11.1 Empirical and Theoretical Probabilities 665 In any experiment, an event must either occur or not occur. The sum of the probability that an event will occur and the probability that it will not occur is 1. Thus, for any event E we conclude that The Sum of the Probabilities Equals 1 + = = − P E P E P E P E ( ) (not ) 1 or (not ) 1 () For example, if the probability that event E will occur is , 5 12 the probability that event E will not occur is −1 , 5 12 or . 7 12 Similarly, if the probability that event E will not occur is 0.3, the probability that event E will occur is − = 1 0.3 0.7, or . 7 10 We make use of this concept in Example 5. Example 5 Drawing One Card from a Deck A standard deck of 52 playing cards is shown in Fig. 11.3. The deck consists of four suits: hearts, clubs, diamonds, and spades. Each suit has 13 cards, including numbered cards ace (1) through 10 and three face cards, the jack, the queen, and the king. Hearts and diamonds are red cards; clubs and spades are black cards. There are 12 face cards, consisting of 4 jacks, 4 queens, and 4 kings. One card is to be randomly drawn from the deck of cards. Determine the probability that the card drawn is a) an 8. b) not an 8. c) a club. d) a jack or queen or king (a face card). e) a heart and a spade. f) a card greater than 5 and less than 9. Face cards Jacks Queens Kings Aces Hearts Clubs Diamonds Spades Figure 11.3 Solution a) There are four 8’s in a deck of 52 cards. = = P(8) 4 52 1 13 b) = − = − = P P (not an 8) 1 (8) 1 1 13 12 13 StatCrunch Applets Experiment Draw Cards
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