Transcript Some Probability Essentials

Business Statistics
QM 2113 - Spring 2002
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 SP( . . . ) = 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(Ac)
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