1. L Berkley Davis Copyright 2009 MER301: Engineering Reliability1 LECTURE 1: Basic Probability Theory

2. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 2 Summary-Probability  Basic Definitions Random Experiments, Outcomes, Sample Spaces, Events  Probability Properties Limits and Definitions  The Laws of Chance Classical Probability Relative Frequency Definition Subjective or Bayesian Probability  Probability Rules Addition/Multiplication Conditional Probabiity

3. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 3 Probability  Probability is used to quantify the likelihood, or chance, that the outcome of an “event” falls within some specified range of values of a specified random variable Random variable can be discrete or continuous  Probabilities are used in many fields of both everyday and professional life Weather forecasting,insurance, investing, medicine, genetics, “can I make that green light” Reliability analysis, safety analysis, strength of materials, quantum mechanics, commercial guarantee policies

4. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 4 Probability- Basic Definitions  Random Experiment An experiment that can result in different outcomes when repeated in the same manner  Outcomes of a Random Experiment Outcome-single result of a random experiment, Elementary Outcomes- all possible results of a random experiment  Sample Space Set or collection of all of the elementary outcomes  Event A collection of outcomes A that share a specified characteristic Complement of an event A is an event comprising all outcomes not belonging to A (not A)

5. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 5 A Random Experiment – Example 1.1  Coin Toss Exercise- flip 3 times and record the results Outcome Sample Space Event Properties of Probability

6. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 6 Probability –Example 1.1(Solution)  Coin Toss Exercise- flip 3 times and record the results Outcome  A single sequence of Heads/Tails(HTT,etc) Sample Space  The eight possible Outcomes from three coin flips Event  The collection of outcomes with,eg, at least one head Properties of Probability-for three coin flips

7. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 7 Properties of Probability  Probability of any particular Elementary Outcome or Event is greater than or equal to zero and is less than or equal to one Probability is always non-negative If an outcome/event cannot occur, probability is zero If an outcome/event is certain to occur,its probability is one  Sum of the Probabilities of all the possible Elementary Outcomes of a Random Experiment is equal to one(All possible Elementary Outcomes by definition equal the Sample Space)

8. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 8 Properties of Probability  The probability that an Elementary Outcome/Event will occur is one minus the probability that the Elementary Outcome/Event will not occur Complement of an event A is an event comprising all outcomes not belonging to A (not A). For the 3 coin toss Example 2.1

9. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 9 Probability- Basic Definitions  Random Experiment An experiment that can result in different outcomes when repeated in the same manner  Outcomes of a Random Experiment Outcome-single result of a random experiment, Elementary Outcomes- all possible results of a random experiment  Sample Space Set or collection of all of the elementary outcomes  Event A collection of outcomes A that share a specified characteristic Complement of an event A is an event comprising all outcomes not belonging to A (not A)

10. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 10 Sample Spaces/Populations  A Sample Space or Population is the set of all possible values of a random variable, called the elementary outcomes, for a given experiment A Sample is a subset of a Sample Space  Definition of a Sample Space depends on what characteristic is to be observed  Types of Sample Spaces Finite Countable Uncountable

11. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 11 Example 1.2 – Finite Sample Space  Consider the random experiment of tossing a coin three times and recording the results  Two of the possible sample spaces for this experiment are The exact sequence of heads (H) and tails (T) in each outcome  S={TTT, TTH, THT, HTT, HHT, HTH, THH, HHH} The number of heads (or tails) in each outcome  S={0, 1, 2, 3}  Binomial Distribution applies to this case

12. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 12 Example1.3–Countable Sample Spaces  The random experiment consists of rolling a dice until a 6 is obtained(so a 6 is obtained by definition in each outcome)  Two of the possible sample spaces are The exact values on the dice in each outcome  S={6, 16, 26, 36, 46, 56, 116, 126, 136, 146, 156, 216, …}  If N=1,2,3,4,5 then S={6, N6, NN6,…} The number of throws needed to get a 6 in each outcome  S= {1,2,3,4, …}

13. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 13 Example1.4 Uncountable Sample Space  The experiment consists of throwing a dart onto a circular dart board marked with three concentric rings. inner ring is worth 3 points middle ring worth 2 points outside ring worth 1 point  Describe two possible Sample Spaces One Sample Space is the exact location of the dart in each outcome  S={(r,  )|0  2 , 0  r  R}  The r and theta distributions are continuous A second is the number of points scored in each outcome  S={1,2,3}

14. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 14 Probability- Basic Definitions  Random Experiment An experiment that can result in different outcomes when repeated in the same manner  Outcomes of a Random Experiment Outcome-single result of a random experiment, Elementary Outcomes- all possible results of a random experiment  Sample Space Set or collection of all of the elementary outcomes  Event A collection of outcomes A that share a specified characteristic Complement of an event A is an event comprising all outcomes not belonging to A (not A)

15. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 15 Probability- Definitions of an Event  Union of two events A and B is an event comprising all outcomes in A or B or both (A or B)  Intersection of two events A and B is an event comprising outcomes common to both A and B (A and B)  Empty Event (Null Set) is one containing no outcomes  If A and B have no outcomes in common then they are Mutually Exclusive or Disjoint

16. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 16 Non-Constant Probability - Example 1.5  Consider a group of five potential blood donors – A, B, C, D, and E – of whom only A and B have type O + blood. Five blood samples, one from each individual, will be typed in random order until an O + individual is identified. Let X=the number of typings necessary to identify an O + individual Determine the probability that an O + individual will be identified in three typings

17. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 17 Summary of Probability Properties  Rule 1 The probability of an elementary outcome/event will be a number between zero and one  Rule 2 If an elementary outcome/event cannot occur, the probability is zero  Rule 3 If an elementary outcome/event is certain, the probability is one  Rule 4 The sum of probabilities of all the elementary outcomes or events in a sample space is one  Rule 5 The probability an elementary outcome/event will not occur is one minus the probability that the outcome/event will occur

18. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 18 Summary of Probability Properties-Text  Rule 1 The probability on an elementary outcome/event will be a number between zero and one  Rule 2 If an elementary outcome/event cannot occur, the probability is zero  Rule 3 If an elementary outcome/event is certain, the probability is one  Rule 4 The sum of probabilities of all the elementary outcomes or events in a sample space is one  Rule 5 The probability an elementary outcome/event will not occur is one minus the probability that the outcome/event will occur

19. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 19 Probability-what is it?  Probability is used to quantify the likelihood, or chance, that the outcome of an “event” falls within some specified range of values of a specified random variable Random variable can be discrete or continuous  Probabilities are used in many fields of both everyday and professional life Weather forecasting,insurance, investing, medicine, genetics, “can I make that green light” Reliability analysis, safety analysis, strength of materials, quantum mechanics, commercial guarantee policies

20. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 20 Probability:The Laws of Chance….  Objective Definitions Classical Probability assumes the game is “fair” and all elementary outcomes have the same probability Relative Frequency Probability of a result is proportional to the number of times the result occurs in repeated experiments  Subjective or Bayesian Definition Bayesian Probability is an assessment of the likelihood of the truth of each of several competing hypotheses,given data and some additional assumptions.

21. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 21 Probability Rules  Addition ( A or B) A and B are mutually exclusive A and B are not mutually exclusive  Multiplication( A and B) A and B are independent A and B are not independent/conditional probability

22. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 22 Dice Example 1.6  36 Elementary Outcomes Probability of a specific outcome is 1/36 Probability of the event “sum of dice equals 7” is 6/36  Addition-P(A or B) Events “sum=7” and “sum=11” mutually exclusive P=(6/36+2/36) Events “sum=7” and “dice 1=3” not mutually exclusive P=(11/36)  Multiplication-P(A and B) Events A=(6,6) and B=(6,6 repeated) are independent P=(1/36) 2 Event A= (6,6) given B=(n 1 =n 2 =even) are not independent P=(1/3)

23. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 23 Dice Example 1.6-Probabilities  For the dice experiment, determine the probability for each event (A, B, C)  A = {the sum on the two dice is 6}  B = {both dice show the same number}  C = {at least one of the faces is divisible by 2}  The Sample Space and Events are given by  For the Sample Space S, N=36 and the probability of an event is the number of outcomes in that event divided by N A=5/36 B=6/36 C=27/36

24. L Berkley Davis Copyright 2009 The Dice Game of Craps and Probability Rules  Player has two Dice  1 st Roll 7 or 11 wins immediately 2,3, or 12 loses immediately Rolls of 4,5,6,8,9,10 Continue  Subsequent Rolls Continue to roll until get the same number as on 1 st roll (win) or a 7 (lose)  Probability of Winning Overall MER301: Engineering Reliability Lecture 1 24

25. L Berkley Davis Copyright 2009 The Dice Game of Craps  Define the following probabilities Let A= probability of 7 or 11 on 1 st roll Let B= probability of 4 on 1 st roll and then another 4 before a 7 Let C= probability of 5 on 1 st roll and then another 5 before a 7 Let D= probability of 6 on 1 st roll and then another 6 before a 7 Let E= probability of 8 on 1 st roll and then another 8 before a 7 Let F= probability of 9 on 1 st roll and then another 9 before a 7 Let G= probability of 10 on 1 st roll and then another 10 before a 7  These are mutually exclusive events so the probability of winning is or  The House always wins… MER301: Engineering Reliability Lecture 1 25

26. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 26 Probability Rules - Conditional Probability  The probability of A given that B has already occurred is called a conditional probability  Conditional Probability is calculated from  This can be written as  If A and B are Independent then so that

27. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 27 Dice Example 1.6  36 Elementary Outcomes Probability of a specific outcome is 1/36 Probability of the event “sum of dice equals 7” is 6/36  Addition-P(A or B) Events “sum=7” and “sum=10” mutually exclusive P=(6/36+3/36) Events “sum=7” and “dice 1=3” not mutually exclusive P=(11/36)  Multiplication-P(A and B) Events A=(6,6) and B=(6,6 repeated) are independent P=(1/36) 2 Event A= (6,6) given B=(n 1 =n 2 =even) are not independent

28. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 28 Medical testing and False positives..  A certain disease affects 1 out of every 1000 people. There is a test that will give a positive result 99% of the time if an individual has the disease. It will also show a positive result 2% of the time for individuals who do not have the disease.  If you test positive, what is the probability that you have the disease?

29. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 29 Medical testing and False positives….What do we know? What do we want to know?  Define the events as A : person has the disease B : person tests positive  The known information can be written as 1/1000 has the disease Probability of a positive result for a person with the disease Probability of a false positive for a person without the disease  We want to know

30. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 30 Medical testing and False positives….  Sample space is four mutually exclusive events…

31. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 31 Medical testing and False positives….  Sample space is four mutually exclusive events..the known quantities are

32. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 32 Medical testing and False positives….  The rows and columns must add up so

33. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 33 Medical testing and False positives….  The probability of actually having the disease given a positive test is then

34. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 34 Medical testing and False positives….  Even though the test is accurate,less than 5% of those who test positive actually have the disease. This “False Positive Paradox” is one reason repeat or alternative medical tests are often required to establish if a person really has a particular disease.

35. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 1 35 Summary-Probability  Basic Definitions Random Experiments,Outcomes,Sample Spaces,Events  Probability Properties  The Laws of Chance Classical Probability Relative Frequency Definition Subjective or Bayesian Probability  Probability Rules Addition/Multiplication Conditional Probabiity