(μ = 1) with concentrated firing, all ammunition hits the same point, and therefore, the probability of damaging the target with several shots Wn is equal to the probability of damaging with one shot W1:
The other extreme case of shooting is independent shots (μ = 0). In this case, the probability of damaging the target with n shots will be
(I.24)
where Wi is the probability of hitting the target with an i‐shot.
With the equal probability of damaging the target with each of the n independent shots,
Example
Five hunters shoot one duck at a time and independently of each other. Each hunter would kill a duck with a probability W1 = 0.2. How likely would all five hunters be to kill a duck?
Solution
Since the shots are independent, the probability can be calculated by the formula (I.25):
To calculate the probability of the target damage with dependent shots, the approximate methods of calculation are usually used, in which the dependence of shots is either ignored or taken into account approximately. One of the simplest methods of taking into account the dependence of the shots is to calculate the probability of damaging the target using the approximate formula
where
The relative error of determining the probability of damaging the target with this method at nW1 ≈ 0.5–5 does not exceed several percents.
Example
One of the hunters from the previous example takes five shots at a duck in a row. The correlation ratio between the shots is μ = 0.5. What's the equal probability of damaging the duck?
Solution
Shots are dependent. The value of nW1 = 5 ⋅ 0.2 = 1. So, we can use the formula (I.26). Let's put the calculated values from the previous example into the formula
The dependence of the shots always leads to a reduced probability of damaging the target compared to the case of independent shots. This effect is more pronounced with more shots fired and a bigger correlation ratio μ. With a small number of shots (n = 2–4) and relatively small correlation ratio values (μ < 0.5), the correction for the dependence of shots is small and the probability of damaging the target can be calculated as in the case of independent shots. If there are a large number of shots and their correlation is significant, correction for the dependence of shots should be taken into account.
An extremely important issue in efficiency theory is to establish a rational relationship between the group error of firing (Ex0, Ey0) and individual or technical dispersion (Bd, Bs). The optimal characteristics of the individual dispersion must be commensurate with the characteristics of the aiming error (Ex0 ≅ Вd); (Ey0 ≅ Вs). Unfortunately, this is not always the case in existing weapon systems; in particular, for artillery weapon systems and small‐caliber unguided aircraft missiles, the group error of firing often far exceeds the individual dispersion.
The optimal ratio between individual and group firing errors can be achieved either by increasing the aiming accuracy or by increasing the individual dispersion of the ammunition. In the practical case of a firing situation, the effectiveness of the firing was increased by up to five times when the technical dispersion was increased to its optimum value without increasing the aiming accuracy.
Another way to increase the effectiveness of the dependent shots, which is widely used for ground artillery, is to shoot with an artificial dispersion of the shots, which is achieved by shooting one target at several aiming positions.
I.3.6 Evaluation of the Effectiveness of Firing on a Group Target
Most often the task of shooting at a group target is to damage the largest possible number of units in the group. As an indicator of the effectiveness of firing in this case, the average number of damaged units from the group is used.
(I.27)
where the random value of Xd is the number of units damaged.
We will deduce the most general formula for Md, suitable for all cases without exception when shooting at a group target. For this purpose, let us consider a group target consisting of N units of T1, T2, …, TN (Figure I.9). This target is to be fired with any ammunition (contact or remote), consisting of any number of shots – dependent or independent, under constant or variable conditions. The aim may be to target individual units or a group as a whole.
Figure I.9 Group target.
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