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Reloading Supplies
Reloading Data
Probability Of Hitting A Distant Target
By Joe Huffman
Copyright 2000
 | Summary of the Problem |
| Solution |
| Alternate sizes and target orientations |
| Conclusions |
Summary of the Problem:
The Boomershoot targets are small -- the 1/2-pint milk cartons represent a rectangle 2.25 x 3.0 inches at a minimum distance of 200 yards. And the targets are as far away as 700 yards. 2.25 inches at 700 yards is 0.31 MOA. For the 2001 Boomersoot one pint, one quart and two quart cartons will be used for some of the more distant targets. These appear as 2.75 x 3.8, 2.75 x 7.5, and 3.75 x 7.6 inch rectangles. At 700 yards 2.75 inches is only 0.38 MOA. Given a particular level of accuracy, say 1 MOA, with a particular cartridge on a windless day what are the odds of connecting with one of these targets at various ranges? And what if there is some wind?
Solution:
The solution was much more difficult that I expected. The explicit derivation looked too complicated. Even if I didn't make an error in the transformation of the probability density functions I was likely to make an error in all the algebraic transformations. I decided to run simulations on the computer. It took several weeks of part time effort before I was able to cobble up something to do the job. Each of the subsections have been tested independently and I'm fairly certain the results are correct within the assumptions made and the trustworthiness of the random number generator (from Knuth).
I assumed that the shooter aims perfectly with the exact correction for the estimated wind and the exact range. Range estimation errors are basically not a problem at the Boomershoots.
I assumed there are three different types of errors made.
1) Under zero wind and zero muzzle velocity variations the cartridge/firearm combination will still have some error. This error is an angle of deviation from the axis of the barrel which is normally distributed (a bell-shaped probability density curve) angle. The direction of the angle is equally likely.
2) Muzzle velocity variations. This error is is normally distributed.
3) Wind estimation error. This error is normally distributed.
I assumed it takes at least a one-half bullet diameter hit to be effective. Just touching is not good enough.
In addition to the three types of random errors there are the variables of range, absolute wind velocity (yes, it makes a difference), and target size. To keep this page from becoming a huge (instead of merely large) collection of tables I have given the results for some common cases and if you want numbers for specific cases send me email and I will generate the numbers for you.
The entries in each cell below is the percent chance of hitting a one-half pint milk carton target at various ranges with the specified cartridge with a single shot. As with all statistical calculations there are errors involved in the estimation. I haven't done calculations on the exact error range, but I expect the numbers are within about +/- 3 or 4%.
Except for the .50 BMG the cartridges assume factory specified muzzle velocity and ballistic coefficients for Federal match ammo. The BC used for .50 BMG is 1.18 (Barnes 800 grain, VLD Borerider) with a MV of 2700 fps. Unless otherwise stated the wind is assumed to be 10 MPH at 90 degrees to the direction of fire. Temperature and other factors are typical Boomershoot conditions of 70F and 3000 feet above sea level. The accuracy of the cartridge is specified as the average 5-shot group size with no muzzle velocity variations or wind. The wind error is the +/- variation such that 90% of the measurements/estimates will be within this range. So if the wind error is 1 MPH, then 90% of the time the wind will be between 9 and 11 MPH.
Probability of a hit on one-half pint carton
No wind
Cartridge accuracy: 0.5 MOA
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 99% | 92% | 85% | 77% | 58% | 46% |
| 308 Win | 99% | 92% | 85% | 76% | 57% | 44% |
| 30.06 | 99% | 92% | 85% | 76% | 56% | 45% |
| 300 Win | 99% | 92% | 85% | 76% | 60% | 48% |
| .50 BMG | 99% | 92% | 85% | 77% | 60% | 47% |
Wind error: 0.5 MPH
Cartridge accuracy: 0.5 MOA.
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 99% | 93% | 72% | 54% | 37% | 24% |
| 308 Win | 100% | 92% | 77% | 60% | 40% | 29% |
| 30.06 | 100% | 92% | 81% | 62% | 41% | 31% |
| 300 Win | 100% | 92% | 83% | 66% | 52% | 37% |
| .50 BMG | 100% | 92% | 83% | 72% | 58% | 46% |
Wind error: 1 MPH
Cartridge accuracy: 0.5 MOA.
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 99% | 86% | 55% | 39% | 23% | 16% |
| 308 Win | 99% | 91% | 63% | 47% | 29% | 18% |
| 30.06 | 99% | 91% | 66% | 48% | 30% | 20% |
| 300 Win | 99% | 93% | 73% | 53% | 38% | 26% |
| .50 BMG | 100% | 92% | 83% | 67% | 53% | 39% |
Wind error: 2 MPH
Cartridge accuracy: 0.5 MOA.
