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Probability & Statistics I
IE 254 Exam I - Reminder
Reminder: Test 1 - June 21 (see syllabus) Chapters 1, 2, Appendix BI
HW Chapter 1 due Monday at first of class!
Probability & Statistics I
Sample Spaces and Events
Random Experiments (with/without sampling replacement)
Sample Space Discrete Events Mutually Exclusive
Understand the definitions from text, not memorize!
Review set operations Tree Diagrams
Probability & Statistics I
Counting Techniques (Appendix BI) Why?
Sometimes, determining the number of outcomes (events is fairly difficult in more complex situations)
Multiplication Rule (tree diagrams) Permutations (I-1) Permutations (arrangements) (I-2) Arrangements (not all different) (I-3) Combinations (I-4) (order not important)
Probability & Statistics I
Probability
Probability Interpretations Degree of Belief / Relative Frequency Equally Likely Outcomes Probability of an Event P(E)
Probability Axioms P(S) = 1 0 P(E) 1 For two events E1 and E2 with E1E2 = ,
P(E1E2) = P(E1) + P(E2)
Probability & Statistics I
Probability Rules
Addition Rules P(AB) = P(A) + P(B) - P(AB) If A & B are mutually exclusive events,
then P(AB) = P(A) + P(B) A collection of events, E1, E2, . . ., Ek, is
said to be mutually exclusive if for all pairs, Ei Ej =
For a collection of mutually exclusive events, P(E1 E2 . . . Ek) = P(E1)+P(E2)+. . .+P(Ek)
Probability & Statistics I
Probability Rules cont’d...
Conditional Probability of an event A given an event B is denoted
as P(A|B) = P(AB) / P(B)
Multiplication Rule P(AB) = P(A|B)P(B) = P(B|A)P(A)
Probability & Statistics I
Probability Rules cont’d...
Total Probability Rule (two events)For any events A & B , P(B) = P(BA) + P(B A’) = P(B|A)P(A) + (B|
A’)P(A’) Total Probability Rule (multiple events)
Assume E1, E2, . . ., Ek, are k mutually exclusive and
exhaustive sets. Then P(B) = P(B E1) + P(B E2)
+ . . . + P(B Ek) = P(B| E1)P(E1) + P(B| E2)P(E2) + . . . +
P(B| Ek)P(Ek)
Probability & Statistics I
Probability Rules cont’d...
Independence: Two events are independent if & only if, any one of the following is true. P(A|B) = P(A) P(B|A) = P(B) P(AB) = P(A)P(B)
The events E1, E2, . . ., Ek, are independent iff for any subset Ei1, Ei2, . . ., Eik
P(Ei1 Ei2 . . . Eik) = P(Ei1 )P(Ei2 ). . .P(Eik)
Probability & Statistics I
Bayes Theorem
Bayes Theorem P(A|B) = P(B|A)P(A)/ P(B)
If E1, E2, . . ., Ek, are k mutually exclusive and exhaustive events and B is any event, then P(E1|B) = P(B|E1)P(E1)/
P(B|E1)P(E1)+ P(B|E2)P(E2)+. . . + P(B|Ek)P(Ek)
Probability & Statistics I
IE254 Chapter 2 and Appendix BI HW
Homework Assignment:Chapter 2 #’s 21, 23, 27, 29, 31, 35, 43, 47,
50, 51, 53, 57, 63, 67, 71, 79, 83, 91, 99Appendix BI #’s 1, 5, 11, 15
All due Friday June 18, 1999
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