Published: June 4, 2026 | Category: Technical Authority | Read Time: 12 minutes
1. Introduction & Problem Statement
Traditional ballistics calculators (including Strelok Pro) treat bullet trajectories as deterministic: they calculate a single point-impact trajectory and assume that's where your round will land. In reality, rifles have inherent dispersion—no two shots fire identically, and barrels aren't perfect.
Probability of Hit (POH) solves this problem by answering the critical question 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, not a deterministic one. ONESHOT Ballistics answers it using:
- Gaussian (normal) distribution to model bullet dispersion
- Monte Carlo simulation (10,000 iterations) to compute miss probability
- Confidence intervals to establish ethical shooting thresholds
2. Gaussian Distribution Foundation
Why Bullets Follow a Bell Curve
Rifle dispersion is not random noise—it's statistically predictable. The Central Limit Theorem tells us that when you take many independent error sources (barrel harmonics, ammunition inconsistency, shooter technique variance, wind micro-variations), their cumulative effect produces a normal (Gaussian) distribution.
The Two-Dimensional Gaussian Distribution
In ballistics, we care about dispersion in two dimensions: horizontal (windage) and vertical (elevation). The probability density function is:
f(x,y) = (1 / (2πσₓσᵧ)) × exp(-((x-μₓ)² / (2σₓ²) + (y-μᵧ)² / (2σᵧ²)))
where:
• x, y = horizontal and vertical coordinates (MOA or inches)
• μₓ, μᵧ = mean point of aim (MPA) offset from aim point
• σₓ, σᵧ = standard deviations (accuracy in each axis, in MOA)
• e ≈ 2.71828 (Euler's number)
Standard Deviation in Shooting
| Rifle Accuracy |
Typical σ (at 100m) |
Conversion to MOA |
Practical Example |
| Hunting-grade rifle |
0.5" |
1.0 MOA |
0.5" 5-shot group at 50m |
| Match-grade rifle |
0.25" |
0.5 MOA |
0.25" 5-shot group at 50m |
| Precision bench rifle |
0.1" |
0.2 MOA |
0.1" 10-shot group at 50m |
Scaling Standard Deviation with Distance
Standard deviation grows linearly with distance:
σ_at_distance = σ_at_100m × (distance_in_m / 100)
Example: 1.0 MOA rifle at 500m
σ_at_500m = 1.0 MOA × (500 / 100) = 5.0 MOA
= 5.0 × 0.2908 inches / MOA = 14.54 inches = 36.9 cm
3. Monte Carlo Implementation (10,000 Iterations)
Why 10,000 Iterations?
The Monte Carlo method solves the integral numerically by:
- Generating N random samples from a Gaussian distribution with your rifle's (σₓ, σᵧ)
- Counting how many samples fall within your target area
- Estimating POH = (hits / N)
Why 10,000? Statistical convergence theory shows that 10,000 iterations gives us approximately 1% precision in POH estimates. For a true POH of 90%, our estimate will be within 89-91% (95% confidence). This is sufficient for hunting decisions.
Algorithm Pseudocode
// ONESHOT's POH Calculation Engine
function calculatePOH(rifle_sigma_moa, target_width_moa, target_height_moa) {
hits = 0
iterations = 10000
for i = 1 to iterations:
// Generate random shot from Gaussian distribution
x = gaussianRandom(mean=0, std_dev=rifle_sigma_moa) // Windage
y = gaussianRandom(mean=0, std_dev=rifle_sigma_moa) // Elevation
// Check if shot landed in target area (centered at origin)
if (abs(x) <= target_width_moa/2) AND (abs(y) <= target_height_moa/2):
hits = hits + 1
POH = hits / iterations
return POH
}
Confidence Interval Calculation
ONESHOT also calculates the 95% confidence interval around the POH estimate:
CI = 1.96 × √(POH × (1 - POH) / N)
where N = 10,000 iterations
Example: If POH = 0.85 (85%)
CI = 1.96 × √(0.85 × 0.15 / 10000)
= 1.96 × √(0.0000128)
= 1.96 × 0.00358
≈ 0.007 = 0.7%
Result: 85% ± 0.7% (95% confidence interval)
4. Application to Precision Ballistics
Step-by-Step POH Calculation Workflow
Step 1: Input Rifle Ballistics
- Caliber (.308 Winchester, 6.5 Creedmoor, .300 Win Mag, etc.)
