Statistics 3601 Dr. Staffan Fredricsson Probability Page 1

STATISTICS and PROBABILITY for

SCIENCE and ENGINEERING

Statistics 3601/5601/7001

Probability

Statistics 3601 Dr. Staffan Fredricsson Probability Page 2

Course Logistics

Exams:- Midterm 1: ?- Midterm 2: ?- Final: Th 12/8

Blackboard: Problems?

Other Questions or Issues?

Note:

Thursday, November 24: Thanksgiving Day

 

Statistics 3601 Dr. Staffan Fredricsson Intro & Probability Page 3

Homework 1Note: Text for 2-14,2-62,2-78,2-82 on Blackboard/Course Materials/Miscellaneous

Homework: Chapter 2.1 to 2.4 – Due Thursday 9/29 Solve Problems 2-14, 2-62, 2-78, [either 2-82 or 2-94]

o Write “3601 Homework 1; <your last name, first name>” on top of page 1o Show sufficient work, not just the answero Write in longhand, or use computer with MS-word or Excel (Equation Editor is

optionally available under “Insert/Object”)o Turn it in during the lectureo If you have problems, see me during office hours, or send me an email question

A solution to all problems will be posted on Blackboard after the Due Date

I will randomly pick one of the problems for scoring.

No credit for submissions after the Due Date

Recommended Exercises:2-15, 27, 55, 63, 67, 79, 89, 97

Statistics 3601 Dr. Staffan Fredricsson Probability Page 4

Probability - Overview• Why would the Probability concept be useful

• Axioms of Probability

• Addition Rules

• Conditional Probability

• Statistical Independence

• Total Probability Rule

• Bayes’ Formula

Statistics 3601 Dr. Staffan Fredricsson Probability Page 5

“Random Experiment” We earlier showed how the concepts of Outcome, Event and

Sample Space could be useful in describing a “Random Experiment”

However, in order to distinguish between e.g. a fair and a loaded die, we need and additional concept, related to the likelihood of different Outcomes/Events. This is the concept of Probability

Outcome Sample Space

Event

Statistics 3601 Dr. Staffan Fredricsson Probability Page 6

Probability (a man-made concept) Probability P(E)

o An assigned number associated with any Outcome and Event E in Sample Space

o Used to quantify the “likelihood” that an Outcome O and an event E will occur when we perform the Experiment

o If several Outcomes are “equally likely” to occur as a result of an experiment, we want to assign identical “Probabilities” to these Outcomes

Outcome OP(O)

Sample Space SP(S)=1

Event EP(E)

Statistics 3601 Dr. Staffan Fredricsson Probability Page 7

Axioms of Probability

Axiom = “Self-evident Property”

Some Derived Consequences:

) ()() (

with and events For two )3(

event any for 1)(0 )2(

1)( )1(

2121

2121

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EEEE

EEP

SP

)()( then , If

)(1)'(

0)(

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P

E

E1

E

E2

E’

E1 E2

Statistics 3601 Dr. Staffan Fredricsson Probability Page 8

How do we assign Probability?Probability P(event E)

An assigned number associated with any event E in Sample Space Used to quantify the “likelihood” that the event E will occur when we perform the Experiment

The “Classical Approach” (for Discrete Sample Space): We have N possible, mutually exclusive, equally likely Outcomes (flip coin, roll dice…) We are interested in the “event E” consisting of Ne different Outcomes

The “Relative Frequency Approach” (for Arbitrary Sample Space): We repeat an Experiment n times The “event E” occurs in ne of the trials

In the limit, as n infinity, the relative frequency becomes stable, and we can replace the approximation with “=“

The “Subjective Probability Approach”: Used by the common man for “Experiments” that can not be repeated “The probability that I will receive an A in this course”

