**By Cameron Curtis, **

The answer is – at least two, probably three or more. Period.

When an Aegis destroyer fires six Standard 3 ABMs at 300 incoming Iranian missiles, it is *not* going to shoot down six of the threats. It will be lucky to shoot down three. It might shoot down two. On many occasions, it might not shoot down any.

The same goes for all our air defense missiles. Patriot PAC-3s, THAAD, Israeli Arrow 3. It is standard air defense doctrine to fire at least two, and often three air defense missiles, for each incoming ballistic missile threat.

Why?

The purpose of this short article is to show the mathematics (don’t be scared, I promise it will be painless) behind the doctrine. I will also show that the doctrine is fundamentally flawed because of a basic statistical truth: The Law of Large Numbers.

Think Las Vegas

The mainstream media makes you think we fire a Patriot and bring down an enemy missile. The Pentagon doesn’t help if they say they fired twelve Standard 3 missiles and brought down twelve Iranian ballistic missiles on October 1. *Of course* they want their weapons to look good. I don’t believe it for one simple reason. They *had* to target specific threats, and they wouldn’t have targeted incoming threats one-to-one. They would have targeted threats two or three-to-one. That’s standard doctrine. That means at best they would have shot down six. At best.

When you think about air defense, think about odds. It’s a game, like Las Vegas. Like the horse races. It’s gambling. How hard is shooting down an incoming missile? Our Anti-Ballistic Missiles (ABMs) are hit-to-kill. The experts say it’s like shooting a bullet with another bullet where both objects are racing toward each other at between six and ten miles per second closure. Can we do that *every* time? Really?

Let’s take the Navy’s Standard 3 ABM as an example. It’s the same argument for all the other missiles. Suppose you do one thousand tests where you fire an incoming missile and one Standard 3. Suppose out of all those tests (just suppose) you score 800 hits. That means you have an estimated hit probability of 800/1000 = 0.8 or 80%. That’s pretty good, and that’s what standard doctrine assumes.

Figure 2. Manning the Aegis Combat System

Now that means if you have an incoming Iranian missile and you fire one Standard 3, you only have a 0.8 or 80% chance of hitting it. An 80% chance of killing it. We say the hit probability is 0.80 and the kill probability is 0.80

If that incoming Iranian missile is carrying an A-Bomb, that leaves a 1.0 – 0.8= 0.20 or 20% chance you will miss. That means there is a 20% chance an A-Bomb is going to blow up Tel Aviv.

That’s not good enough.

You need to fire more than one Standard 3 interceptor missile. This is Las Vegas, remember? You can never be 100% sure you’ll stop that incoming missile. Would you be happy with 95%?

As it happens, there is some simple math that tells the Navy how many to fire to have a 95% chance of stopping that incoming missile. There are more realistic and complicated models, but I’m going to give you the simple Las Vegas one, because it’s actually pretty good at explaining air defense doctrine, and showing *the weaknesses* of air defense doctrine. Don’t be put off by the math, because I’ll do the math for you. Just try to understand the ideas.

Let’s call N the number of missiles we have to fire to achieve a desired Probability(Kill). Put another way, first we decide how sure we want to be of killing the threat and then we work out how many missiles we need to fire. Then:

Ln[1-Probability(Kill)]

N = —————————–

Ln[1-Probability(Hit)]

So, if we want to be 95% sure of killing the missile, Probability(Kill) is 0.95

And we are assuming a chance of one Standard 3 hitting the target is 0.80

Let me do the math for you. This means the number of Standard 3s we have to fire at a single threat is:

Ln[1-0.95]

N = ————- = 1.86135 which we can round up to 2 Standard 3 missiles

Ln[1-0.80]

That’s where the 2 air defense missiles for one threat comes from.

Is 95% kill probability good enough? Maybe we want to be more sure. For 99% kill probability, we use:

Ln[1-0.99]

N = ————- = 2.86135 which we can round up to 3 missiles

Ln[1-0.80]

That’s where standard air defense doctrine comes from. If you want a probability of kill between 95% and 99%, you fire two to three ABMs for each incoming threat. Notice that there is always a chance all three interceptors you fire will miss.

If you don’t understand the ideas in this article, and somebody tells you the rule of thumb is “Fire two or three for every threat,” that is what you will do.

If all you want to know is where standard doctrine comes from, read no further. But, as we’ve hinted, it is not that simple. If you want to know where standard doctrine goes wrong and why we are missing so many targets, read on.

Why are we missing so much? The issue of Realized Hit Probability

Remember our first experiment. We fire an incoming missile 1,000 times and fire an interceptor missile 1,000 times. We score 800 hits, so our probability of a hit is 800/1000 = 0.80 or 80%. With 1,000 trials, that’s a reasonably good estimate.

Each of these Standard 3 missiles costs about $20 million. That’s a lot of cash. What Raytheon does is conduct one test, then another, then another. They conduct maybe half a dozen tests. Let’s be generous and say they do two dozen. That’s nowhere near 1,000.

In statistics, there’s a principle called The Law of Large Numbers. In simple terms, it says that for your guess of the hit probability to be close to the real hit probability, you have to do lots and lots of tests. A thousand isn’t too many. Ten thousand is not too many. One thing is for sure. Two dozen is too little. What does that mean? It means your Estimated Probability [Hit] is not going to be very close to the Realized (True) Probability[Hit]. It means your 0.80 is going to be either too low, or too high. You don’t know. If there’s an A-Bomb coming, you don’t want it to be too high, but it probably is.