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 98% | 62% | 35% | 23% | 11% | 8% |
| 308 Win | 99% | 72% | 44% | 31% | 17% | 10% |
| 30.06 | 99% | 73% | 47% | 31% | 18% | 10% |
| 300 Win | 99% | 87% | 52% | 36% | 24% | 14% |
| .50 BMG | 100% | 93% | 72% | 54% | 39% | 27% |
Wind error: 0.5 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 86% | 62% | 43% | 28% | 23% | 16% |
| 308 Win | 87% | 61% | 44% | 35% | 26% | 20% |
| 30.06 | 87% | 61% | 45% | 34% | 26% | 21% |
| 300 Win | 87% | 62% | 46% | 35% | 31% | 23% |
| .50 BMG | 87% | 62% | 46% | 36% | 32% | 28% |
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 85% | 57% | 33 | 21% | 15% | 8% |
| 308 Win | 85% | 59% | 39% | 25% | 17% | 12% |
| 30.06 | 85% | 59% | 39% | 26% | 19% | 13% |
| 300 Win | 85% | 62% | 42% | 29% | 24% | 17% |
| .50 BMG | 87% | 60% | 46% | 35% | 30% | 22% |
Wind error: 2 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 5 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 75% | 44% | 21% | 13% | 8% | 4% |
| 308 Win | 82% | 48% | 28% | 15% | 10% | 7% |
| 30.06 | 84% | 48% | 30% | 16% | 10% | 7% |
| 300 Win | 85% | 57% | 31% | 22% | 13% | 10% |
| .50 BMG | 86% | 60% | 43% | 28% | 25% | 18% |
Wind error: 0.5 MPH
Cartridge accuracy: 0.5 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 99% | 93% | 73% | 49% | 32% | 18% |
| 308 Win | 100% | 92% | 76% | 53% | 34% | 20% |
| 30.06 | 100% | 92% | 81% | 56% | 26% | 24% |
| 300 Win | 100% | 92% | 82% | 65% | 48% | 28% |
| .50 BMG | 100% | 92% | 84% | 71% | 50% | 37% |
Wind error: 1 MPH
Cartridge accuracy: 0.5 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 99% | 85% | 54% | 38% | 22% | 11% |
| 308 Win | 99% | 91% | 63% | 40% | 25% | 13% |
| 30.06 | 99% | 92% | 65% | 43% | 27% | 15% |
| 300 Win | 99% | 93% | 71% | 52% | 35% | 21% |
| .50 BMG | 100% | 92% | 83% | 64% | 47% | 32% |
Wind error: 2 MPH
Cartridge accuracy: 0.5 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 98% | 61% | 37% | 20% | 11% | 6% |
| 308 Win | 99% | 72% | 45% | 28% | 14% | 6% |
| 30.06 | 99% | 73% | 47% | 30% | 15% | 7% |
| 300 Win | 99% | 85% | 54% | 36% | 24% | 13% |
| .50 BMG | 99% | 93% | 72% | 53% | 34% | 22% |
Wind error: 0.5 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 87% | 60% | 43% | 31% | 19% | 14% |
| 308 Win | 87% | 60% | 42% | 33% | 20% | 17% |
| 30.06 | 87% | 60% | 43% | 33% | 23% | 18% |
| 300 Win | 87% | 61% | 46% | 37% | 29% | 20% |
| .50 BMG | 87% | 62% | 46% | 37% | 28% | 22% |
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 85% | 55% | 33% | 21% | 13% | 6% |
| 308 Win | 84% | 59% | 37% | 24% | 15% | 12% |
| 30.06 | 84% | 59% | 38% | 23% | 17% | 11% |
| 300 Win | 85% | 60% | 43% | 30% | 21% | 14% |
| .50 BMG | 87% | 61% | 46% | 37% | 28% | 19% |
Wind error: 2 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 75% | 42% | 23% | 12% | 7% | 4% |
| 308 Win | 83% | 48% | 29% | 16% | 7% | 5% |
| 30.06 | 84% | 49% | 29% | 16% | 8% | 5% |
| 300 Win | 85% | 55% | 32% | 21% | 13% | 8% |
| .50 BMG | 85% | 60% | 43% | 30% | 22% | 14% |
Alternate sizes and target orientations
One-half pint milk carton - horizontal
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 82% | 54% | 34% | 22% | 11% | 8% |
| 308 Win | 81% | 55% | 39% | 24% | 18% | 11% |
| 30.06 | 81% | 57% | 41% | 24% | 17% | 12% |
| 300 Win | 82% | 59% | 41% | 30% | 18% | 16% |
| .50 BMG | 82% | 58% | 42% | 32% | 22% | 20% |
One pint milk carton - vertical
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 91% | 69% | 44% | 29% | 19% | 11% |
| 308 Win | 91% | 68% | 47% | 31% | 22% | 12% |
| 30.06 | 91% | 69% | 49% | 33% | 23% | 14% |
| 300 Win | 91% | 73% | 53% | 38% | 28% | 19% |
| .50 BMG | 91% | 80% | 56% | 44% | 34% | 29% |
One pint milk carton - horizontal
