704 CHAPTER 11 Probability And Problems A second type of probability problem is the and probability problem, which requires obtaining a favorable outcome in each of the given events. For example, suppose that two cards are to be drawn from a standard 52-card deck and we are interested in the probability of drawing two aces (one ace and then a second ace). Only if both cards drawn are aces would this experiment be considered successful. A formula for determining the probability of events A and B, symbolized P(A and B), follows. c) There are 26 red cards and 26 black cards in a standard 52-card deck. It is impossible to draw one card that is both a red card and a black card. Therefore, the events are mutually exclusive. P P P (red or black) (red) (black) 26 52 26 52 52 52 1 = + = + = = Since P(red or black) 1, = a red card or a black card must be drawn. d) There are 12 face cards in a standard 52-card deck. Six of the 12 face cards are red (the jacks, queens, and kings of hearts and diamonds). Thus, drawing a face card and a red card are not mutually exclusive. P P P P face card or red card face card red card face card and red card 12 52 26 52 6 52 32 52 8 13 ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ − ⎛ ⎝⎜ ⎞ ⎠⎟ = + − = = 7 Now try Exercise 25 Probability of A and B To determine the probability of A and B, use the following formula. P A B P A P B ( and ) ( ) ( ) = ⋅ Since we multiply to determine P A B ( and ), this formula is sometimes referred to as the multiplication formula. When using the multiplication formula, we always assume that event A has occurred when calculating P B( ) because we are determining the probability of obtaining a favorable outcome in both of the given events. We will further discuss this type of probability in Section 11.6 when we discuss conditional probability. Unless we specify otherwise, P A B ( and ) indicates that we are determining the probability that event A occurs and then event B occurs (in that order). Consider a bag that contains three chips: 1 red chip (r), 1 blue chip (b), and 1 green chip (g). Suppose that two chips are selected from the bag with replacement. The tree diagram and sample space for the experiment are shown in Fig. 11.14. There are nine possible outcomes for the two selections, as indicated in the sample space. Sample Space Tree Diagram r rr rg rb g b r br bg bb g b r gr gg gb g b r g b Figure 11.14
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