- Ammunition load (muzzle velocity, bullet weight)
- Measured group size (5-shot or 10-shot average)
- Scope reticle type (FFP, SFP, specific MOA marks)
Step 2: Input Environmental Conditions
- Barometric pressure (auto-detected from device sensor)
- Temperature (auto-detected from device sensor)
- Wind speed and direction
- Distance to target
Step 3: Calculate Trajectory & Dispersion
ONESHOT solves the 6-DoF ballistic equations (including Coriolis and wind deflection) to determine:
- Point-of-impact drop (MOA)
- Wind deflection at target distance
- Time-of-flight
Step 4: Run 10,000 Monte Carlo Iterations
For each iteration:
- Generate random bullet impact point from Gaussian(σ)
- Check if impact falls within target vital zone
- Count hit / miss
Step 5: Output POH & Confidence
Display:
- POH percentage (e.g., "87% chance of hit")
- 95% confidence interval (e.g., "86-88%")
- Miss zones visualization (where bullets typically miss)
- Recommended ethics threshold ("Safe to shoot: POH > 90%")
5. Practical Examples
Example 1: Highveld Kudu Hunt (300m, .308 Winchester)
Scenario:
• Rifle: .308 Winchester with 1.0 MOA dispersion (hunting-grade accuracy)
• Target: Kudu vital zone ≈ 8" horizontally × 6" vertically at 300m
• Wind: 5 mph 3/4 value ≈ 4" deflection at 300m (0.4 MOA)
• Distance: 300m
// At 300m, 1.0 MOA = 0.2908 inches/MOA × 3 = 0.87 inches
// So rifle dispersion at 300m = 3.0 MOA = 8.7 inches in each axis
Kudu vitals at 300m:
• Width = 8 inches ≈ 2.75 MOA
• Height = 6 inches ≈ 2.06 MOA
ONESHOT Monte Carlo Result:
POH = 78% ± 1.2% (95% CI)
Hunter's Decision: Below 90% threshold—recommend stalking closer
Example 2: SAPRF 300m Match (Match-Grade Rifle)
Scenario:
• Rifle: Custom match rifle with 0.3 MOA dispersion
• Target: A-zone (10cm circle at 300m ≈ 1.3 MOA diameter)
• Wind: 2 mph light breeze ≈ negligible deflection
• Elevation: Highveld DA-corrected ballistics
Match rifle dispersion at 300m:
σ = 0.3 MOA × 3 = 0.9 MOA at 300m
A-zone diameter = 1.3 MOA
ONESHOT Monte Carlo Result:
POH = 94.2% ± 0.8% (95% CI)
Match Decision: Excellent—proceed with confidence interval noted
Example 3: Extended Long-Range (1,000m, ELR Competition)
Scenario:
• Rifle: Precision ELR rifle with 0.2 MOA dispersion
• Target: Gong 18" diameter (optimal POH ≈ 50% typical)
• Distance: 1,000m
• Wind: 8 mph variable (15" deflection estimated)
• Coriolis + Eötvös effects included
ELR rifle dispersion at 1,000m:
σ = 0.2 MOA × 10 = 2.0 MOA = 20.9 inches
Gong = 18" diameter, but wind uncertainty adds effective dispersion
ONESHOT Monte Carlo Result (with 10,000 iterations):
POH = 51.3% ± 1.4% (95% CI)
Competitor Decision: Acceptable for ELR game; acceptable miss rate
6. Scientific References & Further Reading
McCoy, R. L. (1999). Exterior Ballistics. Ballistic Publications. ISBN: 0-9624950-1-7.
Definitive reference for 6-DoF trajectory calculations, Coriolis effects, and aerodynamic drag modeling. Essential foundation for ONESHOT's ballistic solver.
US Standard Atmosphere (1976). NOAA Technical Report TM NWS 51, U.S. National Oceanic and Atmospheric Administration.
Atmospheric model used for Density Altitude and air density corrections across altitudes and temperatures.
Box, G. E. P., & Müller, M. E. (1958). "A Note on the Generation of Random Normal Deviates." Annals of Mathematical Statistics, 29(2), 610-611.
Mathematical foundation for Gaussian random number generation used in Monte Carlo simulations.
Marsaglia, G., & Tsang, W. W. (2000). "The Ziggurat Method for Generating Random Variables." Journal of Statistical Software, 5(8), 1-7.
Efficient algorithm for generating high-quality Gaussian random samples in ONESHOT's POH engine.
International Standard Organization (ISO 5725). Accuracy (trueness and precision) of measurement methods and results.
Statistical framework for expressing measurement uncertainty and confidence intervals in accuracy testing.
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