# Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY

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1 Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY 1. Jack and Jill do not like washing dishes. They decide to use a random method to select whose turn it is. They put some red and blue tiles in a container. Without looking, each draws one tile. If they match, Jack washes. If they are different colors, Jill washes. How many of each color tile could be in the container to make this method "fair"? ("Fair" means they have the same chance of washing every night.) 3. A security specialist is designing a code for a security system. The code will use only the letters A, B, C. If the specialist wants the probability of guessing the code at random to be less than 1/1,000,000, how long must the code be? 3. Suppose you re on a game show, and you are given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then says to you, Do you want to pick door No. 2? Is it to your advantage to switch your choice? Int. Algebra_Ch. 12 Summary pg 1 of 17

2 COUNTING PRINCIPLE 1. You are at your school cafeteria that allows you to choose a lunch meal from a set menu. You have two choices for the main course (a hamburger or a pizza), two choices of a drink (orange juice, apple juice) and three choices of dessert (cookies, ice-cream, jello). How many different meal combos (one main course, one drink, and one dessert) can you select? Explain. 2. An ice-cream shop offers 2 types of cones, 31 flavors of ice cream, and 12 toppings. How many different ice-cream cones can a customer order if each order only has one cone, one flavor and one topping? 3. How many different ways can a license plate be formed a) if 7 letters are used and no letter can be repeated? b) if 4 letters followed by 3 digits are used, no letter or digit can be repeated? c) if 7 letters or digits are used, each letter or digit can be repeated, and the first character must be a number? 4. A multiple-choice exam consists of 15 questions, each of which has 4 possible answers. If a student guesses on all the questions, how many different sets of answers are possible? Counting Principle If event M can occur in m ways and event N can occur in n ways, then the event M followed by event N can occur in ways. 5. A restaurant offers 5 choices of appetizer, 10 choices of main meal and 4 choices of dessert. A customer can choose to eat just one course, or two different courses, or all three courses. Assuming all choices are available, how many different possible meals does the restaurant offer? Write on a separate sheet and attach to this packet. Int. Algebra_Ch. 12 Summary pg 2 of 17

3 Permutations Linear Permutations with repetitions: 1. How many different codes are possible for this combination lock? 2. A password is constructed using the following criteria: It must have six characters with only letters (A-Z) or digits (0-9). The first character must be a letter. The characters can be repeated. How many different passwords are possible? Linear Permutations without repetitions: 1. How many ways can ten people be arranged for a photo lineup in one horizontal line? 2. How many different batting orders are there in a 9-person softball team? 3. Lynn has 59 books to arrange on a shelf. How many different ways are there to arrange all the books? Suppose you have n objects to be arranged in a line, how many different permutations (ordered arrangements are possible? 4. There are 10 finalists in a figure skating competition. How many ways can a gold, silver, and bronze medal be awarded? 5. A special type of password consists of four different letters of the alphabet, where each letter is used only once. How many different possible passwords are there? 6. You have 100 pictures in your iphone. You want to select 24 of them to arrange in your Facebook page. How many different arrangements of pictures are possible? Suppose you are choosing r objects from n objects to be arranged in a line, how many different permutations (ordered arrangements) are possible? Int. Algebra_Ch. 12 Summary pg 3 of 17

4 7. How many different arrangements of the letters in the word MATH are possible? 8. How many different arrangements of the letters in the word ALGEBRA are possible? 9. How many different arrangements of the letters in the word MATHEMATICS are possible? 10. How many different arrangements of the letters in the word DEPENDENT are possible? Circular Permutations: 11. Four people are seated around a circular table. How many different seating arrangements are possible? Note: The following arrangements are considered the same. D C A C D B B A 12. Becky is making a bracelet with 12 beads of different colors, how many different bracelets are possible? Suppose you have n objects to be arranged in a circle, how many different circular permutations are possible? Int. Algebra_Ch. 12 Summary pg 4 of 17

5 Combinations Combinations without repetitions: 1. Lily is picking two types of fruits for her fruit bowl. She has three choices available: strawberries, grapes, and peaches. How many different fruit combinations can she make? 2. Tim has 5 shirts in his closet. He is choosing 3 of them to take on a vacation. How many different ways can he do this? 3. In how many ways can four people be selected from a group of six to serve on a committee? 4. At Joe's Pizza Parlor, in addition to cheese there are 8 different toppings. a) If you want to order a pizza with only 3 toppings, how many different ways can you order your pizza? b) If you can order any number of those 8 toppings, then how many different combinations of toppings could you possibly order? Suppose you are choosing a group of r objects from n objects, how many different combinations are possible? (Order does not matter.) Int. Algebra_Ch. 12 Summary pg 5 of 17

