Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations

Transcription

1 Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684) III. Factorial: (see Glossary) IV. Permutations of n Objects Taken r at a Time: V. Permutations with Repetition:

2 Examples: Notes 10.1 page 2 1. At a blood drive, blood can be labeled one of four types (A, B, AB, or O), one of two Rh factors (+ or ), and one of two genders (F or M). How many different ways can blood be labeled? 2. At a used book sale, you are interested in 5 novels, 3 books of nonfiction, and 7 comic books. If you buy one of each kind, how many different choices do you have? 3. The digits 0, 1, 2, 3, and 4 are used to generate four-digit customer codes. How many different codes are possible if digits a. can be repeated? b. cannot be repeated? 4. A license plate consists of two digits followed by four letters. How many different license plates can be made if the letters O and I cannot be used? 5. A television news director has 8 news stories to present on the evening news. a. How many different ways can the stories be presented? b. If only 3 of the stories will be presented, how many possible ways can a lead story, a second story, and a closing story be presented? 6. How many different ways can 4 raffle tickets be selected from 50 tickets if each ticket wins a different prize? 7. Find the number of distinguishable permutations of the letters in the following: a. CARROLL b. RAPP c. SPAGNUOLO

3 Algebra 2 Notes Section 10.2: Use Combinations and the Binomial Theorem Objective(s): Vocabulary: I. Combination: II. Combinations of n Objects Taken r at a Time: III. Multiple Events: When finding the number of ways. When finding the number of ways. IV. Subtracting Possibilities: Counting problems that involve phrases like are sometimes easier to solve by or possibilities you. V. Pascal's Triangle:

4 Notes 10.2 page 2 VI. Binomial Theorem: Examples: 1. Parents have 10 books that they can read to their children this week. Five of the books are nonfiction and 5 are fiction. a. If the order in which they read the books is not important, how many different sets of 4 books can they choose? b. In how many groups of 4 books are all the books either nonfiction or fiction? 2. The Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshmen. a. How many different committees of exactly 2 seniors and 2 juniors can be chosen? b. How many different committees of at most 4 students can be chosen? 3. You are going to toss 10 different coins. How many different ways will at least 4 of the coins show heads?

5 Notes 10.2 page 3 4. From a collection of 7 baseball caps, you want to trade 3. Use Pascal's triangle to find the number of combinations of 3 caps that can be traded. 5. Use the binomial theorem to write the binomial expansion of (x + y) Use the binomial theorem to write the binomial expansion of (x y) You toss a coin 10 times. In how many ways can you get exactly 8 heads? HINT: Use the binomial expansion of (H + T) 10 to help with the counting.

6

7 Algebra 2 Notes Section 10.3: Define and Use Probability Objective(s): Vocabulary: I. Outcome: II. Event: III. Sample Space: IV. Probability: V. Theoretical Probability of an Event: VI. Experimental Probability of an Event: P(A) = VII. Geometric probabilities:

8 Examples: 1. In an experiment, three coins are flipped. List the possible outcomes in the sample space of the experiment. Notes 10.3 page 2 2. You pick a card from a standard deck of 52 playing cards. Find the probability of a. picking an 8 b. picking a red king 3. When two tiles with numbers between 1 and 10 are chosen from two different bags, there are 100 possible outcomes. Find the probability that a. the sum of the two numbers is not 10 b. the product of the numbers is greater than At a school dance, the parents sell pizza slices. The table shows the number of pizza slices that are available. A student chooses a slice at random. What is the probability that the student chooses a thin crust slice with pepperoni? Pepperoni Plain Cheese Thin Crust Thick Crust Two standard six-sided dice are rolled. Find the probability of a. rolling a 6. b. rolling a number larger than 9.

9 Notes 10.3 page 3 6. A cereal company plans to put 5 new cereals on the market: a wheat cereal, a rice cereal, a corn cereal, an oat cereal, and a multigrain cereal. The order in which the cereals are introduced will be randomly selected. Each cereal will have a different price. a. What is the probability that the cereals are introduced in order of their suggested retail price? b. What is the probability that the first cereal introduced will be the multigrain cereal? 7. The blood types for a sample of donors at a blood drive are displayed in the bar graph. Find the experimental probability that a randomly selected blood donor would have blood type O. 8. The sections of a spinner are numbered 1 through 12. Each section of the spinner has the same area. You spin the spinner 180 times. The table shows the results. For which number is the experimental probability of stopping on the number the same as the theoretical probability? Spinner Results

10 9. Find the probability that the polynomial Notes 10.3 page 4 2 x x c can be factored if c is a randomly chosen integer from 1 to Find the probability that a dart thrown at the rectangular board hits one of the triangles. Assume that the dart is equally likely to hit any point inside the board.

