1. MEGN 537 – Probabilistic Biomechanics Ch. 1 – Introduction Ch
MEGN 537 – Probabilistic Biomechanics Ch.1 – Introduction Ch.2 – Mathematics of Probability
Anthony J Petrella, PhD

2. Ch.1 - Introduction

3. Uncertainty Uncertainty present in physical systems
Repeated measurement yields variability
Dimensional tolerances, respiration rate, tissue material properties, joint loading, etc.
What impact does this uncertainty have on performance?

4. Strength-Based Reliability
Safety factor shows acceptable design
Some percentage of the time, stress may exceed strength

5. Reliability Definitions
Probability of Failure
POF = 0.001,
Probability of Survival or Reliability
Reliability = (three 9s), (four 9s)
POF + POS = 1

6. Reliability-based Design
Design for Six Sigma
Concept developed by Bill Smith in 1993
Motorola owns six sigma trademark
Six sigma corresponds to 3.4 failures per 1,000,000
POF = 0.000,003,4 or Reliability = 0.999,997,6
Design Excellence or BlackBelt programs
Many companies have implemented their versions
GE and Honeywell boast 100s of millions of dollars saved

7. Uncertainty Sources of uncertainty Inherent / repeated measurement
Statistical uncertainty – limited availability of sampling size means actual distribution unknown
Modeling uncertainty – how good is the model?
Cognitive or qualitative sources – intellectual abstraction of reality, human factors

8. Course Objectives
Ability to understand and apply probability theory and probabilistic analysis methods
To assess impact of uncertainty in parameters (inputs) on performance (outcomes)
Determine the appropriate distribution to represent a dataset
To apply this knowledge to real biomechanical systems

9. Ch.2 - Mathematics of Probability

10. Definitions Probability: The likelihood of an event occurring
Event: Represents the outcome of a single experiment (or single simulation)
Experiment: An occurrence that has an uncertain outcome (die toss , coin toss, tensile test) – usually based on a physical model
Simulation: An occurrence that has an uncertain outcome – usually based on an analytical or computational model

11. Example – Coin Toss
OR = add, AND = multiply
If you flip a coin two times, what is the probability of:
seeing “heads” one time?
seeing “heads” two times?

12. Example – Coin Toss
OR = add, AND = multiply
If you flip a coin two times, what is the probability of :
seeing “heads” one time?
option 1: heads (0.5) AND tails (0.5) = 0.25 option 2: tails (0.5) AND heads (0.5) = 0.25 option 1 OR option 2 = = 0.5
seeing “heads” two times?
option 1: heads (0.5) AND heads (0.5) = 0.25

13. Example – TKR Casting
A knee implant casting process is known to produce a defective part 5% of the time If 10 castings were tested, find the probability of:
a) no defective parts
b) exactly one defective part
c) at least one defective part
d) no more than one defective part

14. Permutations & Combinations
Number of permutations of r objects from a set of n distinct objects (ordered sequence)
Number of combinations in which r objects can be selected from a set of n distinct objects
n objects taken r at a time
Independent of order

15. Example – Answers Must consider combinations for each # of defects
Probability
Combinations 1 2 3 4 5 6 7 8 9 10
0.95
0.05 45
120
210
252
E-08
E-09
E-11
E-14
Sum
1.00

16. Example - Answers no defective parts b) exactly one defective part
P(0 defects) = P(part 1 no defect)*P(part 2 no defect)… = (1-0.05)^10 = 0.598
P(1 defect) = P(part 1 defect)*P(part 2 no defect)…
= (0.05)*(0.95)^9 *10 = 0.315

17. Example - Answers
c) at least one defective part d) no more than one defective part
P(≥ 1 defect) = P(1defect) + P(2 defects) + P(3 defects)…
= 1- P(0 defects) = = 0.402
P(≤ 1 defect) = P(0 defects) + P(1 defect)
= = 0.913

18. Definitions
Sample Space (S): The set of all basic outcomes of an experiment
Mutually Exclusive: Events that preclude occurrence of one another
Collectively Exhaustive: No other events are possible S A B

19. Probability Relations
Experimental outcomes can be represented by set theory relationships
Union: A1A3, elements belong to A1 or A3 or both
P(A1A3) = P(A1) + P(A3) - P(A1A3) = A1+A3-A2
Intersection: A1A3, elements belong to A1 and A3
P(A1A3) = P(A3|A1) * P(A1) = A2 (multiplication rule)
Complement: A’, elements that do not belong to A
P(A’) = 1 – P(A) S

20. Special Cases If the events are statistically independent
P(AB) = P(B|A) * P(A) = P(B) * P(A)
If the events are mutually exclusive
P(AB) = 0
P(AB) = P(A) + P(B) - P(AB)
= P(A) + P(B) S A B

21. Example
For a randomly chosen automobile: Let A={car has 4 cylinders} B={car has 6 cylinders}. Since events are mutually exclusive, if B occurs, then A cannot occur. So P(A|B) = 0 ≠ P(A).
If 2 events are mutually exclusive, they cannot be independent…when A & B are mutually exclusive, the information that A occurred says something about B (it cannot have occurred), so independence is precluded

22. Rules of Set Theory Commutative: AB = BA, AB = BA
Associative: (AB)C = A(BC)
Distributive: (AB)C = (AC)(BC)
Complementary: P(A) + P(A’) = 1
de Morgan’s Rule:
(AB)’ = A’B’ Complement of union = intersection of complements
(A  B)’ = A’  B’ Complement of intersection = union of complements

23. Conditional Probability
The likelihood that event B will occur if event A has already occurred
P(AB) = P(B|A) * P(A)
P(B|A) = P(AB) / P(A)
Requires that P(A) ≠ 0
Multiplication Rule:
P(AB) = P(A|B) * P(B) = P(B|A) * P(A) S A B

24. Example
Common knee injuries include: PCL tear (A), MCL sprain (B), meniscus tear (C)
Injury statistics as reported by epidemiology literature:
a) What does the Venn Diagram look like?
Injury A B C
A  B
A  C
B  C
A  B  C
Probability
0.14
0.23
0.37
0.08
0.09
0.13
0.05

25. Example
Injury A B C
A  B
A  C
B  C
A  B  C
Probability
0.14
0.23
0.37
0.08
0.09
0.13
0.05 A B C S
0.02
0.03
0.04
0.07
0.05
0.08
0.20
0.51

26. Example A B C S
0.02
0.03
0.04
0.07
0.05
0.08
0.20
0.51
b) What is the probability that a patient with an MCL sprain (B) will later sustain a PCL tear (A)?
P(A|B) = P(A  B) = = P(B)

27. Example A B C S
0.02
0.03
0.04
0.07
0.05
0.08
0.20
0.51
c) If a patient has sustained either an MCL sprain (B) or a meniscus tear (C) or both, what is the probability of a later PCL tear (A)?
P(A| BC) = P(A  (BC) = = P(BC)

28. Example
d) If a patient has sustained at least one knee injury in the past, what is the probability he will later tear his PCL?
P(A|at least one) = P(A | ABC) = P(A  (ABC) P(ABC)
P(A  (ABC) = P(A) = = P(ABC) P(ABC)