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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

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1 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §11.1 Probability & Random-Vars

2 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §10.3 Power & Taylor Series  Any QUESTIONS About HomeWork §10.3 → HW-19 10.3

3 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 5 Bruce Mayer, PE Chabot College Mathematics 6.5 Random Experiment…  A Random Experiment is an action or process that leads to one of several possible outcomes.  Some examples: ExperimentOutComes Flip a coin Heads, Tails Exam Scores Numbers: 0, 1, 2,..., 100 Assembly Time t > 0 seconds Course Grades F, D, C, B, A, A+

6 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 6 Bruce Mayer, PE Chabot College Mathematics OutComes, Events, SampleSpace  OutCome → is a particular result 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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 1 st Roll2 nd RollTotal OutComes 11234566 21234566 31234566 41234566 51234566 61234566 GRAND Total OutComes36

8 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  OutComes, etc.  One outcome: 1 st Roll = a five, 2 nd Roll = a two → which can be represented by the ordered 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 9 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 10 Bruce Mayer, PE Chabot College Mathematics 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 Probability of an Event = Total Number of SPECIFIED OutComes Total Number of POSSIBLE OutComes Probability= ONE OutCome=1 =2.78% 36 OutComes36 Probability= FIVE OutComes=5 =13.89% 36 OutComes36

11 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 12 Bruce Mayer, PE Chabot College Mathematics AND Probability  Probability  The LIKELYHOOD that a Specified OutCome Will be Realized The “Odds” Run from 0% to 100% What are the Odds of winning the California MEGA-MILLIONS Lottery?  Class Question: What are the Odds of winning the California MEGA-MILLIONS Lottery?

13 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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 1 st ping-pong Ball = 5 out of 75  The Odds for the 2 nd ping-pong Ball = 4 out of 75, and so On  The Odds for the MEGA are 1 out of 15

14 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 15 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 17 Bruce Mayer, PE Chabot College Mathematics Probability Distribution Function  A Probability Distribution Function (PDF) maps the possible values of x against 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 18 Bruce Mayer, PE Chabot College Mathematics Discrete Example: Roll The Die 1/6 145623 xp(x) 1p(x=1)=1/6 2p(x=2)=1/6 3p(x=3)=1/6 4p(x=4)=1/6 5p(x=5)=1/6 6p(x=6)=1/6

19 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 21 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 22 Bruce Mayer, PE Chabot College Mathematics Expected Value  The EXPECTED 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, µ X

23 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Coin-Tossing µ X  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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 24 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 25 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 26 Bruce Mayer, PE Chabot College Mathematics 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 ) = µ X  The Variance Calculation

27 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 27 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 28 Bruce Mayer, PE Chabot College Mathematics Geometric Random Variable  Consider Again Coin Tossing  Take a fair coin and toss as many times as needed to Produce the 1 st Heads.  Let X ≡ number of tosses needed for FIRST Heads.  Sample points={H, TH, TTH, TTTH, …}  The Probability Distribution of X X 1 (H)2 (TH)3 (TTH)4 (TTTH) N (TTT…TH) P(X)P(X)1/21/41/81/161/2 N

29 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 29 Bruce Mayer, PE Chabot College Mathematics Geometric Random Variable  Consider now an UNfair Coin Tossing  Flip until the 1 st 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 X 1 (H)2 (TH)3 (TTH)4 (TTTH) N (TTT…TH) P(X)P(X)0.30.7·0.30.7·0.7·0.30.7·0.7·0.7·0.30.7·0.7·…·0.7·0.3

30 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 30 Bruce Mayer, PE Chabot College Mathematics 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 1 st Success n −1 FailuresSuccess

31 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 32 Bruce Mayer, PE Chabot College Mathematics Geometric Random Variable  After Some Algebraic Analysis Find for a GeoMetric Random Variable Expected Value Standard Deviation

33 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 33 Bruce Mayer, PE Chabot College Mathematics WhiteBoard PPT Work  Problems From §11.1 P31 → HighWay Safety Stats Telsa Model S

34 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 34 Bruce Mayer, PE Chabot College Mathematics All Done for Today Roll the Dice

35 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 35 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

36 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 36 Bruce Mayer, PE Chabot College Mathematics HiWay Safety Stats  The Data  Probability of any Given No. of Accidents Per Day No. Accidents per Day, X No. Days of Observation Total Accidents 060 188 2714 326 4312 500 62 700 818 919 Σtotals =3069 No. Accidents per Day, X P(X)P(X)P(X)·X 06/30 =0.20 18/30 =0.2667 27/30 =0.23330.4667 32/30 =0.06670.2 43/30 =0.10.4 50/30 =00 62/30 =0.06670.4 70/30 =00 81/30 =0.03330.2667 91/30 =0.03330.3 Σtotals =12.3

37 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 37 Bruce Mayer, PE Chabot College Mathematics HiWay Safety Stats  The Expected Value  µ X = 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 No. Accidents per Day, X P(X)P(X)P(X)·X 06/30 =0.20 18/30 =0.2667 27/30 =0.23330.4667 32/30 =0.06670.2 43/30 =0.10.4 50/30 =00 62/30 =0.06670.4 70/30 =00 81/30 =0.03330.2667 91/30 =0.03330.3 Σtotals =12.3

38 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 38 Bruce Mayer, PE Chabot College Mathematics HiWay Safety Stats HistoGram

39 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 39 Bruce Mayer, PE Chabot College Mathematics HiWay Safety Stats  The σ 2 Calc →  Then the Std Deviation from the Variance No. Accidents per Day, x k (x k − µ X ) 2 P(xk)P(xk)(x k − µ X ) 2 ·P(x k ) 05.290.20001.0580 11.690.26670.4507 20.090.23330.0210 30.490.06670.0327 42.890.10000.2890 57.290.0000 613.690.06670.9127 722.090.0000 832.490.03331.0830 944.890.03331.4963 Σtotals =1.0005.3433

40 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 40 Bruce Mayer, PE Chabot College Mathematics

41 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 41 Bruce Mayer, PE Chabot College Mathematics Exam1st TimersRepeats Chemical67%40% Civil64%29% Electrical and Computer63%28% Environmental63%35% Mechanical72%41% Structural Engineering (SE) Vertical Component50%34% Structural Engineering (SE) Lateral Component38%43% PE Exam Pass Rates Group 1 PE Exams, October 2013 Pass Rates

42 BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 42 Bruce Mayer, PE Chabot College Mathematics Exam 1st Timers Repeats Agriculture (October 2013)69%50% Architectural74%43% Control Systems (October 2013)76%53% Fire Protection (October 2013)69%37% Industrial72%50% Metallurgical and Materials (October 2013)62%0% Mining and Mineral Processing (October 2013)71%37% Naval Architecture and Marine Engineering58%46% Nuclear (October 2013)54%44% Petroleum (October 2013)75%53% Software50%NA 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.

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