An important quantity when comparing missiles is the CEP (Circular Error Probable). It is defined as the radius of the circle in which 50 % of the fired missiles land. The smaller it is, the better the accuracy of the missile. The German V2 rockets for example had a CEP of about 17 km. So there was a 50/50 chance of a V2 landing within 17 km of its target. Targeting smaller cities or even complexes was next to impossible with this accuracy, one could only aim for a general area in which it would land rather randomly.

Today’s missiles are significantly more accurate. The latest version of China’s DF-21 has a CEP about 40 m, allowing the accurate targeting of small complexes or large buildings, while CEP of the American made Hellfire is as low as 4 m, enabling precision strikes on small buildings or even tanks.

Assuming the impacts are normally distributed, one can derive a formula for the probability of striking a circular target of Radius R using a missile with a given CEP:

p = 1 – exp( -0.41 · R² / CEP² )

This quantity is also called the “single shot kill probability” (SSKP). Let’s include some numerical values. Assume a small complex with the dimensions 100 m by 100 m is targeted with a missile having a CEP of 150 m. Converting the rectangular area into a circle of equal area gives us a radius of about 56 m. Thus the SSKP is:

p = 1 – exp( -0.41 · 56² / 150² ) = 0.056 = 5.6 %

So the chances of hitting the target are relatively low. But the lack in accuracy can be compensated by firing several missiles in succession. What is the chance of at least one missile hitting the target if ten missiles are fired? First we look at the odds of all missiles missing the target and answer the question from that. One missile misses with 0.944 probability, the chance of having this event occur ten times in a row is:

p(all miss) = 0.944^{10} = 0.562

Thus the chance of at least one hit is:

p(at least one hit) = 1 – 0.562 = 0.438 = 43.8 %

Still not great considering that a single missile easily costs 10000 $ upwards. How many missiles of this kind must be fired at the complex to have a 90 % chance at a hit? A 90 % chance at a hit means that the chance of all missiles missing is 10 %. So we can turn the above formula for p(all miss) into an equation by inserting p(all miss) = 0.1 and leaving the number of missiles n undetermined:

0.1 = 0.944^{n}

All that’s left is doing the algebra. Applying the natural logarithm to both sides and solving for n results in:

n = ln(0.1) / ln(0.944) = 40

So forty missiles with a CEP of 150 m are required to have a 90 % chance at hitting the complex. As you can verify by doing the appropriate calculations, three DF-21 missiles would have achieved the same result.

Liked the excerpt? Get the book “Statistical Snacks” by Metin Bektas here: http://www.amazon.com/Statistical-Snacks-ebook/dp/B00DWJZ9Z2. For more excerpts see The Probability of Becoming a Homicide Victim and How To Use the Expected Value.

Could you explain where the factor 0.41 comes from?

I have open in front of me “Naval Operations Analysis”, 3rd edn, by Wagner, Mylander and Sanders. Pages 284-285 give formulae for P(hit) based on both standard deviation of miss distance and CEP (50th percentile of miss distance). Their formula for CEP is

p(hit) = 1 – (1/2)^(R^2/CEP^2)

Their s.d.-based formula agrees with one I have from another source, and they give the conversion factor CEP = 1.1774 s.d. If I apply this conversion factor to their CEP-based formula, it agrees with the s.d.-based one. Both give an answer to your problem of 9.2%, not 5.6%.

All the best,

John

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(Dr) John D Salt