This document provides an overview of probability theory and its applications in defense modeling. It begins with the evolution of probability from its origins in gambling to its modern applications. It then defines key probability concepts like random experiments, sample spaces, and events. The rest of the document discusses probability distributions and operations, conditional probabilities, Bayes' theorem, and examples of probability applications in defense modeling like detection modeling, damage estimation, and land mine modeling. Validation approaches for statistical models are also covered.

1. § Evolution of Probability
§ Basic Definitions
§ Characteristics of Probability
§ Approaches for Defining Probability
§ Basic Operations
§ Conditional Probabilities
§ Independence And Multiplication Rule
§ Bayes Theorem
§ Defense Application
PROBABILITY THEORY
Dr. S. k. das
Email:- skdas@issa.drdo.in, ph: 9811525928

2. Evolution of PROBABILITY
OTHING IN LIFE IS CERTAIN. IN EVERYTHING WE DO,
WE GAUDGE THE CHANCES OF SUCCESSFUL OUTCOMES,
FROM BUSINESS TO MEDICINE TO THE weather. BUT FOR
MOST OF HUMAN HISTORY, PROBABILITY, THE FORMAL STUDY OF
CHANCE WAS USED FOR ONLY ONE THING: Gambling
N
1

3. CHEVALIER DE MERE
FOUR SIDED ASTRAGAli Roman Emperor CLAUDIUS
BLAISE PASCAL (1623-1666)
2
Evolution of PROBABILITY

4. BASIC DEFINITIONS
AS OUR GAMBLER PLAYS A GAME, WE PLAY SCIENTIST,
OBSERVING THE OUTCOME:
A RANDOM EXPERIMENT : IS THE
PROCESS OF OBSERVING THE OUTCOME OF A
CHANCE EVENT.
THE ELEMENTARY OUTCOMES ARE ALL
POSSIBLE RESULTS OF THE RANDOM
EXPERIMENT.
THE SAMPLE SPACE IS THE SET OR
COLLECTION OF ALL THE ELEMENTARY
OUTCOMES.
3

5. SAMPLE SPACE
ONE DIE
PAIR OF DICE PAIR OF SIX DICE
4

6. 5

7. CHARACTERISTICS OF PROBABILITY
6

8. LIKE A CLEVER POLITICIAN,
WE HAVE AVOIDED CERTAIN
UNPLESENT QUESTIONS, SUCH AS
A) WHAT DOES PROBABILITY
MEAN? AND
B) HOW DO WE ASSIGN
PROBABILITIES TO OUTCOMES?
7

10. EVENT
9

11. 10

12. 11

13. SUBTRACTION RULE
12

14. 9 13

15. 14

16. 15

17. 16

18. 17

19. 18

20. 19

21. 20

22. 21

23. 22

24. 23

25. 24

26. 25

27. 26

28. 27

29. 28

30. 30 29

31. 1 30

32. 1 31

33. 1 32

34. 1 33

35. Case Studies

36. Detection model
TO DEVELOP MODELS FOR DETECTION AND ACQUISITION OF TARGETS (i.e.
AGGREGATED COMBAT ENTITIES :- COY, Bn, Sqn., Rgt, etc.)
Single Soldier
Multiple Soldiers
Pd=0.7, m=2 km, L= Day Light, s = clear sky
B=Desert, a=naked eye, LOS=10km
Pd=0.7, m= 5 km, L= Day Light, s = clear sky
B=Desert, a=naked eye, LOS=10km

37. Random experiments

38. Detection Probability Computation of Different Military Targets
1 35

39. Detection of pers in coy.
Coy. Against Coy. In Daylight and Desert Condition

40. detection of veh. In coy
Coy. Against Coy. In Daylight and Desert Condition

41. Detection of pers in coy.
Coy. Against Coy. In New Moon and Desert Condition

42. implementation

43. implementation

44. ArtY Fire: Damage Estimation

45. Deflection
Range
Aim Point
Fall of Shot
Deflection error
Rangeerror
Firing
Direction
Line Error & Deflection Error

46. Deflection
Range
Target
Fall of Shots
Simulation Of Fall Of Shots

47. Target Impact Point
y = v
Damage To
the target
Distribution of
Damage Function
c(y)
Distribution of
Fall of Shots
g(y)
c(y1)
c(yn)
E[c(y)]
Fall of Shots And Damage
Function