6 5. There are 12 boys and 14 girls in Mrs. Smith s math class. a) Find the number of ways that she can select a team of 3 students from the class to work on a group project. b) Find the number of ways that she can select a team of 2 girls and 2 boys from the class to be on a committee. c) Find the number of ways that she can arrange the students in a class photo. 6. In the California Super Lotto Plus lottery, each ticket consists of a set of five numbers chosen from 1-47, in which no number is repeated and the order does not matter; and one MEGA number chosen from a) How many different tickets can be formed? You win the jackpot if your ticket has all numbers matching the winning numbers that are randomly drawn by the lottery official. What is your probability of winning the jackpot if you buy only one ticket? b) A ticket matches all five of the first five winning numbers but not the MEGA number. How many different tickets of this kind are possible? What is the probability to get such a ticket? c) You win \$1 if your ticket matches three of the first five numbers only but not the MEGA number. How many different tickets of this kind are possible? What is the probability of winning \$1 if you only buy one ticket? d) What is the probability that a ticket would match only four of the first five numbers and the MEGA number? Int. Algebra_Ch. 12 Summary pg 6 of 17

7 6. A multiple-choice exam consists of 15 questions, each of which has 4 possible answers with only one right answer. a) If a student guesses on all the questions, what is the probability that the student will have all the questions right? b) If a student guesses on all the questions, what is the probability that the student will have exactly 10 of the questions right? 7. Five cards are selected from a deck of 52 cards to form a poker hand. a) How many different ways can this be done? b) A flush is a poker hand such as Q , where all five cards are of the same suit, but not necessarily in sequence. What is the probability of getting a flush? c) A full house is a hand such as , that contains three matching cards of one rank and two matching cards of another rank. What is the probability of getting a full house? d) Find the probability that a poker hand would have exactly one pair. Find the probability that a poker hand would have exactly two pairs. Int. Algebra_Ch. 12 Summary pg 7 of 17

8 A Plinko Game In this game, a ball is dropped from the top. Player wins the amount that the ball lands in at the bottom. \$1000 \$500 \$0 \$100 \$0 \$500 \$1000 Determine the probability of winning for each amount. Explain your thinking process. Int. Algebra_Ch. 12 Summary pg 8 of 17

9 Pascal s Triangle Determine the number of possible ways to get to each opening in the Plinko s game board. \$1000 \$500 \$0 \$100 \$0 \$500 \$1000 Add up all the numbers in each row. What do you notice? Describe other patterns that you observe in the Pascal s Triangle. Int. Algebra_Ch. 12 Summary pg 9 of 17

10 Consider the following games: Game 1: Flip two coins. If you get exactly two heads, you win. Game 2: Flip three coins. If you get exactly two heads, you win. Game 3: Flip four coins. If you get exactly two heads, you win. Game 4: Flip five coins. If you get exactly two heads, you win. Which game gives you the greatest chance of winning? Explain. Int. Algebra_Ch. 12 Summary pg 10 of 17

11 Binomial Expansion 1. Expand each binomial by multiplying and using the distributive property. (H + T) 0 (H + T) 1 + (H + T) (H + T) (H + T) Describe all the patterns that you observe in the binomial expansions. 3. Write the expansion of each of the following: a) (H + T) 5 b) (H + T) 8 c) If you expand (H + T) 20, what would be the coefficient of the term with H 20? What would be the 2 nd term? 15 th term? 20 th term? What would be the term that includes T 3? Int. Algebra_Ch. 12 Summary pg 11 of 17

12 Write the coefficients of the binomial expansion (Pascal s Triangle) in terms of the number of combinations n! choose k, n C k or n \$ # & " k % : (H + T) 0 (H + T) 1 + (H + T) (H + T) (H + T) Write the general expansion for (H + T) n, where n is any whole number. If you flip a coin n times, what is the probability of getting exactly r heads, for r n? More practice: 1. Suppose you flip a coin three times. What is the probability that will have no head? 1 head? 2 heads? 3 heads? 2. Of four children, what are the possibilities of having boys & girls? How many of those possibilities will have no boy? 1 boy? 2 boys? 3 boys? 4 boys? Int. Algebra_Ch. 12 Summary pg 12 of 17

