IE 254 Exam I - Reminder

1
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!

2
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

3
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)

4
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 E1?E2 ?,
• P(E1?E2) P(E1) P(E2)

5
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, 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)

6
Probability Rules contd...

• Conditional Probability
• of an event A given an event B is denoted as
  P(AB) P(A?B) / P(B)
• Multiplication Rule
• P(A?B) P(AB)P(B) P(BA)P(A)

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Probability Rules contd...

• Total Probability Rule (two events)
• For any events A B ,
• P(B) P(B?A) P(B ? A) P(BA)P(A)
  (BA)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)

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Probability Rules contd...

• Independence Two events are independent if
  only if, any one of the following is true.
• P(AB) P(A)
• P(BA) P(B)
• P(A?B) 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)

9
Bayes Theorem

• Bayes Theorem
• P(AB) P(BA)P(A)/ P(B)
• If E1, E2, . . ., Ek, are k mutually exclusive
  and exhaustive events and B is any event, then
• P(E1B) P(BE1)P(E1)/
• P(BE1)P(E1) P(BE2)P(E2). . . P(BEk)P(Ek)

10
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