Each round's initial conditions are drawn from a probability distribution instead of a single fixed value: muzzle velocity, crosswind and the drag coefficient all vary shot to shot. Integrating Newton's second law for every sampled round —
m·dV_x/dt = -D_x − m·g·sin(θ) (horizontal)
m·dV_y/dt = -D_y − m·g·cos(θ) (vertical, drag D ∝ v·|v|)
— produces one simulated trajectory. Repeating this thousands of times (Monte Carlo sampling) turns a single deterministic shot into a cloud of plausible outcomes.
Range sensitivity to velocity error:
Δx = √(Σ δv_i² · cos²(θ_i))
Dispersion of the impact points (std. deviation):
σ = √( (1/N) · Σ (x_i − μ)² )
The wider the impact cloud, the larger σ — this is the same statistic used to draw a circular/elliptical error probable (CEP) around the mean point of impact. Sensitivity analysis (which slider moves σ the most) tells you which input is worth controlling first to tighten accuracy.