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Introduction to Probability

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  1. Introduction to Probability • Experiments, Outcomes, Events and Sample Spaces • What is probability? • Basic Rules of Probability • Probabilities of Compound Events

  2. 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: • Flip a coin • Flip a coin 3 times • Roll a die • Draw a SRS of size 50 from a population • Trial: A physical action , the result of which cannot be predetermined

  3. 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:

  4. 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:

  5. 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.): • Let B be the event that the city selected is in the US • Let A be the event that the city selected is in California Venn Diagram: Graphical representation of sample space and events

  6. 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:

  7. 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

  8. 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: • Roll a fair die • 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

  9. If the odds that an event occurs is a:b, then Odds Example: If the odds of Came Home winning the Derby are 9:2, what is the subjective probability that he will win?

  10. Probabilities of Events Let A be the event A = {o1, o2, …, ok}, where o1, o2, …, 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?

  11. 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.” • Additive Law of Probability: A S

  12. 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? 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.”

  13. A B S • 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 ofTwo 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.” Unions and Intersections of Two Events

  14. 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 Additive Law of Probability

  15. 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?

  16. 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 B A Mutually Exclusive Events Mutually Exclusive Events: Events that have no basic outcomes in common, or equivalently, their intersection is the empty set. S

  17. 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” ?