Download Introduction to Probability

Introduction to Probability
• Experiments, Outcomes, Events and Sample Spaces
• What is probability?
• Basic Rules of Probability
• Probabilities of Compound Events
Experiments, Outcomes, Events and Sample
Spaces
Experiment: An experiment is any activity from which results
are obtained. A random experiment is one in which the outcomes, or
results, cannot be predicted with certainty.
Examples:
1. Flip a coin
2. Flip a coin 3 times
3. Roll a die
4. Draw a SRS of size 50 from a population
Trial: A physical action , the result of which cannot be predetermined
Basic Outcomes and Sample Spaces
Basic Outcome (o): A possible outcome of the experiment
Sample Space: The set of all possible outcomes of an
experiment
Example: A company has offices in six cities, San Diego, Los
Angeles, San Francisco, Denver, Paris, and London. A new
employee will be randomly assigned to work in on of these
offices.
Outcomes:
Sample Space:
Example #2: A random sample of size two is to be selected from
the list of six cities, San Diego, Los Angeles, San Francisco,
Denver, Paris, and London.
Outcomes:
Sample Space:
Events and Venn Diagrams
Events: Collections of basic outcomes from the sample space. We say
that an event occurs if any one of the basic outcomes in the event
occurs.
Example #1 (cont.):
1. Let B be the event that the city selected is in the US
2. Let A be the event that the city selected is in California
Venn Diagram: Graphical representation of sample space and events
Example #2: A random sample of size two is to be selected from
the list of six cities, San Diego, Los Angeles, San Francisco,
Denver, Paris, and London.
Let E be the event that both cities selected are in the US
E=
Sample Space and Venn Diagram:
Assigning Probabilities to Events
Probability of an event P(E): “Chance” that an event will occur
• Must lie between 0 and 1
• “0” implies that the event will not occur
• “1” implies that the event will occur
Types of Probability:
• Objective
 Relative Frequency Approach
 Equally-likely Approach
• Subjective
Relative Frequency Approach: Relative frequency of an event
occurring in an infinitely large number of trials
Time Period
Number of Male
Live Births
Total Number of Live
Births
Relative Frequency of
Live Male Birth
1965
1927.054
3760.358
0.51247
1965-1969
9219.202
17989.360
0.51248
1965-1974
17857.860
34832.050
0.51268
Equally-likely Approach: If an experiment must result in n equally
likely outcomes, then each possible outcome must have
probability 1/n of occurring.
Examples:
1. Roll a fair die
2. Select a SRS of size 2 from a population
Subjective Probability: A number between 0 and 1 that reflects a
person’s degree of belief that an event will occur
Example: Predictions for rain
Odds
If the odds that an event occurs is a:b, then
a
P ( A) 
ab
Example: If the odds of Came Home winning the Derby are
9:2, what is the subjective probability that he will win?
Probabilities of Events
Let A be the event A = {o1, o2, …, ok}, where o1, o2, …, ok are k
different outcomes. Then
P( A)  P(o1 )  P(o2 ) 
 P(ok )
Problem 5.3.4: The number on a license plate is any digit
between 0 and 9. What is the probability that the first digit is a 3?
What is the probability that the first digit is less than 4?
Probabilities of Compound Events
• Law of Complements:
“If A is an event, then the complement of A, denoted by A ,
represents the event composed of all basic outcomes in S
that do not belong to A.”
A
S
• Additive Law of Probability:
Law of Complements
“If A is an event, then the complement of A, denoted by ,
represents the event composed of all basic outcomes in S that
do not belong to A.”
A
A
S
Law of Complements:
P( A)  1  P( A)
Example: If the probability of getting a “working” computer is ).7,
What is the probability of getting a defective computer?
Unions and Intersections of Two Events
• Unions of Two Events
“If A and B are events, then the union of A and B, denoted
by AB, represents the event composed of all basic
outcomes in A or B.”
• Intersections of Two Events
“If A and B are events, then the intersection of A and B,
denoted by AB, represents the event composed of all
basic outcomes in A and B.”
A
B
S
Additive Law of Probability
Let A and B be two events in a sample space S. The probability
of the union of A and B is
P( A  B)  P( A)  P( B)  P( A  B).
A
B
S
Using Additive Law of Probability
Example: At State U, all first-year students must take
chemistry and math. Suppose 15% fail chemistry, 12% fail
math, and 5% fail both. Suppose a first-year student is selected
at random. What is the probability that student selected failed
at least one of the courses?
A
B
S
Mutually Exclusive Events
Mutually Exclusive Events: Events that have no basic
outcomes in common, or equivalently, their intersection is the
empty set.
Let A and B be two events in a sample space S. The probability
of the union of two mutually exclusive events A and B is
P( A  B)  P( A)  P( B).
A
B
S
Multiplication Rule and
Independent Events
Multiplication Rule for Independent Events: Let A and B
be two independent events, then
P( A  B)  P( A) P( B).
Examples:
• Flip a coin twice. What is the probability of observing two heads?
• Flip a coin twice. What is the probability of getting a head and then a tail? A
tail and then a head? One head?
• Three computers are ordered. If the probability of getting a “working” computer
is .7, what is the probability that all three are “working” ?