uncertainty in y location for a uniformly random point in the unit square.
y
FORMULARadius Squared
x^2 + y^2
Squared distance from the origin used to test whether the point is inside the circle.
x
y
r2
FORMULAInside Quarter Circle Indicator
if(r2 <= 1, 1, 0)
Indicator for whether the sampled point lands inside the quarter circle area.
r2
inside
FORMULAPi Estimate Per Sample
4 * inside
Single draw contributes 4 if inside, else 0; the simulation mean converges to pi.
inside
pi_sample
OUTPUTPi Estimate
Pi estimate (mean across samples)
The mean of this output across iterations is the Monte Carlo estimate of pi.
OUTPUTFraction Inside
Inside quarter circle (0 to 1)
The mean of this equals pi/4; multiply by 4 to estimate pi.
What is Carlo?
Carlo is a visual tool for Monte Carlo simulation. Model uncertainty by dragging probability distributions, connecting them visually, and running thousands of scenarios instantly.