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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 - PowerPoint PPT Presentation
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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 coin2. Flip a coin 3 times3. Roll a die4. Draw a SRS of size 50 from a population
Trial: A physical action , the result of which cannot be predetermined
Basic Outcomes and Sample SpacesBasic 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 DiagramsEvents: 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 USE=
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
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 die2. 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
Time Period Number of MaleLive Births
Total Number of LiveBirths
Relative Frequency ofLive 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
Odds
If the odds that an event occurs is a:b, then
( )a
P Aa 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
1 2( ) ( ) ( ) ( )kP A P o P o P o
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?
• 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.”
• Additive Law of Probability:
Probabilities of Compound Events
A
A
S
“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.”
Law of Complements
A
A
S
Law of Complements:
Example: If the probability of getting a “working” computer is ).7,What is the probability of getting a defective computer?
( ) 1 ( )P A P A
• Unions of Two Events
“If A and B are events, then the union of A and B, denoted by AB, 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 AB, represents the event composed of all basic outcomes in A and B.”
Unions and Intersections of Two Events
S
BA
Additive Law of ProbabilityLet 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
S
BA
Using Additive Law of Probability
S
BA
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?
Mutually Exclusive EventsMutually Exclusive Events: Events that have no basic outcomes in common, or equivalently, their intersection is the empty set.
S
BA
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
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” ?