N

NEeventP

e) (

n

nEeventP

e)(

Statistics 3601 Dr. Staffan Fredricsson Probability Page 9

Exercise 2-54

N

NEeventP

e) (

use will We:Note

2.54

Statistics 3601 Dr. Staffan Fredricsson Probability Page 10

Exercise 2-66 a, b, d

)(1)'(

) (

use will We:Note

EPEPN

NEeventP

e

2.66

Statistics 3601 Dr. Staffan Fredricsson Probability Page 11

Addition/Union Rules

AA BBA

)( )( )( )(

)'()()()()'(

)'()()'(

)}'( and )( and )'( events exclusive{mutually

)]'()()'[()(

BPBAPAPBAP

BAPBAPBAPBAPBAP

BAPBAPBAP

BABABA

BABABAPBAP

)()()()()()()()(

:shown that becan it Similarly,

CBAPCBPCAPBAPCPBPAPCBAP

)()()(then

),( events exclusivemutually are and If :yprobabilit of axiom 3rd heRemember t

BPAPBAP

BABA

Statistics 3601 Dr. Staffan Fredricsson Probability Page 12

Exercise 2-78

BABA

BPBAPAPBAPN

NEeventP

e

ExclusiveMutually ,

)()( )( )(

) (

use will We:Note

2.78

Statistics 3601 Dr. Staffan Fredricsson Probability Page 13

Conditional Probability To this point, we have discussed the Probability of Events that may result

from a Random Experiment

It is sometimes useful to discuss the “Conditional Probability” of an Event A that may result from a Random Experiment, given that the Event B occurred.

For example, when considering epidemic illness in a population, consider the “Random Experiment” that we randomly draw a single individual from the population of a county. We may be interested in the “Conditional Probability” of the Event “Individual has illness”, given that the Event “Individual was inoculated” has occurred.

“Conditional Probability” can be viewed as the Probability in a modified Random Experiment, consisting only of the Outcomes that satisfy the Condition

A BBA

Statistics 3601 Dr. Staffan Fredricsson Probability Page 14

Conditional Probability P(A|B)

Tree diagram for classified parts:

05.0360

18)'|(

25.040

10)|(

1.0400

40)(

07.0400

28)(

FDP

FDP

FP

DP

A BBA

Example: 400 parts classified by (visual) surface flaws and (functionally) defective

372

28

Statistics 3601 Dr. Staffan Fredricsson Probability Page 15

Conditional Sample Space Redefined

)|()()|()()(

:)Rule"tion Multiplica" (aka on"Intersecti theofy Probabilit" calculate toused becan This

)(

)()|(

)(

)()|(

:yProbabilit lConditiona

BAPBPABPAPBAP

AP

BAPABP

BP

BAPBAP

A BBA

A BBA

Statistics 3601 Dr. Staffan Fredricsson Probability Page 16

Exercise 2-90 a, d, e

)(

)( )|(

) (

use will We:Note

BP

BAPBAP

N

NEeventP

e

2-90.

Statistics 3601 Dr. Staffan Fredricsson Probability Page 17

(Statistically) Independent Events

false allor truealleither are equations These :Note

)()()(

or )()|(

or )()|(

:iff t"Independen" called are Events Two

tIndependen ally)(Statistic are and Events the

say that We.Event in is Outcome y that theProbabilit the

affect not does Event in is Outcome that theknowledge case, In this

])(

)( [generally )()|(

yProbabilit lConditiona thecases, someIn

BPAPBAP

BPABP

APBAP

BA

B

A

AP

ABPBPABP

Statistics 3601 Dr. Staffan Fredricsson Probability Page 18

Exercise 2-136 (p. 55)

)( )()()(

)()()( t Independen

use will We:Note

BPBAPAPBAP

BPAPBAP

2-136.

Statistics 3601 Dr. Staffan Fredricsson Probability Page 19

Total Probability Rule

)'()'|()()|()(

)'()()(

exclusive][mutually )'()(

APABPAPABPBP

ABPABPBP

ABABB

)()|( ... )() |()()|(

)( ... ) ()()(

events exhaustive and exclusivemutually ,, .... , ,for Similarly,

2211

21

21

kk

k

k

EPEBPEPEBPEPEBP

EBPEBPEBPBP

EEE

Statistics 3601 Dr. Staffan Fredricsson Probability Page 20

Exercise 2-105

)'()'|()()|()(

Ruley Probabilit Total theuse shall We

:

APABPAPABPBP

Note

2.105.

Statistics 3601 Dr. Staffan Fredricsson Probability Page 21

Probability - Summary

)()|( ... )() |()()|()(

: thenevents, exhaustive and exclusivemutually ,, .... , , If

)()()(

: theneventst independen B andA If

)()()(

: thenevents, exclusivemutually B andA If

)()()()(

)(

)()|(

:Generally

2211

21

kk

k

EPEBPEPEBPEPEBPBP

EEE

BPAPBAP

BPAPBAP

BAPBPAPBAP

BP

BAPBAP

Statistics 3601 Dr. Staffan Fredricsson Probability Page 22

Next Lecture

Plan for next lecture: • Cover Sections 2.7, 3.1-3.4

(Bayes’ Theorem, Discrete Random Variables)

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