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 E 1 and E 2 with E 1  E 2 = , P(E 1  E 2 ) = P(E 1 ) + P(E 2 )

Probability & Statistics I Probability Rules  Addition Rules  P(A  B) = P(A) + P(B) - P(A  B)  If A & B are mutually exclusive events, then P(A  B) = P(A) + P(B)  A collection of events, E 1, E 2,..., E k, is said to be mutually exclusive if for all pairs, E i  E j =   For a collection of mutually exclusive events, P(E 1  E 2 ...  E k ) = P(E 1 )+P(E 2 )+...+P(E k )

Probability & Statistics I Probability Rules cont’d...  Conditional Probability  of an event A given an event B is denoted as P(A|B) = P(A  B) / P(B)  Multiplication Rule  P(A  B) = 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(B  A) + P(B  A’) = P(B|A)P(A) + (B|A’)P(A’)  Total Probability Rule (multiple events) Assume E 1, E 2,..., E k, are k mutually exclusive and exhaustive sets. Then P(B) = P(B  E 1 ) + P(B  E 2 ) P(B  E k ) = P(B| E 1 )P(E 1 ) + P(B| E 2 )P(E 2 ) P(B| E k )P(E k )

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(A  B) = P(A)P(B)  The events E 1, E 2,..., E k, are independent iff for any subset E i1, E i2,..., E ik P(E i1  E i2 ...  E ik ) = P(E i1 )P(E i2 )...P(E ik )

Probability & Statistics I Bayes Theorem  Bayes Theorem  P(A|B) = P(B|A)P(A)/ P(B)  If E 1, E 2,..., E k, are k mutually exclusive and exhaustive events and B is any event, then P(E 1 |B) = P(B|E 1 )P(E 1 )/ P(B|E 1 )P(E 1 )+ P(B|E 2 )P(E 2 ) P(B|E k )P(E k )

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, 99 Appendix BI #’s 1, 5, 11, 15 All due Friday June 18, 1999