Post on 16-Dec-2015
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics§11.1
Probability& Random-
Vars
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
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
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
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:
Experiment OutComes
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+
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.
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
1st Roll 2nd Roll Total OutComes1 1 2 3 4 5 6 62 1 2 3 4 5 6 63 1 2 3 4 5 6 64 1 2 3 4 5 6 65 1 2 3 4 5 6 66 1 2 3 4 5 6 6
GRAND Total OutComes 36
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 8
Bruce Mayer, PE Chabot College Mathematics
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)
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.
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
Probabilityof an Event
=
Total Number of SPECIFIED OutComes
Total Number of POSSIBLE OutComes
Probability =ONE OutCome = 1
=2.78%36 OutComes 36
Probability =FIVE OutComes = 5
= 13.89%36 OutComes 36
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
36
5
36
1
36
1
36
1
36
1
36
19
6,46,54,65,66,69
P
PPPPPP
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%
Class Question: What are the Odds of winning the California MEGA-MILLIONS Lottery?
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 1st ping-pong Ball = 5 out of 75
The Odds for the 2nd ping-pong Ball = 4 out of 75, and so On
The Odds for the MEGA are 1 out of 15
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)
15
1
!75
!70!5
15
1
71
1
72
2
73
3
74
4
75
5
Odds
• This is Technically a COMBINATION
085,890,258
1
000,902,066,31
120
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:
Current theX120000,902,066,31
1!7115
!70
15
1
71
1
72
1
73
1
74
1
75
1
Odds
• This is a PERMUTATION
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
xXPxp
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%).
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
1 4 5 62 3
x
xp all
1
x
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
xp
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?
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:
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
%8.14148.0
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, µX k
n
kkk
n
kkX xpxxXPxXE
11
)(
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 23
Bruce Mayer, PE Chabot College Mathematics
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:
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%:
kkX xPxEV
8
151111111 kk xPxEV
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
4
1
8
2
8
175
kkX xPxEV
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) = µX
The Variance Calculation
2
1
2
1
2Var
n
kkXk
n
kkk xpµxxpXExX
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
n
kkXk xpµxX
1
22 Var
046.2
125.0)25.05(
125.0)25.01(...125.0)25.1(
2
times7
22
X
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 1st 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) 1/2 1/4 1/8 1/16 1/2N
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 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
X 1 (H) 2 (TH) 3 (TTH) 4 (TTTH) N (TTT…TH) P(X) 0.3 0.7·0.3 0.7·0.7·0.3 0.7·0.7·0.7·0.3 0.7·0.7·…·0.7·0.3
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 1st Success
pppppppnXP n 111111
n−1 Failures Success
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
%3.23233.063.063.012 12 XP
%2.3032.063.063.014 14 XP
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
p
µXEV X
1
p
pX
1
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
TelsaModel S
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 34
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
RolltheDice
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 35
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
–
srsrsr 22
a2 b2
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. Accidentsper Day, X
No. Days of Observation
TotalAccidents
0 6 01 8 82 7 143 2 64 3 125 0 06 2 127 0 08 1 89 1 9
Σtotals = 30 69
No. Accidentsper Day, X P(X) P(X)·X
0 6/30 = 0.2 01 8/30 = 0.2667 0.26672 7/30 = 0.2333 0.46673 2/30 = 0.0667 0.24 3/30 = 0.1 0.45 0/30 = 0 06 2/30 = 0.0667 0.47 0/30 = 0 08 1/30 = 0.0333 0.26679 1/30 = 0.0333 0.3
Σtotals = 1 2.3
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
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
No. Accidentsper Day, X P(X) P(X)·X
0 6/30 = 0.2 01 8/30 = 0.2667 0.26672 7/30 = 0.2333 0.46673 2/30 = 0.0667 0.24 3/30 = 0.1 0.45 0/30 = 0 06 2/30 = 0.0667 0.47 0/30 = 0 08 1/30 = 0.0333 0.26679 1/30 = 0.0333 0.3
Σtotals = 1 2.3
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 38
Bruce Mayer, PE Chabot College Mathematics
HiWay Safety Stats HistoGram
0 1 2 3 4 5 6 7 8 90
1
2
3
4
5
6
7
8
No. of Accidents/Day
No
. of D
ays
MTH16 • P11.1-31 HiWay Safety
0 1 2 3 4 5 6 7 8 90
5
10
15
20
25
30
No. of Accidents/Day
% o
f No
. of D
ays
(N
= 3
0 d
ays
)
MTH16 • P11.1-31 HiWay Safety
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 39
Bruce Mayer, PE Chabot College Mathematics
HiWay Safety Stats
The σ2 Calc →
Then the StdDeviation fromthe Variance
No. Accidentsper Day, xk
(xk − µX)2 P(xk) (xk − µX)2·P(xk)
0 5.29 0.2000 1.05801 1.69 0.2667 0.45072 0.09 0.2333 0.02103 0.49 0.0667 0.03274 2.89 0.1000 0.28905 7.29 0.0000 0.00006 13.69 0.0667 0.91277 22.09 0.0000 0.00008 32.49 0.0333 1.08309 44.89 0.0333 1.4963
Σtotals = 1.000 5.3433
Day
Accidents331.2
Day
Accidents 433.5
2
2
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 40
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 41
Bruce Mayer, PE Chabot College Mathematics
Exam 1st Timers RepeatsChemical 67% 40%Civil 64% 29%Electrical and Computer 63% 28%Environmental 63% 35%Mechanical 72% 41%Structural Engineering (SE) Vertical Component 50% 34%Structural Engineering (SE) Lateral Component 38% 43%
PE Exam Pass RatesGroup 1 PE Exams, October 2013 Pass Rates
BMayer@ChabotCollege.edu • MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 42
Bruce Mayer, PE Chabot College Mathematics
Exam1st
TimersRepeats
Agriculture (October 2013) 69% 50%
Architectural 74% 43%
Control Systems (October 2013) 76% 53%
Fire Protection (October 2013) 69% 37%
Industrial 72% 50%
Metallurgical and Materials (October 2013) 62% 0%
Mining and Mineral Processing (October 2013) 71% 37%
Naval Architecture and Marine Engineering 58% 46%
Nuclear (October 2013) 54% 44%
Petroleum (October 2013) 75% 53%
Software 50% NA
Group 2 PE Exams, October 2013 and April 2013 Pass RatesIn 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.