Probability of Hit (POH): Statistical Model
Deep Dive into Gaussian Distributions, Monte Carlo Simulation, and Confidence Intervals for Precision Ballistics.
1. Introduction & Problem Statement
Traditional ballistics calculators treat bullet trajectories as deterministic: they calculate a single point-impact trajectory and assume that's where your round will land. Probability of Hit (POH) replaces this with a stochastic model using 10,000 Monte Carlo iterations to map a two-dimensional error ellipse on target indicators.
Every precision hunter must ask: How likely am I to hit the vitals on this animal from this distance, given my rifle's accuracy and the target size?
This is fundamentally a statistical question. ONESHOT Ballistics answers it using Gaussian (normal) distributions to model bullet dispersion and Monte Carlo simulations to establish ethical shooting thresholds.
2. Gaussian Distribution Foundation
Rifle dispersion follows a normal (Gaussian) distribution due to the Central Limit Theorem acting on independent error sources such as barrel harmonics, ammunition inconsistency, and micro-wind variations. ONESHOT models this in two dimensions using a probability density function defined by horizontal and vertical standard deviations (σ).
f(x,y) = (1 / (2πσₓσᵧ)) * exp(-((x-μₓ)² / (2σₓ²) + (y-μᵧ)² / (2σᵧ²)))
Standard Deviation in Shooting
| Rifle Accuracy | Typical σ (at 100m) | Conversion to MOA |
|---|---|---|
| Hunting-grade rifle | 0.5" | 1.0 MOA |
| Match-grade rifle | 0.25" | 0.5 MOA |
| Precision bench rifle | 0.1" | 0.2 MOA |
3. Monte Carlo Implementation (10,000 Iterations)
The Monte Carlo method solves the impact probability integral numerically by generating 10,000 random shot samples from the rifle's Gaussian distribution. By counting hits against the target area, the engine estimates POH with a 1% precision threshold and calculates a 95% confidence interval for the result.
CI = 1.96 * √(POH * (1 - POH) / N)
Statistical convergence theory shows that 10,000 iterations provides sufficient precision for ethical hunting decisions. For a true POH of 90%, the ONESHOT estimate will typically fall within a tight 89-91% range.
4. Interactive POH Visualizer
Simulate 1,000 shots on target using Gaussian dispersion. Adjust the rifle's precision (σ) and target size to see how Probability of Hit changes.
5. Scientific References
McCoy, R. L. (1999)
Exterior Ballistics. Ballistic Publications. The definitive reference for 6-DoF trajectory calculations and aerodynamic drag modeling used in the ONESHOT core.
Marsaglia, G., & Tsang, W. W. (2000)
"The Ziggurat Method for Generating Random Variables." Journal of Statistical Software. The efficient algorithm used for generating high-quality Gaussian random samples in the POH engine.