Introduction to Probability Experiments Outcomes Events and Sample

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 ü Relative Frequency Approach ü 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 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 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 = {o 1, o 2, …, ok}, where o 1, o 2, …, ok are k different outcomes. Then 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 , 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 S Law of Complements: 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 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: 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 A B S

Multiplication Rule and Independent Events Multiplication Rule for Independent Events: Let A and B be two independent events, then 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” ?