Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Chabot Mathematics §11.1 Probability& Random-Vars Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 10.3 Review § • Any QUESTIONS About • §10.3 Power & Taylor Series • Any QUESTIONS About HomeWork • §10.3 → HW-19

3. §11.1 Learning Goals • Define outcome, sample space, random variable, and other basic concepts of probability • Study histograms, expected value, and variance of discrete random variables • Examine and use geometric distributions

4. Random Experiment • A Random Experiment is a PROCESS • repetitive in nature • the outcome of any trial is uncertain • well-defined set of possible outcomes • each outcome has an associated probability • Examples • Tossing Dice • Flipping Coins • Measuring Speeds of Cars On Hesperian

5. Random Experiment… • A Random Experiment is an action or process that leads to one of several possible outcomes. • Some examples:

6. OutComes, Events, SampleSpace • OutCome → is a particularresult of a Random Experiment. • Event → is the collection of one or more outcomes of a Random Experiment. • Sample Space → is the collection or set of all possible outcomes of a random experiment.

7. Example  OutComes, etc. • Roll one fair die twice and record the sum of the results. • The Sample Space is all 36 combinations of two die rolls

8. Example  OutComes, etc. • One outcome: 1st Roll = a five,2nd Roll = a two→ which canbe represented by theordered pair (5,2) • One Event (or Specified Set of OutComes) is that the sum is greater than nine (9), which consists of the (permutation) outcomes (6,4), (6,5), (6,6), (4,6), and (5,6)

9. Random Variable • A Random Variable is a function X that assigns a numerical value to each outcome of a random experiment. • A DISCRETE Random Variable takes on values from a finite set of numbers or an infinite succession of numbers such as the positive integers • A CONTINUOUS Random Variable takes on values from an entire interval of real numbers.

10. Probability • Probability is a Quotient of the form • Example: Consider 2 rolls of a Fair Die • Probability of (3, 4) • Probability that the Sum > 9

11. Probability OR (U) vs AND (∩) • The Sum>9 is an example of the OR Condition. • The OR Probability is the SUM of the INDIVIDUAL Probabilities • The AND Probability is the MULTIPLICATION of the INDIVIDUAL Probabilities

12. AND Probability • Probability  The LIKELYHOOD that a Specified OutCome Will be Realized • The “Odds” Run from 0% to 100% • Class Question: What are the Odds of winning the California MEGA-MILLIONS Lottery? Exactly! 258 890 085 : 1

13. 258 890 085 ... EXACTLY???!!! • To Win the MegaMillions Lottery • Pick five numbers from 1 to 75 • Pick a “MEGA” number from 1 to 15 • The Odds for the 1stping-pong Ball = 5 out of 75 • The Odds for the 2ndping-pong Ball = 4 out of 75, and so On • The Odds for the MEGA are 1 out of 15

14. 258 890 085... Calculated • Calc the OverAll Odds as the PRODUCT of each of the Individual OutComes(AND situation) • This is Technically a COMBINATION

15. 258 890 085... is a DEAL! • The ORDER in Which the Ping-Pong Balls are Drawn Does NOT affect the Winning Odds • If we Had to Match the Pull-Order: • This is a PERMUTATION

16. Probability Distribution Function • A probability assignment has been made for the Sample Space, S, of a Particular Random Experiment, and now let X be a Discrete Random Variable Defined on S. Then the Function p such that:for each value x assumed by X is called a Probability Distribution Function

17. Probability Distribution Function • A Probability Distribution Function (PDF) maps the possible values of xagainst their respective probabilities of occurrence, p(x) • p(x) is a number from 0 to 1.0, or alternatively, from 0% to 100%. • The area under a probability distribution function Curve or BarChart is always 1 (or 100%).

18. x p(x) 1 p(x=1)=1/6 2 p(x=2)=1/6 3 p(x=3)=1/6 4 p(x=4)=1/6 5 p(x=5)=1/6 6 p(x=6)=1/6 Discrete Example: Roll The Die 1/6 1 2 3 4 5 6

19. Example B-School Admission • A business school’s application process awards two points to applications for each grade of A, one point for each grade of B or C, and zero points for lower grades • If Each category of grades is equally likely, what is the probability that a given student meets the admission requirement of five total points from grades from 3 different courses?

