158 CHAPTER 3 Probability B A A and B Outcomes here are double counted by P(A) + P(B) 4♣ 4♦ 4♥ 4♠ A♣ A♦ A♥ A♠ 44 other cards Deck of 52 Cards Odd Less than three 3 5 1 2 4 6 Roll a Die TRY IT YOURSELF 1 Determine whether the events are mutually exclusive. Explain your reasoning. 1. Event A: Randomly select a jack from a standard deck of 52 playing cards. Event B: Randomly select a face card from a standard deck of 52 playing cards. 2. Event A: Randomly select a vehicle that is a Ford. Event B: Randomly select a vehicle that is a Toyota. Answer: Page A38 The Addition Rule The probability that event A or B will occur, P1A or B2, is given by P1A or B2 = P1A2 + P1B2 - P1A and B2. If events A and B are mutually exclusive, then the rule can be simplified to P1A or B2 = P1A2 + P1B2. Events A and B are mutually exclusive. This simplified rule can be extended to any number of mutually exclusive events. The Addition Rule for the Probability of A or B In words, to find the probability that one event or the other will occur, add the individual probabilities of each event and subtract the probability that they both occur. As shown in the Venn diagram at the left, subtracting P1A and B2 avoids double counting the probability of outcomes that occur in both A and B. Using the Addition Rule to Find Probabilities 1. You select a card from a standard deck of 52 playing cards. Find the probability that the card is a 4 or an ace. 2. You roll a die. Find the probability of rolling a number less than 3 or rolling an odd number. SOLUTION 1. A card that is a 4 cannot be an ace. So, the events are mutually exclusive, as shown in the Venn diagram. The probability of selecting a 4 or an ace is P14 or ace2 = P142 + P1ace2 = 4 52 + 4 52 = 8 52 = 2 13 ≈ 0.154. 2. The events are not mutually exclusive because 1 is an outcome of both events, as shown in the Venn diagram. So, the probability of rolling a number less than 3 or an odd number is P1less than 3 or odd2 = P1less than 32 + P1odd2 - P1less than 3 and odd2 = 2 6 + 3 6 - 1 6 = 4 6 = 2 3 ≈ 0.667. EXAMPLE 2 To explore this topic further, see Activity 3.3 on page 166. 3.3
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