11.1 Empirical and Theoretical Probabilities 663 In the remainder of this chapter, the word probability will refer to theoretical probability. Example 3 illustrates how to use the theoretical probability formula. Example 3 Determining Theoretical Probabilities A fair die is rolled. Determine the probability of rolling a) a 2. b) an even number. c) a number greater than 3. d) a 7. e) a number less than 7. Solution a) There are six possible equally likely outcomes: 1, 2, 3, 4, 5, or 6. The event of rolling a 2 can occur in only one outcome. = = P(2) number of outcomes that will result in a 2 total number of possible outcomes 1 6 b) The event of rolling an even number can occur in three outcomes: 2, 4, or 6. = = = P(even number) number of outcomes that result in an even number total number of possible outcomes 3 6 1 2 c) The event of rolling a number greater than 3 can occur in three outcomes: 4, 5, or 6. = = P(number greater than 3) 3 6 1 2 d) The event of rolling a 7 can occur in zero outcomes. Thus, the event cannot occur and the probability is 0. = = P(7) 0 6 0 e) The event of rolling a number less than 7 can occur in 6 outcomes: 1, 2, 3, 4, 5, or 6. Thus, the event must occur and the probability is 1. = = P(number less than 7) 6 6 1 ■ Now try Exercise 23 Four important facts about probability follow. Important Probability Facts 1. The probability of an event that cannot occur is 0. 2. The probability of an event that must occur is 1. 3. Every probability is a number between 0 and 1 inclusive; that is, ≤ ≤ P E 0 ( ) 1. 4. The sum of the probabilities of all possible outcomes of an experiment is 1. StatCrunch Applets Experiment Roll Die
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