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      Theoretically, there can be two extreme cases of dependence between shot errors: the first case is when there is no common error source, i.e. no repeating component (μ = 0); the second case is when all error sources are common, i.e. all components are repeated as many times as the shot is fired (μ = 1). In the first case, the shots will be independent, and in the second case, the shots will be functionally dependent. With concentrated firing and independent shots, the ammunition hits into different points randomly spread around the center of aim, and with functionally dependent shots, all ammunition hits into the same point at a random distance from the center of aim.

      I.3.5 Probability of Damaging a Single Target

      The probability of damaging a single target W is one of the key indicators of combat effectiveness in the theory of firing, which underlies the calculation of many other indicators. A single target is damaged by one or more shots. Most methods of estimating the effectiveness of multiple shots are based on a preliminary determination of the probability of a single shot damaging the target, so the methods of calculating this probability greatly contribute to the simplicity and accuracy of estimating the effectiveness of the shooting in more complex cases.

      I.3.5.1 Damaging the Target with a Single Shot

      Let us first consider damaging the target with a single shot. The single‐shot probability of damaging the target W₁ is defined as the probability of a complex event, which consists of hitting the target and damaging it if the target is hit:

      (I.17)

      where p₁ is the probability of hitting the target with one shot; G(1) is the probability of damaging the target with one shot (the value of the conditional damage law G(m) at m = 1).

      With a known average required number of hits ω for a target that has no damage accumulation,

      (I.18)

The probability of hitting the target with one shot p₁ can be calculated as follows. If the target is projected on the dispersion plane by lines parallel to the direction of firing, then p₁ is the probability that the point of a hit on the picture plane will be within some region of the target's projection (Figure I.7). This probability is expressed with the formula

Figure I.7 Projecting of a target on a picture plane.

      Source: From Wentzel [2].

(I.19)

where φ(x, y) is the law of dispersion (I.11).

In some cases, the integral (I.19) is expressed through known functions. Let us give the corresponding calculation formulas.

      1 The target is a rectangle with sides parallel to the principle dispersion axes (Figure I.8).(I.20) where α, β, γ, δ – coordinates of target's boundaries on 0x and 0y axes; is the normalized function of Laplace:(I.21) (the values of this function for different values of the argument are given in the Appendix, Table A.2).Figure I.8 Coordinates of target boundaries.Source: From Wentzel [2].

      2 Target – circle, dispersion – circular (Ex = Ey = E). If the dispersion center coincides with the target center (there is no systematic error), the probability of hitting within the circle of radius rt is expressed by the formula(I.22) If the value and configuration of the vulnerable target area Sv are known, the probability of damaging the target can be calculated simply as the probability of hitting the vulnerable target area.Let us now consider what will change if the target is damaged by remote warheads. The probability of damaging the target, in this case, will be determined by the following formula:(I.23) where G(x, y) – coordinate law of damage.If the area of the specified damage zone Ssp and its specified sizes 2lx and 2ly are known, the probability of damaging can be defined simply as a probability of hitting the specified target area Ssp, for example, by the formula (I.20).Thus, whether by contact or remote ammunition, the probability of damaging a single target can always be calculated as the probability of hitting either a vulnerable area of the target Sv or the specified damage zone of the target Ssp .

      I.3.5.2 Damaging the Target with Multiple Shots

      When shooting with multiple shots, the probability of damaging the target is affected by the dependence of the shots due to repeated errors.

In the extreme case of functionally dependent shots (μ = 1) with concentrated firing, all ammunition hits the same point, and therefore, the probability of damaging the target with several shots W_n 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 W_i 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,

(I.25)

      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

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