Chapter 4: Basic Probability

Chapter 4: Basic Probability
• Chapter Goal:
– Explain basic probability concepts and definitions
– Use contingency tables to view a sample space
– Apply common rules of probability
– Compute conditional probabilities
– Determine whether events are statistically independent
• Probability – the chance or likelihood that an uncertain
(particular) event will occur
• Probability is always between 0 and 1, inclusive
• There are three approaches to assessing the probability
of un uncertain event:
1. a priori classical probability– based of a prior knowledge
prob. of occurrence 
number of occurance of the event
total number of possible outcomes
2. empirical classical probability– based on observed data
prob. of occurrence 
number of favorable outcomes observed
total number of outcomes observed
3. subjective probability -- an individual judgment or
opinion about the probability of occurrence
Basic Concepts:
• Sample Space – the collection of all possible events
• An Event – Each possible type of occurrence or outcome from
the sample space
• Simple Event – an event that can be described by a single
• Complement of an event A -- All outcomes that are not part of
event A
• Joint event --Involves events that can be described by two or
more characteristics simultaneously
Basic Concepts: Continued
• Mutually exclusive events: Events that cannot occur together
• Collectively exhaustive events
– One of the events must occur
– The set of events covers the entire sample space
• The probability of any event must be between 0 and 1,
inclusively. That is: 0 ≤ P(A) ≤ 1 for any event A
• The sum of the probabilities of all mutually exclusive and
collectively exhaustive events is 1 exhaustive. That is, if A, B,
and C are mutually exclusive and collectively exhaustive event,
then (the entire sample space)
P(A)  P(B)  P(C)  1
Contingency Tables:
• A sample space can be presented by a C.T.
• It is very useful for the study of empirical probabilities
• Example: Let’s say 400 managers were surveyed about
booking airline tickets and researching prices of tickets in the
Booked Airline Ticket in the
Sample Space: Total Number of
Managers Surveyed
Yes, B1
Researched Prices in the Internet
No, B2
Yes, A1
No, A2
• Computing Simple (Marginal) Probobilities.
P(A)  P(A and B1)  P(A and B2 )   P(A and Bk )
• Where B1, B2, …, Bk are k mutually exclusive and collectively exhaustive
• Probability of a joint event, A and B:
P( A and B) 
number of outcomes satisfying A and B
total number of possible outcomes
• How many simple events are in my example?
• How many joint events are in my example?
Rules of Probability:
1. General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
If A and B are mutually exclusive, then
P(A and B) = 0, so the rule can be simplified
P(A or B) = P(A) + P(B)
If A and B are collectively exhaustive, then
P(A or B) = P(A) + P(B)=1
2. A conditional probability is the probability of
one event, given that another event has
P(A and B)
P(A | B) 
P(A and B)
P(B | A) 
The conditional
probability of A given
that B has occurred
The conditional
probability of B given
that A has occurred
Where P(A and B) = joint probability of A and B
P(A) = marginal probability of A
P(B) = marginal probability of B
3. Multiplication Rule– for two events A and B
P(A and B)  P(A | B) P(B)
4. Two events A and B are statistically independent if
the probability of one event is unchanged by the
knowledge that other even occurred. That is:
P(A | B)  P(A) or P(B | A)  P(B)
5. Then the multiplication rule for two statistically
independent events is:
P(A and B)  P(A) P(B)
• Let look at problem 4.26, page 148