Download Basic Probability Theory • chance experiment situation whose

Basic Probability Theory
• chance experiment
situation whose ultimate outcome is uncertain
• sample space
set of all possible outcomes of a chance experiement
• event ( A, B, … )
set of outcomes from the sample space
• simple event
event consisting of a single outcome
• complement of an event ( not A, Ac )
the event consisting of all outcomes not in A
• union (disjunction) of events ( A or B, A » B)
the event made up of all events in either A or B or
both
• intersection (conjunction) of events ( A and B,
A!« B)
the event made up of all events in both A and B
• disjoint or mutually exclusive events
events whose intersection is empty
Basic Properties
• probability of an event E is denoted P(E)
• P(E) lies between 0 and 1
• measures the likelihood that E will occur as the
result of the chance experiment
• P(S) = 1 if S is the entire sample space
• P(E) + P(not E) = 1
• if events E and F are disjoint, then
P(E or F) = P(E) + P(F)
Different Approaches to Probability
Classical Approach
• assumes outcomes are all equally likely
#outcomes in E
• P(E)=
# outcomes in sample space
†
†
Empirical Approach
• probability of E is the relative frequency of the
occurrence of E in a very long series of trials of the
chance experiment
• based on the Law of Large Numbers: as the number
of trials of an experiment increases, the relative
frequency of the occurrence of E approaches P(E)
# occurrences in E
•
Æ P(E)
# of trials
Subjective Approach
• probability of E is a personal judgment based on
experience and other facts
conditional probability P(E|F)
• probability of event E under the assumption that F
is known to have occurred
P(E «F )
P(E |F )=
P(F )
independent events
†
• E and F are independent only if P(E|F) = P(E);
otherwise they are dependent
• E and F are independent when P(E « F) = P(E)P(F)
addition rule
• P(E » F) = P(E) + P(F) – P(E « F)
multiplication rule
• P(E « F) = P(E|F)P(F)
Bayes’ Rule
• If E and F are disjoint complementary events, then
for any event A,
P(A |E)P(E)
P(E | A)=
P(A |E)P(E)+P(A | F)P(F)
†