20. Example B-School Admission • SOLUTION: • The sample space is the set of 27 outcomes (using “A” to represent a grade of A, “B” to represent a B or C, and “N” to represent a lower grade) • The Entire Sample Space Listed:

21. Example B-School Admission • The event “student meets admission requirement of five points” consists of any outcomes that total at least five points according to the given scale. i.e. the outcomes • This acceptance Criteria Thus has a Probability

22. Expected Value • TheEXPECTED VALUE (or mean) of a discrete Random Variable, X,with PDF p(x) gives the value that we would expect to observe on average in a large number of repetitions of the experiment • That is, the Expected Value, E(X) is a Probability-Weighted Average, µX

23. Example  Coin-Tossing µX • A “friend” offers to play a game with you: You flip a fair coin three times and she pays you $5 if you get all tails, whereas you pay her $1 otherwise • Find this Game’s Expected Value • SOLUTION: • The sample space for the experiment of flipping a coin three times:

24. Example  Coin-Tossing µX • The expected value is the sum of the product of each probability with its “value” to you in the game: • Since Each outcome is equally likely, All the Probabilities are 1/8=0.125=12.5%:

25. Example  Coin-Tossing µX • Calculating the Probability Weighted Sum Find • Thus, in the long run of playing this game with your friend, you can expect to LOSE 25¢ per 8-Trial Game

26. Discrete Random Var Spread • The Expected Value is the “Central Location” or Center of a symmetrical Probability Distribution Function • The VARIANCE is a measure of how the values of X “Spread Out” from the mean value E(X) = µX • The Variance Calculation

27. Discrete Random Var Spread • The Square Root of the Variance is called the STANDARD DEVIATION • Quick Example → The standard deviation of the random variable in the coin-flipping game

28. Geometric Random Variable • Consider Again Coin Tossing • Take a fair coin and toss as many times as needed to Produce the 1st Heads. • Let X ≡ number of tosses needed for FIRST Heads. • Sample points={H, TH, TTH, TTTH, …} • The Probability Distribution of X

29. Geometric Random Variable • Consider now an UNfair Coin Tossing • Flip until the 1st head a biased coin with 70% of getting a tail and 30% of seeing a head, • Let X ≡ number of tosses needed to get the first head. • The Probability Distribution of X

30. Geometric Random Variable • In these two example cases the OutCome value can be interpreted as “the probability of achieving the first success directly after n-1 failures.” Let: • p ≡ Probability of SUCCESS • Then (1-p) = Probability of FAILURE • Then the OverAll Probability of 1st Success n−1 Failures Success

31. Example  Exam Pass Rate • The Electrical Engineering Version of the Professional Engineer’s Exam has a Pass (Success) Rate of about 63% • Find the probability of Passing on the • SECOND Try • FOURTH Try • Assuming GeoMetric Behavior

32. Geometric Random Variable • After Some Algebraic Analysis Find for a GeoMetric Random Variable • Expected Value • Standard Deviation

33. WhiteBoard PPT Work • Problems From §11.1 • P31 → HighWay Safety Stats TelsaModel S

34. All Done for Today RolltheDice

35. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

36. HiWay Safety Stats • The Data • Probability of any Given No. of Accidents Per Day

37. HiWay Safety Stats • The Expected Value • µX = 2.3 Accidents/Day • EV(X) Interpretation • The Expected Value of 2.3 Accidents per Day is, on Average, the No. of Accidents likely to occur on any random day of observations

38. HiWay Safety Stats HistoGram

39. HiWay Safety Stats • The σ2 Calc → • Then the StdDeviation fromthe Variance

41. PE Exam Pass Rates Group 1 PE Exams, October 2013 Pass Rates

42. Group 2 PE Exams, October 2013 and April 2013 Pass Rates In most states, and for most exams, the Group 2 exams are given only in October, as indicated in parentheses. The following table shows pass rates of Group 2 examinees.