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Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 (For help, go to Lesson 6-4.) A bag has 5 blue (B) chips, 4 red (R) chips, and 3 tan (T) chips. Find each probability for choosing a chip at random from the bag. 1. P(R) 2. P(not R) 3. P(B) 4. P(R or B) 5. P(T) 6. P(B or T) Check Skills You’ll Need 12-4 Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 Solutions 1. favorable outcomes drawing a red chip 4 1 = = = all possible outcomes 12 12 3 2. favorable outcomes drawing a chip that is not red 8 2 = = = all possible outcomes 12 12 3 3. favorable outcomes drawing a blue chip 5 = = all possible outcomes 12 12 favorable outcomes 4. all possible outcomes = drawing a red or blue chip 9 3 = = 12 12 4 5. favorable outcomes drawing a tan chip 3 1 = = = all possible outcomes 12 12 4 6. favorable outcomes drawing a blue or tan chip 8 2 = = = all possible outcomes 12 12 3 12-4 Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 The school cafeteria sells sandwiches for which you can choose one item from each of the following categories: two breads (wheat or white), two meats (ham or turkey), and two condiments (mayonnaise or mustard). Draw a tree diagram to find the number of sandwich choices. ham wheat turkey ham white turkey mayonnaise mustard mayonnaise mustard mayonnaise mustard mayonnaise mustard There are 8 possible sandwich choices. 12-4 Each branch of the “tree” represents one choice—for example, wheat-hammayonnaise. Quick Check Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 In some state lotteries, the winning number is made up of five digits chosen at random. Suppose a player buys 5 tickets with different numbers. What is the probability that the player has a winning number? First find the number of possible outcomes. For each digit, there are 10 possible outcomes, 0 through 9. 1st digit 2nd digit 3rd digit 4th digit 5th digit outcomes outcomes outcomes outcomes outcomes • • • • 10 10 10 10 10 total outcomes = 100,000 Then find the probability when there are five favorable outcomes. number of favorable outcomes 5 P(winning number) = number of possible outcomes = 100,000 The probability is 5 1 , or . 100,000 20,000 12-4 Quick Check Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 Use the following information for Questions 1 and 2. In a game, a number cube is tossed to determine the number of spaces to move, and a coin is tossed to determine forward or backward movement. 1. How many possible outcomes are there? 12 2. What is the theoretical probability you will move four spaces? 1 6 3. How many different three-digit whole numbers are possible using the digits 1, 2, 3, 4, and 5? 125 12-4 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 (For help, go to Lesson 5-4.) Multiply. 1. 3 1 • 5 5 4. 5 • 4 9 8 2. 1 2 • 4 4 5. 4 • 3 7 6 3. 4 2 • 10 10 6. 9 • 8 10 9 Check Skills You’ll Need 12-5 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Solutions 3 20 2 1 12 2 2. 16 = 8 1. 25 5 4. 72 = 18 5. 42 = 7 12-5 8 2 3. 100 = 25 72 4 6. 90 = 5 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Quick Check You roll a number cube once. Then you roll it again. What is the probability that you get 5 on the first roll and a number less than 4 on the second roll? P(5) = 1 6 There is one 5 among 6 numbers on a number cube. P(less than 4) = 3 6 There are three numbers less than 4 on a number cube. P(5, then less than 4) = P(5) • P(less than 4) 1 3 = 6 • 6 = 3 1 , or 36 12 The probability of rolling 5 and then a number less than 4 is 1 . 12 12-5 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Three girls and two boys volunteer to represent their class at a school assembly. The teacher selects one name and then another from a bag containing the five students’ names. What is the probability that both representatives will be boys? P(boy) = 2 5 Two of five students are boys. P(boy after boy) = 1 4 If a boy’s name is drawn, one of the four remaining students is a boy. P(boy, then boy) = P(boy) • P(boy after boy) 2 1 = 5 • 4 = 2 1 , or 20 10 Substitute. Simplify. The probability that both representatives will be boys is 12-5 1 . 10 Quick Check Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Solve. 1. You roll a number cube once. Then you roll it again. What is the probability that you get 6 on the first roll and a number greater than 3 on the second roll? 1 12 2. Suppose there are three white marbles and three black marbles in a bag and you want to remove two marbles. What is the probability that you will select a white marble and then a black marble? Express your answer as a percent. 30% 12-5 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Solve. 3. Each of five girls and seven boys wants to be one of the two announcers for a variety show. To be fair, a teacher puts the names of the twelve students in a hat and draws two. What is the probability that the teacher will draw the names of two boys? Of two girls? 7 ; 5 22 33 12-5