1101 11.7 Basics of Probability Probability concepts can be illustrated using Venn diagrams, as shown in Figure 16. The rectangle there represents the sample space in an experiment. The area inside the circle represents event E, and the area inside the rectangle, but outside the circle, represents event E′. E E9 Figure 16 EXAMPLE 2 Using the Complement of an Event In the experiment of drawing a card from a standard deck, find the probabilities of event E, “the card is an ace,” and of event E′. SOLUTION There are 4 aces in a standard deck of 52 cards, so n1E2 = 4 and n1S2 = 52. P1E2 = n1E2 n1S2 = 4 52 = 1 13 Write in lowest terms. Of the 52 cards, 48 are not aces, so n1E′2 = 48. P1E′2 = n1E′2 n1S2 = 48 52 = 12 13 Write in lowest terms. S Now Try Exercises 23(a) and (b). In Example 2, P1E2 + P1E′2 = 1 13 + 12 13 = 1. This is always true for any event E and its complement E′. Rules for Complementary Events If events E and E′ are complements, then the following hold true. P1E2 +P1E′2 =1 P1E2 =1 −P1E′2 P1E′2 =1 −P1E2 These equations suggest an alternative way to compute the probability of an event. For example, if it is known that P1E2 = 1 13 , then P1E′2 = 1 - 1 13 = 12 13 . Odds Probability statements can be expressed in terms of odds, a comparison of P1E2 with P1E′2. The odds in favor of an event E are expressed as the ratio of P1E2 to P1E′2, or as the quotient P1E2 P1E′2 . Standard deck of 52 cards NOTE A standard deck of 52 cards has four suits: hearts 1, diamonds ◆, spades 0, and clubs 3. There are 13 cards in each suit, including a jack, a queen, and a king (sometimes called the “face cards”), an ace, and cards numbered from 2 to 10. The hearts and diamonds are red, and the spades and clubs are black. We refer to this standard deck of cards in this section.
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