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 82% | 54% | 34% | 22% | 11% | 8% |
| 308 Win | 91% | 70% | 49% | 34% | 20% | 18% |
| 30.06 | 91% | 71% | 48% | 34% | 20% | 17% |
| 300 Win | 92% | 73% | 48% | 40% | 25% | 17% |
| .50 BMG | 92% | 72% | 54% | 41% | 32% | 23% |
One quart milk carton - vertical
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 91% | 74% | 55% | 38% | 27% | 14% |
| 308 Win | 91% | 75% | 62% | 41% | 31% | 18% |
| 30.06 | 91% | 76% | 63% | 43% | 31% | 21% |
| 300 Win | 91% | 78% | 66% | 50% | 39% | 28% |
| .50 BMG | 91% | 85% | 69% | 59% | 50% | 41% |
One quart milk carton - horizontal
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 95% | 78% | 60% | 50% | 32% | 21% |
| 308 Win | 95% | 76% | 62% | 49% | 36% | 27% |
| 30.06 | 95% | 77% | 61% | 49% | 38% | 25% |
| 300 Win | 95% | 79% | 60% | 53% | 43% | 28% |
| .50 BMG | 95% | 78% | 61% | 52% | 45% | 34% |
Two quart milk carton - vertical
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 96% | 90% | 69% | 49% | 34% | 23% |
| 308 Win | 96% | 90% | 74% | 57% | 41% | 28% |
| 30.06 | 96% | 90% | 74% | 60% | 42% | 29% |
| 300 Win | 96% | 91% | 75% | 64% | 49% | 34% |
| .50 BMG | 97% | 91% | 85% | 70% | 58% | 49% |
Two quart milk cartons - horizontal
Wind error: 1 MPH
Cartridge accuracy: 1.0 MOA.
Muzzle velocity standard deviation: 10 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 100% | 90% | 75% | 61% | 42% | 27% |
| 308 Win | 100% | 92% | 76% | 61% | 48% | 31% |
| 30.06 | 100% | 91% | 76% | 61% | 50% | 31% |
| 300 Win | 100% | 91% | 77% | 63% | 56% | 40% |
| .50 BMG | 100% | 91% | 78% | 61% | 54% | 46% |
Less Accurate Guns
Suppose your gun is not a 1 MOA gun. Suppose further that you have ideal wind conditions -- maybe it's worth a shot (or ten).
One-half pint milk carton - vertical
Wind error: 0 MPH
Wind velocity: 0 MPH
Cartridge accuracy: 2.0 MOA
Muzzle velocity standard deviation: 20 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 50% | 34% | 27% | 19% | 13% | 10% |
| 308 Win | 49% | 34% | 26% | 18% | 12% | 9% |
| 30.06 | 49% | 34% | 26% | 19% | 13% | 10% |
| 300 Win | 50% | 35% | 27% | 20% | 15% | 11% |
| .50 BMG | 49% | 34% | 27% | 19% | 15% | 11% |
One-half pint milk carton - vertical
Wind error: 0 MPH
Wind velocity: 0 MPH
Cartridge accuracy: 4.0 MOA (can you say "SKS" quality?)
Muzzle velocity standard deviation: 20 fps
| Cartridge | 200 yards | 300 yards | 400 yards | 500 yards | 600 yards | 700 yards |
| .223 | 29% | 19% | 13% | 10% | 6% | 4% |
| 308 Win | 29% | 18% | 13% | 9% | 5% | 4% |
| 30.06 | 29% | 19% | 13% | 10% | 6% | 4% |
| 300 Win | 29% | 19% | 14% | 10% | 7% | 4% |
| .50 BMG | 29% | 19% | 13% | 10% | 7% | 4% |
Conclusions:
Interesting exercise. I was surprised at how little difference it made to have a wide short target compared to the same size target oriented in the vertical direction. Only with the .223 did it really make a (barely) statistical difference at the greater ranges. The .50 BMG was better overall, as expected, but it didn't really have that big of an advantage until the wind error started getting large. The 300 Win Mag was better than all the others, but it didn't have that much better performance than the .308, just barely above the "noise". What was really surprising to me was that with no wind, excellent ammo (and shooter) that one of the half-pint cartons were subject to nearly 50% chance of getting hit with any of the cartridges at 700 yards. And furthermore just a 0.5 MPH error in reading a 10 MPH wind will drop that 10 to 20 percentage points.
The simulation also confirmed that 200 yards is close enough for just about anyone to connect with and that 700 yards, even with the larger targets will give everyone a challenge while still being possible.
Email: Joe Huffman
Last updated:
February 06, 2006