QM Spring 2002 Business Statistics Some Probability Essentials

Student Objectives  Understand concepts of events and probability  Use standard probability notation  Relate probability to relative frequency  Compute simple probabilities – Joint (with independent events) – Conditional – Union (of mutually exclusive events)  Discuss the concept of independence with respect to probability  Define probability distribution

What is Probability?  Just a numeric way of expressing about how certain we feel that a particular event will occur; measures chance – Uses a scale of 0 to 1 (computations) – Conversationally: 0% to 100% – Alternatively, in terms of odds  Can determine probability – Theoretically – Subjectively – Empirically (i.e., using relative frequencies)  Probability allows us to develop inferences based upon descriptive statistics

Some Foundations  Basic notation: P(... ) is the probability that whatever’s inside the parentheses will occur, e.g., P(B) = probability that event B will occur P(x=5) = probability that x will be 5 P(Raise) = probability that JoJo will get a raise P(75) = probability that exam score will be 75  Definitive rules: – 0.00 ≤ P(... ) ≤ 1.00 or 0% ≤ P(... ) ≤ 100% – For exhaustive & mutually exclusive set of events  P(... ) = 1.00 – Keep these in mind when doing calculations (i.e., the voice of reason)

Additional Common Notation  Joint events – P(A and B) = probability that both A and B will occur – Same as P(A ∩ B); intersection of events  Conditional events – P(A | B) = probability that A will occur, given B has occurred – Also interpreted as “if B occurs, the probability that A will”  Union (sorry, can’t think of a more common term) – P(A or B) = probability that either A will occur or B will occur (or both will) – Same as (A U B); union of events

Some Things to Note  Commutative? – Yes: P(A and B) = P(B and A) P(A or B) = P(B or A) – No: P(A | B) ≠ P(B | A) Unless by coincidence  Extensions: – P(A and B and C and... ) – P(A or B or C or... ) – Intersection and union concepts apply to more than just two events  Always: define events ahead of time!

Additional Rules  First, some definitions – Independence: not related; if one event occurs, it doesn’t affect whether another does – Mutually exclusive: if one event occurs, another can’t  Now, the rules: – P(A and B) = P(A) * P(B) Only if events are independent Can be used to determine independence Will occur if and only if P(A | B) = P(A) – P(A or B) = P(A) + P(B) Only if events are mutually exclusive For our purposes, this will always be the case! Leads to the complement rule: P(A) = 1 - P(A c )

Relative Frequency  Regardless of method used to determine probability, it can be interpreted as relative frequency – Recall that relative frequency is observed proportion of time some event has occurred Sites developed in-house Incomes between $10,000 and $20,000 – Probability is just expected proportion of time we expect something to happen in the future given similar circumstances  Note also, proportions are probabilities  Example: ASU Student Demographics

Probability Applications  Statistical inference  Decision analysis  Reliability

Homework  Work probability exercises on handout  Read about discrete distributions (Section 4.3)  Prepare for discussion and analysis of Case 4-B