From what we’ve observed in Iran missile attack overwhelms Israeli air defense, the Realized Probability [Hit] is more like 0.25. If that is true, how many missiles should we fire to take down each incoming threat? Let’s apply the Las Vegas model again with Probability[Hit] = 0.25 or 25% and a desired Probability[Kill] = 0.99 or 99%.

Ln[1-0.99]

N = ————- = 16.0078 which we can round up to 16 Standard 3 missiles

Ln[1-0.25]

That’s what it’s going to take to knock down one incoming threat. How much will it cost? $20 million x 16 missiles = $320 million. That’s a lot of “How are ya’s.”

The more complicated model doesn’t make the results look any better. That’s why I say our simple Las Vegas model is good enough to understand the issues.

If the incoming missile is carrying an A-Bomb, it’s worth it. If it’s carrying two thousand pounds of RDX, certainly not.

But put yourself in the place of our officers. Our air defense was never meant to stop 300 or 1,000 Iranian ballistic missiles coming in at the same time. It was meant to stop a handful of Soviet ballistic missiles. Our officers, faced with a swarm attack, have no choice but to fall back on doctrine. They simply don’t have enough.

It gets worse: The Problem of Decoys

Suppose we assume our Estimated Probability[Hit] of 80% is close to what we would get if we did the 1,000 trial experiment. In other words, suppose it’s pretty good (which we know it is not).

There’s another way for the enemy to reduce the Realized Probability[Hit] to 0.25 and it is not hard. In a previous article, Defeating Anti-Ballistic Missile Defenses we discussed how easy it is for the enemy to deploy decoys to throw off our ABM interceptors. They can either separate the warhead from the booster stage and slice the booster into pieces, or they can deploy balloons. In space, where there is no air resistance, balloons will fly along with the warhead at the same speed and on the same trajectory. The balloons can be of any size or reflectivity we desire.

Figure 3. In space, balloons travel at the same speed and on the same trajectory as a warhead.

Let’s say we deploy three decoys. They appear to the seeker of the ABM’s kill vehicle exactly the same as the warhead. That means that when we fire one missile at one threat, and that threat deploys three decoys, our missile will suddenly be confronted with four threats, of which only one is real. It won’t know which is the real threat.

Figure 4 shows what our ABM will see:

Figure 4. Our interceptor sees a warhead and three decoys.

Our interceptor can’t tell the warhead from the decoys. There is AI we can program to help, but there is a way to defeat every AI rule. For example, balloons can be made as reflective as we like or as large or small as we like compared to the size of the warhead. The chances of our interceptor hitting the correct threat are immediately reduced to one-in-four, or 0.25 or 25%.

And it can get even worse.

The last example assumed our Estimated Probability[Hit] of 0.80 was close to true. If, in fact, Realized Probability [Hit] is 0.25 and the enemy deploys three decoys as above, the Realized Probability [Hit With Decoys] = 0.25 * 0.25 = 0.0625 or 6.25%.

How many missiles do we need to have a 99% chance of bringing down a threat that deploys 3 decoys? Let’s see what the Las Vegas model tells us:

Ln[1-0.99]

N = —————– = 71.3554 which we can round to 71 missiles

Ln[1-0.0625]

In other words, forget about it. An Arleigh Burke has only 90 VLS cells. A THAAD battery has 48 missiles. If the Iranians fire 300 or 1,000 missiles, each deploying three decoys, it’s all over. If they slip half a dozen A-Bombs in with the conventional warheads, we face a catastrophe.

Evidence for this is all around us because the wars in Ukraine and Israel are providing live experiments. Of course, the hit data collected by the military is classified, but we can follow the evidence of our eyes. We’ve seen video of Iranian missiles raining on Israeli targets – unopposed. Every day, we are treated to Ukraine’s requests for more air defense systems and missiles. They are obviously running out. Russia has been firing two thousand missiles a year at Ukraine, and they are getting through. Today, Ukraine has almost no air defense left.

Conclusion

Thinking about these issues from first principles is rather sobering. Our ballistic missile defense was never designed to defend against enemy missile swarms.

We need to be very selective about which threats we engage.

For example, Iron Dome is useless against ballistic missiles. We should hold back Iron Dome missiles for use against drones. Patriot PAC-3 will not be able to cope with modern IRBMs, especially if they deploy decoys. Patriot PAC-3, however, is *extremely* effective against aircraft and drones. We should hold back Patriot PAC-3 interceptors to deal with cruise missiles, which are the Iranians’ most accurate weapons. Their cruise missiles are the biggest threat to our high-value targets.

There isn’t much to say about their ballistic missiles. Russian ballistic missiles are highly accurate, Iranian missiles less so. We can do the best we can with THAAD, Arrow 3, and Standard 3. We have to be as selective as possible, and those decisions have to be left to our commanders in the field. They will be the ones with the best situational awareness.

Interested readers with a bit of Excel skill can put the Las Vegas model on a spreadsheet and play with it. This is not just theory. It is the practical reason our air defenses are struggling with enemy missile attacks.

This essay first appeared in SOFREP

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About the Author

Cameron Curtis

*You can reach the author at: **[email protected]*