11.5 Or and And Problems 703 In Example 2, it is impossible to select a number that is both less than 5 and greater than 8 when only one number is to be selected. Events such as these are said to be mutually exclusive. 2 1 3 4 10 9 B A 5, 6, 7, 8 U Less than 5 Greater than 8 Less than 5 and greater than 8 Figure 11.13 Since there are no numbers that are both less than 5 and greater than 8, P(number is less than 5 and greater than 8) 0. = Therefore, P P P P number is less than 5 or greater than 8 number is less than 5 number is greater than 8 number is less than 5 and greater than 8 4 10 2 10 0 6 10 3 5 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = + − = = Thus, the probability of selecting a number less than 5 or greater than 8 is . 3 5 The six numbers that are less than 5 or greater than 8 are 1, 2, 3, 4, 9, and 10. 7 Now try Exercise 11 Definition: Mutually Exclusive Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously. Learning Catalytics Keyword: Angel-SOM-11.5 (See Preface for additional details.) If events A and B are mutually exclusive, then P A B ( and ) 0 = and the addition formula simplifies to P A B P A P B ( or ) ( ) ( ). = + Example 3 Probability of A or B One card is drawn from a standard 52-card deck. Determine whether the following pairs of events are mutually exclusive and determine P(A or B). a) A B drawing a jack, drawing a queen = = b) A B drawing an ace, drawing a heart = = c) A B drawing a red card, drawing a black card = = d) A B drawing a face card, drawing a red card = = Solution a) There are four jacks and four queens in a standard deck of 52 cards. It is impossible to draw both a jack and a queen when only one card is drawn. Therefore, these events are mutually exclusive. P P P (jack or queen) (jack) (queen) 4 52 4 52 8 52 2 13 = + = + = = b) There are 4 aces and 13 hearts in a standard 52-card deck. One card, the ace of hearts, is both an ace and a heart. Therefore, these events are not mutually exclusive. P P P P P P P (ace) 4 52 (heart) 13 52 (ace and heart) 1 52 (ace or heart) (ace) (heart) (ace and heart) 4 52 13 52 1 52 16 52 4 13 = = = = + − = + − = = A A The ace of hearts is both an ace and a heart. Mikael Damkier/ Shutterstock
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