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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 4.1 Classical Probability HAWKES LEARNING SYSTEMS Probability, Randomness, and Uncertainty math courseware specialists 4.1 Classical Probability Definitions: • Probability experiment – any process in which the result is random in nature. • Outcome – each individual result that is possible for a given experiment. • Sample space – the set of all possible outcomes for a given experiment. • Event – a subset of the sample space. HAWKES LEARNING SYSTEMS Probability, Randomness, and Uncertainty math courseware specialists 4.1 Classical Probability Sample space and events: Consider an experiment in which a coin is tossed and then a 6-sided die is rolled. a. List the sample space for the experiment. b. List the outcomes in the event “tossing a tail then rolling an odd number”. Solution: a. Each outcome consists of a coin toss and a die roll. b. Choosing the members of the sample space which fit the event “tossing a tail then rolling an odd number” gives: {T1, T3, T5} HAWKES LEARNING SYSTEMS Probability, Randomness, and Uncertainty math courseware specialists 4.1 Classical Probability Three methods for calculating the probability: 1. Subjective – an educated guess regarding the chance that an event will occur. 2. Empirical – if all outcomes are based on experiment. 3. Classical – if all outcomes are equally likely. HAWKES LEARNING SYSTEMS Probability, Randomness, and Uncertainty math courseware specialists 4.1 Classical Probability Determine whether each of the following probabilities is subjective, empirical, or classical: a. The probability of selecting the queen of spades out of a standard deck of cards. (1 out of 52 cards). Classical b. An economist predicts a 20% chance that technology stocks will decrease in value over the next year. Subjective c. A police officer wishes to know the probability that a driver, chosen at random, will be driving under the influence of alcohol on a Friday night. At a roadblock, he records the number of drivers and the number of drivers driving with more than the legal blood alcohol limit. He determines that the probability is 3%. Empirical HAWKES LEARNING SYSTEMS Probability, Randomness, and Uncertainty math courseware specialists 4.1 Classical Probability Calculate the probability: A large jar contains more marbles than you are willing to count. Instead, you draw some coins at random, replacing each coin before the next draw. You record the picks in the following table: Red Blue Green Yellow Purple 15 29 25 31 10 a. What is the probability that on your next draw you will obtain a blue marble? b. What is the probability that on your next draw you will obtain a yellow marble? Click “/” to add denominator 4 out of 6, thus 2 out of 3 Click “/” to add denominator Total = 44 + 57 + 77 = 178 Total = 44 + 49 + 23 + 37 + 22 = 175 The probability that will be an ace is 4/52 Click “/” to add denominator Number of outcomes = 6 results to the power of 2 pairs = 62 = 36 Number of Events (sum of 10) = 5 and 5, 4 and 6, 6 and 4 = 3 Probability (sum of 10) = 3/36 = 1/12 Click “/” to add denominator Number of outcomes = 2 results to the power of 8 tosses = 28 = 256 Number of Events (all tails) = T T T T T T T T = just once = 1 Probability (all tails) = 1/256 Click “/” to add denominator Number of chips above 111 = 434 – 111 = 323 Probability (above 111) = 323/434