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