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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

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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

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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

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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

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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+

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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.

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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

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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)

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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.

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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

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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

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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?

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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

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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

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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

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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

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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%).

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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

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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?

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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:

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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

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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

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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:

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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%:

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 –

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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

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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

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

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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

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

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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

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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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