48. Area target
y1 = v1
D/2-D/2
y2 = v2
y3 = v3
Impact Points
For Area Target

49. ∫
=
−=
−
2/
2/
1)(
Dy
Dy
dyvyc ∫
=
−=
−
2/
2/
2 )(
Dy
Dy
dyvyc ∫
=
−=
−
2/
2/
3 )(
Dy
Dy
dyvyc
Total Damage to the Area Target for
Three Impact points
∫ ∫
+∞=
−∞=
=
−= ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−=
v
v
Dy
Dy
dvvgdyvyc
D
)()(
1
2/
2/
∫
+∞=
−∞=
==
v
v
DDRange dvvgccEEFD )()][(
CD1
___
D
CD2
___
D
CD3
___
D
Area Equivalent
AverageDamage
Depth

50. -40 250 30
Target’s Depth=65
,
332
1
exp
332
1
)(
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
×
=
x
xf
π
Accuracy, N(0,33)
)
10
)30(
exp()10,30,( 2
2
⎥
⎦
⎤
⎢
⎣
⎡ −
−=
x
xc
Damage Function, Carleton (30,10)
dx
x
Area ∫− ⎥
⎦
⎤
⎢
⎣
⎡ −
−=
25
40 2
2
)
10
)30(
exp(
dzz∫
−
−
−=
5.
7
2
)exp(10
=10*[(1-erf(.5))/2-(1-erf(7))/2]*sqrt(pi)
=10*[0.2398 - 0 ] *1.7725
= 10* 0.2398 *1.7725
= 2.398 *1.7725 = 4.2505
Hence, Average Damage =
5460.0
65
2505.4_
===
Depth
Areatotal
cD
332
1
exp
332
1
0.0654)(
2
100
100
100
100
dx
x
dxxfcEFD Dx
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
×
×== ∫∫
+
−
+
−
π
0.0652997525.0.0654)exp(332
332
1
5460.0
14.2
14.2
2
=×=−××
×
×= ∫
+
−
dzzEFDx
π
0042.00652.00652.0 =×=×= EFDEFDEFD yx

51. Hit Probability
Estimation

52. Elevated firing positions also increase the first-round hit probability as shown.
Firing down at a tank from an angle of 20 degrees increases the chance of a
hit by 2/3 at 200 meters. A 45-degree angle doubles the first-round
probability of a hit when compared to a ground level shot.
HIT Probability Computation of tanks
1 34

53. Land Mines model

54. V&V OF
MINES MODEL
Rod Forma=ons
Reducing ENME
Assault Forma=on
Each target in Rod Fn
Sta=s=cal Valida=on
Sta=s=cal Valida=on Sta=s=cal Valida=on
§ Expected Number of
Mines Encountered
§ Expected Frac=onal
Damage
§ Conflic=ng Situa=ons
§ Deadlock Situa=ons
An=-pers Fragmntn Mine
§ Kill Probability
§ Expected Number of Mines
Encountered (ENME)
f ( MD, Tw, Mr )
§ Forma=on
§ Rounding Off
An=-tank Influence Mine
§ Hit Probability
f ( ML ,Mw ,TW , Tl , Mr)
§ Expected Number of
Mines Encountered
§ Forma=on
§ Rounding Off
An=-Pers Blast Mine
MD Casualty
Poisson
(4/1,4)
Gen Pareto
(k=0.19,σ=0.15,
µ=0.16)
Discrete
Uniform
(1,4/1,4)
Gen Pareto
(k=-0.93, σ=16.1,
µ=11.7)
Land Mine Model
Logical Verifica=on Logical Verifica=on Logical Verifica=on
MD Casualty
Poisson
(4/6)
Gen Extr Value
(k=6.45,σ=5.43,
µ=5.43)
Discrete
Uniform
(1,4/6)
Gen Extr Value
(k=-78.93, σ=75.4,
µ=64.8)
MD Casualty
Poisson
(4/1,8)
Inverse Gaussian
(λ=0.34, µ =0.37,
γ=0.74)
Discrete
Uniform
(2, 4/ 1,8)
Inverse Gaussian
(λ=-4.96, µ=47.1,
γ=85.7)
BM IM FM

55. Questions?

56. Thank You