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Chapter02 - Chapter 2 Probability 1 Section 2.1 Basic Ideas...

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1 Chapter 2. Probability
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2 Section 2.1 Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance with certainty. Examples: rolling a die tossing a coin weighing the contents of a box of cereal.
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3 Sample Space Definition: The set of all possible outcomes of an experiment is called the sample space for the experiment. Examples: For rolling a fair die, the sample space is {1, 2, 3, 4, 5, 6}. For a coin toss, the sample space is {heads, tails}. For weighing a cereal box, the sample space is (0, ∞), a more reasonable sample space is (12, 20) for a 16 oz. box.
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4 More Terminology Definition: A subset of a sample space is called an event . A given event is said to have occurred if the outcome of the experiment is one of the outcomes in the event. For example, if a die comes up 2, the events {2, 4, 6} and {1, 2, 3} have both occurred, along with every other event that contains the outcome “2”.
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5 Example 1 An electrical engineer has on hand two boxes of resistors, with four resistors in each box. The resistors in the first box are labeled 10 ohms, but in fact their resistances are 9, 10, 11, and 12 ohms. The resistors in the second box are labeled 20 ohms, but in fact their resistances are 18, 19, 20, and 21 ohms. The engineer chooses one resistor from each box and determines the resistance of each. Let A be the event that the first resistor has a resistance greater than 10, let B be the event that the second resistor has resistance less than 19, and let C be the event that the sum of the resistances is equal to 28. 1. Find the sample space for this experiment. 2. Specify the subsets corresponding to the events A , B , and C .
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6 Combining Events The union of two events A and B , denoted A B, is the set of outcomes that belong either to A , to B , or to both. In words, A B means “ A or B. So the event A or B” occurs whenever either A or B (or both) occurs. U U
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7 Example 2 Let A = {1, 2, 3} and B = {2, 3, 4}. What is A B ? U
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8 Intersections The intersection of two events A and B , denoted by A B , is the set of outcomes that belong to A and to B . In words, A B means “ A and B. Thus the event “ A and B ” occurs whenever both A and B occur. I I
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9 Example 3 Let A = {1, 2, 3} and B = {2, 3, 4}. What is A B ? I
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10 Complements The complement of an event A , denoted A c , is the set of outcomes that do not belong to A . In words, A c means “not A. Thus the event “not A ” occurs whenever A does not occur.
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11 Example 4 Consider rolling a fair sided die. Let A be the event: “rolling a six” = {6}. What is A c = “not rolling a six”?
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12 Mutually Exclusive Events Definition: The events A and B are said to be mutually exclusive if they have no outcomes in common. More generally, a collection of events is said to be mutually exclusive if no two of them have any outcomes in common.
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