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Chapter 7 Sets & Probability

Chapter 7 Sets & Probability. Section 7.3 Introduction to Probability. Why use probability? . A great many problems that come up in the applications of mathematics involve random phenomena – those for which exact prediction is impossible.

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Chapter 7 Sets & Probability

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  1. Chapter 7 Sets & Probability Section 7.3 Introduction to Probability

  2. Why use probability? • A great many problems that come up in the applications of mathematics involve random phenomena – those for which exact prediction is impossible. • The best we can do is determine the probability of the possible outcomes.

  3. Sample Spaces • In probability, an experiment is an activity or occurrence with an observable result. • Each repetition of an experiment is called a trial. • The possible results of each trial are called outcomes. • The set of all possible outcomes for an experiment is the sample space for that experiment.

  4. Example: A spinner that is equally divided up into three spaces is spun and a coin is tossed. Give the sample space for this experiment. Spinner Coin Toss Possible Outcomes H (1, H) 1 T (1, T) H (2, H) 2 T (2, T) H (3, H) 3 T (3, T) S = { (1,H), (1,T), (2,H), (2,T), (3,H), (3,T) }

  5. Events • An event is a subset of a sample space. • An event in which only one outcome is possible is called a simple event. • If the event equals the sample space, then the event is called a certain event. • If the event is equal to the null, or empty, set, then the event is called an impossible event.

  6. Example: Consider rolling a single die. S = { 1, 2, 3, 4, 5, 6} Event A: Rolling a 5 Event B: Rolling an odd number Event C: Rolling a number less than 7 Event D: Rolling a number greater than 6 Which event is a simple event? Event A Which event is an impossible event? Event D Which event is a certain event? Event C

  7. Set Operations for Events • Let E and F be events for a sample space, S. E  F occurs when both E and F occur; E  F occurs when E or F or both occur; E ′ occurs when E does not occur. • Mutually Exclusive Events Events E and F are mutually exclusive events if E  F = Ø

  8. Probability • For sample spaces with equally likely outcomes, the probability of an event is defined as follows. Basic Probability Principle: Let S be a sample space of equally likely outcomes , and let event E be a subset of S. Then the probability that event E occurs is For any event E, 0 ≤ P(E) ≤ 1.

  9. Example: A marble is drawn from a bowl containing 3 yellow, 4 white, and 8 blue marbles. Find the probability of the following events. 1.) A yellow marble is drawn P (yellow) = 3 / 15 = 1 / 5 2.) A blue marble is drawn P (blue) = 8 / 15 3.) A white marble is not drawn P (not white) = 11 / 15

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