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Probability

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Basic Concepts of Probability and Counting

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Probability Experiment: … an action, or trial, through which specific results are obtained. The result of a single trial is called an OUTCOME. The set of all possible outcomes is called the SAMPLE SPACE. An EVENT is a subset of the sample space.

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EX: Identify the sample space and determine the # of outcomes 16. Guessing a student’s letter grade (A, B, C, D, F) in a class. 18. Tossing three coins. (Hint… draw a tree diagram)

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The Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m · n EX: For dinner you select one each from 3 appetizers, 4 entrees, and 2 desserts. How many different ‘meals’ can you make if you choose one from each category?

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3 Types of Probability #1 Classical Probability (AKA Theoretical Probability): used when each outcome in a sample space is equally likely to occur. P(E) = probability of event E P(E) = # of outcomes in event E Total # of outcomes in sample space

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#2 Empirical Probability (AKA Statistical Probability) Based on observations obtained from probability experiments. Same as relative frequency of event. P(E) = Frequency of Event E = f Total Frequency n

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#3 Subjective Probability Result from intuition, educated guesses, and estimates.

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As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical probability of the event. The Law of Large Numbers

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Classify as an example of classical, empirical, or subjective probability. The probability of choosing 6 numbers from 1 to 40 that matches the 6 numbers drawn by a state lottery is 1/3,838,380 ≈ 0.00000026.

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Rules of Probability 0 < P(E) < 1 The probability of an event is between 0 and 1 P(E) = 0 means the event CANNOT occur. P(E) = 1 means the event is CERTAIN. ΣP(E) = 1 The sum of the probabilities of all outcomes in the sample space is one.

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Complementary Events The complement of event E (denoted E’) is the set of all outcomes in the sample space that are NOT part of event E. P(E) + P(E’) = 1 P(E’) = 1 – P(E) P(E) = 1 - P(E’)

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Conditional Probability & the Multiplication Rule

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Conditional Probability … the probability of an event occurring, GIVEN that another event has occurred. The conditional probability of event B occurring given that event A occurred is P(B | A)

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Independent & Dependent Events Two events are INDEPENDENT if the occurrence of one does not affect the probability of the other event. A and B are independent if… P(B | A) = P(B) or if… P(A | B) = P(A)

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Dependent or Independent? 8. Returning a rented movie after the due date and receiving a late fee. 12. A ball numbered 1 through 52 is selected from a bin, replaced, and a second numbered ball is selected from the bin.

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The Multiplication Rule The probability that A and B will occur in sequence is: P(A and B) = P(A) · P(B | A) If A and B are independent, use: P(A and B) = P(A) · P(B)

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