11.5 Or and And Problems 705 Note that the probability of selecting a red chip followed by a blue chip (rb), indicated by P (red and blue), is . 1 9 The probability of selecting a red chip and a blue chip, in any order (rb or br), is . 2 9 In this section, when we ask for P A B ( and ), it means the probability of event A occurring and then event B occurring, in that order. Example 4 An Experiment with Replacement Two cards are to be drawn with replacement from a standard 52-card deck. Determine the probability that two spades will be drawn. Solution Since the deck of 52 cards contains thirteen spades, the probability of drawing a spade on the first draw is . 13 52 The card drawn is then returned to the deck. Therefore, the probability of drawing a spade on the second draw remains . 13 52 If we let A represent the drawing of the first spade and B represent the drawing of the second spade, the formula may be written as follows. P A B P A P B P P and P P ( and ) ( ) ( ) (2 spades) (spade 1 spade 2) (spade 1) (spade 2) 13 52 13 52 1 4 1 4 1 16 = ⋅ = = ⋅ = ⋅ = ⋅ = 7 Now try Exercise 51 Example 5 An Experiment Without Replacement Two cards are to be drawn without replacement from a standard 52-card deck. Determine the probability that two spades will be drawn. Solution This example is similar to Example 4. However, this time we are doing the experiment without replacing the first card drawn to the deck before drawing the second card. The probability of drawing a spade on the first draw is . 13 52 When calculating the probability of drawing the second spade, we must assume that the first spade has been drawn. Once this first spade has been drawn, only 51 cards, including 12 spades, remain in the deck. The probability of drawing a spade on the second draw becomes . 12 51 The probability of drawing two spades without replacement is P P P (2 spades) (spade 1) (spade 2) 13 52 12 51 1 4 4 17 1 17 = ⋅ = ⋅ = ⋅ = 7 Now try Exercise 55 Now we introduce independent events . Definition: Independent Events Event A and event B are independent events if the occurrence of either event in no way affects the probability of occurrence of the other event. Rolling dice and tossing coins are examples of independent events. In Example 4, the events are independent, since the first card was returned to the deck. The probability of drawing a spade on the second draw was not affected by the first drawing. Did You Know? All Boys or All Girls m The Schwandt Family Do you know a family where the children are all boys or the children are all girls? The probability that the children in a family are all boys or all girls decreases as the number of children increases. In general, assuming boys and girls are equally likely, if a family has n children, then the probability that all the children are boys is about 1 2n and the probability that all the children are girls is also about . 1 2n For example, consider the Schwandt family of Rockford, Michigan, who had 14 boys before having a girl. The probability that a family has 14 consecutive boys is about 0.00006. 1 2 1 16,384 14 = ≈ Or consider the Higuera family of Paris, Texas, who have 14 girls and no boys. The probability that a family has 14 consecutive girls is also about 0.00006. 1 2 1 16,384 14 = ≈ However, should either of these families have another child, the probability that the child is a boy is still about 1 2 and the probability that the child is a girl is still about . 1 2 Casey Sykes/The Grand Rapids Press/AP Images
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