Thursday, September 27, 2018

Why Mathematical Models Break Down. And Why Our Life Depends On That-5. What Is Probable.

So, consider now a simple problem (such as it is taught in War Colleges in Russia) for a simple tank (or even ship) scenario of shooting at target. This is an actual problem and it is excellent for demonstrating how non-linear war is. Recall, we already know a little bit about Osipov-Lanchester equations, whose solution is quadratic. So here it is:

In a combat the blue tank (ours) detects red tank (bogey) and the blue crew is given an order to dispose of bogey red tank in three shots max. You may apply this to, for now, ship on ship artillery duel.  Here immediately comes a scientific assumption—we, for the simplicity of experiment, assume that we already know the actual probabilities of each of the three shots. Those probabilities will be described by a much complex interaction (will see how in Salvo Model) of crew's level of training, ballistic computer properties and myriad other things, which are not the point now. So, say we know that the Probability P1 of the first shot hitting the target (red bogey) is 0.6, consequently P2=0.75 and P3=0.85.  We also know our crucial Omega, ω, a mathematical expectation, or, speaking in layman's lingo—the weighted average of hits required for disabling such type of a target as our red bogey. Say, our ω=1.2. So, for this particular tactical task what will be the probability of killing the (red) bastard? It is warranted to say that Probability of this "kill" is also THIS very important and dominating parameter which defines what is known as a decisive element of any commanding decision, be it on one-on-one battle or in a very complex, multi-level operation—Criterion of Effectiveness. Criterion of Effectiveness is, most of the time, a probability of killing enemy SOB thus completing the task and attaining our objective(s).

The solution is very simple:

For this kind of task the Probability of a "kill" Pk from three shots will be:
 We plug in our numbers and get:


So, the probability of out tank killing the evil bastard is very high and pleasant 0.95. Good job! Death to occupants! 

Of course, one can also go exactly 180 degrees and using desired probability, say I want to defeat SOBs who deployed their MLRS launcher about to blow us up with the probability of Pd = 0.97, which is THE Criterion of Effectiveness, and get the number of required forces (tanks, ships, missiles etc.), aka in Russia as Naryad Sil (literally—forces which are "dressed"). Of course, in this case one will have to consider such things as probabilities of, say, tanks hitting the enemy with the first shot. So, say in this case, we need to count how many tanks we need to blow enemy's missile launcher (MLRS) up and then repulse enemy's counterattack. If we consider that the probability of hitting the target with the first shot will be the same (for simplicity of demonstration) for all tanks and is Ps = 0.45, we, using good ol' formula (in real life it will be much more expanded and complex one, we'll get to that too):         

Lower case n here is this number of tanks which we have to solve for.  Solution is easy:                  

After using logarithms (not crucial for now) we get our n=5.87 or, rounding it up, 6 tanks to do the mission. 

These are simple examples. But, as I stated, these examples are easily applied to other forces and here is the trick. These things, and much more, used to be done by staffs—they still are doing this and models and mathematics they operate is very complex. Nowadays these are computer battlefield networks which do most of this job but here is the deal—commanding officer, operational staff continue to matter immensely and especially on a purely human level. Greatest military minds seldom calculated themselves what is above and much more, but secret to their success was having those highly developed synapses which allowed them, very often without calculator or logarithmic ruler, see a larger tactical and operational framework. They had this tactical-operational non-linear intuition which helped them time after time achieve success in a seemingly completely chaotic business of war. From that, also, all Combat Manuals, Tactical and Operational procedures were written. It is true when they say that military manuals are written in blood. War is probabilistic in nature—always was—and that is what many laymen fail to recognize, that excellent tactical and operational level officers are developed in a tremendously rigorous mathematical (and physics) fundamental sciences, which allow them to proceed further into an extremely complex world of modern military technology and its combat use. Without understanding of modern high tech combat and being able to predict outcomes within highly non-linear combat framework no serious discussion is possible...

To Be Continued...

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