13 3. Mrs. Vu gives her class a test of 15 multiple-choice questions. Each question has 4 answer choices, with only one correct choice. A student guesses randomly on all the questions. a) What is the probability of getting all 15 questions answered correctly? b) What is the probability of getting none of the 15 questions answered correctly? c) What is the probability of getting exactly 10 questions answered correctly? d) What is the probability of getting at least 10 questions answered correctly? Bernoulli s principle: Suppose an experiment consists of n independent repetitions of an experiment with two outcomes, called success and failure. Let p(success) = s and p(failure) = f. The probability of getting r successes out of n independent repetitions is Int. Algebra_Ch. 12 Summary pg 13 of 17

14 1. Expand the binomial: a) (x y) 5 b) (3x + 2) 8 c) (2x + y ) 6 2. How many terms are there in the expansion of (2a b) 20? Find the 5 th term in the expansion of (2a b) a) Explain why the expansion (x + y) 12 cannot have a term containing x 6 y 7. b) Explain why 24a 4 b 5 cannot be a term in the expansion of (a + b) 9. Int. Algebra_Ch. 12 Summary pg 14 of 17

15 Mutually Exclusive/Inclusive Events Two events A & B are mutually exclusive if they do not share any outcomes. 1) If a card is randomly selected from a standard deck of 52 cards, a) what is the probability of getting a king or a queen? b) what is the probability of getting a king and a queen? c) what is the probability of getting a king or a heart? d) what is the probability of getting face card or a diamond? e) what is the probability of getting a face card and a diamond? f) what is the probability of getting a face card or a king? Which of the above events are mutually exclusive? 2) Sixteen people study French, 21 study Spanish and there are 30 altogether. Use a Venn diagram to describe the situation. What is the probability that a person chosen random from the group would take both French and Spanish? French only? 3) In a class of 32 children, 16 have a skateboard, 12 have a bicycle and 17 have a scooter. 5 of them have a skateboard and a bicycle. 7 of them have a skateboard and a scooter. 4 of them have a bicycle and a scooter. They all have at least one of the three things. What is the probability that a child chosen at random from the class has a scooter but not a bicycle? Two events are mutually exclusive if If A & B are two inclusive events then P(A or B) = p(a B) = If A & B are mutually exclusive then p(a B) = so P(A or B) = p(a B) = Int. Algebra_Ch. 12 Summary pg 15 of 17

16 Independent Events 1) What is the probability that a family having four boys in a row? 2) Two dice are rolled. What is the probability of getting an even number on one die and a 1 on the other? Two events are independent if the result from one even has no effect on the other. If A & B are two independent events then P(A and B) = p(a B) = Dependent Events & Conditional Probability 3) Two cards are drawn from a standard deck of cards without replacement. a) What is the probability of drawing an ace and then a king? b) What is the probability of drawing two kings? Two events are dependent if If A & B are two dependent events then P(A and B) = p(a B) = 4) In a high school, 54% of the students are girls and 62% of the students play sports. Half of the girls at the school play sports. a) What percentage of the students who play sports are boys? b) If a student is chosen at random, what is the probability that it is a boy who does not play sports? Int. Algebra_Ch. 12 Summary pg 16 of 17

17 5) If it rains tomorrow, the probability is 60% that James will stay home and watch a movie. If it does not rain tomorrow, there is a 10% chance that he will stay home and watch a movie. Suppose that the chance of rain tomorrow is 80%, what is the probability that James will stay home and watch a movie? 6) Two identical jars are on a table. Jar A contains 10 chocolate chip cookies and 5 peanut butter cookies. Jar B contains 5 chocolate chip cookies and 20 peanut butter cookies. What is the probability that a cookie selected at random is a chocolate chip cookie? 7) A particular drug test is 3% likely to produce a false positive result, that is, of 100 people that have not used drug were tested, the test reports would show 3 positive cases. The same drug test is 5% likely to produce a false negative result, that is, of 100 people that have used drug were tested, the test reports would show 5 negative cases. At a particular high school, the probability of a student having used drugs is 2%. What is the probability that a student tested positive actually used drugs? Int. Algebra_Ch. 12 Summary pg 17 of 17

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