11 Algebra 2 Notes Section 10.4: Find Probabilities of Disjoint and Overlapping Events Objective(s): Vocabulary: I. Compound event: II. Overlapping events: III. Disjoint/mutually exclusive: IV. Probability of Compound Events: V. Complement events: VI. Probability of the Complement of an Event: Examples: 1. As a class: A six-sided die is rolled. Draw a Venn diagram that relates the two events. Then decide whether the events are disjoint or overlapping. a. Event A: The result is an even number. b. Event A: The result is 2 or 4. Event B: The result is a prime number. Event B: The result is an odd number.

12 Notes 10.4 page 2 2. Work with a partner. A six-sided die is rolled. For each pair of events, find the probabilities. a. Event A: The result is an even number. Event B: The result is a prime number. a) P(A) = b) P(B) = c) P(A and B) = d) P(A or B) = b. Event A: The result is a 2 or 4. Event B: The result is an odd number. a) P(A) = b) P(B) = c) P(A and B) = d) P(A or B) = 3. Events A and B are disjoint. P A 2 1 and P B. Find P A B 3 6 or. 4. P A 0.8, P B 0.05, and P A or B 0.6. Find P A B and. 5. A six-sided die is rolled. What is the probability that the number rolled is less than 3 or greater than 5? P(n < 3 or n > 5) = 6. A six-sided die is rolled. What is the probability of rolling a number greater than 4 or even? P(n > 4 or n is even) =

13 Notes 10.4 page 3 7. Of 100 students surveyed, 92 own either a car or a computer. Also, 65 own cars and 82 own computers. What is the probability that a randomly selected student owns both a car and a computer? P(car and computer) = 8. A card is drawn from a standard deck of 52 cards. Find the probability of each event. a. The card is not a 7. b. The card is not a red face card. P(not a 7) = P(not a red face card) = 9. Two six-sided dice are rolled. What is the probability that the sum is neither 4 nor 8?

14 Algebra 2 Notes Section 10.5: Find Probabilities of Independent and Dependent Events Objective(s): Vocabulary: I. Independent Events: II. Probability of Independent Events: III. Dependent Events: IV. Conditional Probability: V. Probability of Dependent Events: (top of p. 720) The formula for finding probabilities of dependent events Examples: 1. You are trying to guess a three-letter password that uses only the letters A, E, I, O, U, and Y. Letters can be used more than once. Find the probability that you pick the correct password YOU. 2. You are trying to guess a three-letter password that uses only the letters A, E, I, O, U, and Y. Letters cannot be used more than once. Find the probability that you pick the correct password AIE.

15 Method Age 3. In a survey at a football game, 50 of 75 male fans and 40 of 50 female fans said that they favor the new team mascot. If 1 male and 1 female fan are randomly selected, what is the probability that both favor the new mascot? Notes 10.5 page 2 4. A survey found that 46% of parents surveyed say that they read to their children at least once a week. If 3 parents are selected at random, what is the probability that all 3 will say that they read to their children at least once a week? 5. During each of the 5 days of a particular week, an employee is randomly given 1 of 10 prizes. All prizes are available each day, and one of the prizes is a $500 gift certificate. What is the probability that an employee receives the $500 prize at least once? In Exercises 6 and 7 complete the two-way table Arrival Tardy Walk 22 On Time Tot al City Bus 60 Total Response Yes No Total Under Over Total 30 75

16 Response Notes 10.5 page 3 8. The table shows the status of 200 registered college students. Part Time Full Time Female Male a. What is the probability that a randomly selected student is female? P(Female) = b. What is the probability that a randomly selected student if female, is a full time student? P(Full-time Female) = 9. For financial reasons, a school district is debating about eliminating a Computer Programming class at the high school. The district surveyed parents, students, and teachers. The results, given as joint relative frequencies, are shown in the two-way table. a. What is the probability that a randomly selected parent voted to eliminate the class? Population Parents Students Teachers Yes b. What is the probability that a randomly selected student did not want to eliminate the class? No c. Determine whether voting to eliminate the class and being a teacher are independent events.

17 Notes 10.5 page You randomly select 2 cards from a standard deck of 52 cards. What is the probability that the first card is a heart and the second is a club if a. you replace the first card before selecting the second. b. you do not replace the first card before selecting the second. 11. Suppose your area has 8 different Internet Service Providers (ISPs) and you and 3 friends randomly select your own ISP. What is the probability that you all choose different ISPs? 12. On a manufacturing line, 20% of all the items produced are defective. Although all the items are inspected before they are shipped, 10% of the items are incorrectly classified as either defective or not defective. What percent of the items will be classified as not defective?