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 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” ?