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Probability introduction
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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
EPEPEEP
EEEE
EEP
SP
)()( then , If
)(1)'(
0)(
2121 EPEPEE
EPEP
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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