# Apex High School Laura Duncan Road. Apex, NC Wake County Public School System

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1 Apex High School 1501 Laura Duncan Road Apex, NC Wake County Public School System 1

2 CCM2 Unit 6 Probability Unit Description In this unit, students will investigate theoretical and experimental probability, including independent and dependent events, mutually exclusive events, conditional probability, permutations and combinations. Context of Unit By the end of this unit students will be able to List sample spaces, find intersections of unions of sets, with and without Venn diagrams, find probabilities and odds of simple events. Find probabilities of independent and dependent events using Fundamental Counting Principle. Find probabilities of mutually exclusive and inclusive events using addition rules. Find conditional probabilities using Venn diagrams, two-way frequency tables, and use conditional probability to determine if events are independent. Find probabilities using permutations and combinations. Understand the difference between theoretical and experimental probability and find experimental probabilities using simulations. Essential Questions Why can there be a difference between the actual and expected probability? Why can the results of an experiment be different than expected? How does probability influence opinions and decisions in everyday situations? Why would the probabilities of two similar events differ? Enduring Understandings Randomness, bias, and independence can greatly influence (help determine) the probability of an event occurring. Gathered data can be manipulated, represented, and interpreted to support different claims. Knowing the probability of an event occurring can help make predictions about a future event. Probability helps us understand and describe random phenomena. Unit Facts The probability of an event occurring is the number of successes divided by the total number of possible outcomes. The probability of an event occurring is between 0 and 1 inclusive. A probability of zero means the event will not occur. A probability of one means the event will definitely occur. A sample space is the set of all possible outcomes of an event. A union of two sets consists of all of the members in either set. The union of A and B can be expressed as A B. An intersection of two sets consists of only the members in both sets. The intersection of A and B can be expressed as A B. The complement of a set consists of all of the members not in the set. The complement of A can be expressed as A c Complement of a set is defined as P( B c ) = 1 - P(B). The conditional probability of A given B is expressed as P(A B). 2

3 The conditional probability of P(A B) = P(A and B)/P(B). The probability of either of two independent events, A and B, occurring is denoted as P(A or B). P(A or B) = P(A) + P(B) - P (A and B). This is called the addition rule. The probability of two independent events, A and B, both occurring is denoted as P(A and B). P(A and B)= P(A)P(B). This is the multiplication rule. Two events are independent if and only if the probability of the events occurring together is the product of their probabilities. A and B are independent events if P(A B) = P(A) and P(B A) = P(B). Data can be represented in a two way frequency table when the elements in the data set fall into two different categories. A permutation is defined as an arrangement of those objects into a particular order. The formula for permutation is P(n,r) = n!/(n-r)! A combination is defined as an arrangement of objects when order doesn t matter. The formula for combinations is C(n,r) = n!/ ((n-r)!r!). Probabilities will change depending on if the object(s) are replaced after each pick. The odds of an event occurring is the number of successes divided by the number of failures. 3

4 Guided Notes: Sample Spaces, Subsets, and Basic Probability Sample Space: List the sample space, S, for each of the following: a. Tossing a coin: b. Rolling a six-sided die: c. Drawing a marble from a bag that contains two red, three blue, and one white marble: Intersection of two sets (A B): Union of two sets (A B): Example: Given the following sets, find A B and A B A = {1,3,5,7,9,11,13,15} B = {0,3,6,9,12,15} A B = A B = Venn Diagram: Picture: Example: Use the Venn Diagram to answer the following questions: 1. What are the elements of set A? 2. What are the elements of set B? 3. Why are 1, 2, and 4 in both sets? 4. What is A B? 5. What is A B? 4

5 Example: In a class of 60 students, 21 sign up for chorus, 29 sign up for band, and 5 take both. 15 students in the class are not enrolled in either band or chorus. 6. Put this information into a Venn Diagram. If the sample space, S, is the set of all students in the class, let students in chorus be set A and students in band be set B. 7. What is A B? 8. What is A B? Compliment of a set: Ex: S = { -3,-2,-1,0,1,2,3,4, } A = { -2,0,2,4, } If A is a subset of S, what is A C? Example: Use the Venn Diagram above to find the following: 9. What is A C? B C? 10. What is (A B) C? 11. What is (A B) C? Basic Probability Probability of an Event: P(E) = Note that P(A C ) is every outcome except (or not) A, so we can find P(A C ) by finding. Why do you think this works? Example: An experiment consists of tossing three coins. 12. List the sample space for the outcomes of the experiment. 5

6 13. Find the following probabilities: a. P(all heads) b. P(two tails) c. P(no heads) d. P(at least one tail) e. How could you use compliments to find d? Example: A bag contains six red marbles, four blue marbles, two yellow marbles and 3 white marbles. One marble is drawn at random. 14. List the sample space for this experiment. 15. Find the following probabilities: a. P(red) b. P(blue or white) c. P(not yellow) Note that we could either count all the outcomes that are not yellow or we could think of this as being 1 P(yellow). Why is this? Example: A card is drawn at random from a standard deck of cards. Find each of the following: 16. P(heart) 17. P(black card) 18. P(2 or jack) 19. P(not a heart) Odds: The odds of an event occurring are equal to the ratio of to. Odds = 6

7 20. The weather forecast for Saturday says there is a 75% chance of rain. What are the odds that it will rain on Saturday? What does the 75% in this problem mean? The favorable outcome in this problem is that it rains: Odds(rain) = Should you make outdoor plans for Saturday? 21. What are the odds of drawing an ace at random from a standard deck of cards? 7

8 Name CCM2 Unit 6 Lesson 1 Homework Organize the data into the circles. Factors of 64: 1, 2, 4, 8, 16, 32, 64 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Answer Questions about the diagram below Fall Sports 13 8 Winter Sports Spring Sports 19 1) How many students play sports year-round? 2) How many students play sports in the spring and fall? 3) How many students play sports in the winter and fall? 4) How many students play sports in the winter and spring? 5) How many students play only one sport? 6) How many students play at least two sports? 8

9 7. Suppose you have a standard deck of 52 cards. Let: a. Describe for this experiment, and find the probability of. b. Describe for this experiment, and find the probability of. 8) Suppose a box contains three balls, one red, one blue, and one white. One ball is selected, its color is observed, and then the ball is placed back in the box. The balls are scrambled, and again, a ball is selected and its color is observed. What is the sample space of the experiment? 9) Suppose you have a jar of candies: 4 red, 5 purple and 7 green. Find the following probabilities of the following events: Selecting a red candy. Selecting a purple candy. Selecting a green or red candy. Selecting a yellow candy. Selecting any color except a green candy Find the odds of selecting a red candy Find the odds of selecting a purple or green candy 10) What is the sample space for a single spin of a spinner with red, blue, yellow and green sections spinner? 9

10 What is the sample space for 2 spins of the first spinner? If the spinner is equally likely to land on each color, what is the probability of landing on red in one spin? What is the probability of landing on a primary color in one spin? What is the probability of landing on green both times in two spins? 11) Consider the throw of a die experiment. Assume we define the following events: Describe for this experiment. Describe for this experiment. Calculate and, assuming the die is fair. 10

11 Day 1 and Day 2 Guided Notes: Probability of Independent and Dependent Events Independent Events: Dependent Events: Suppose a die is rolled and then a coin is tossed. Explain why these events are independent. Fill in the table to describe the sample space: Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6 Head Tail How many outcomes are there for rolling the die? How many outcomes are there for tossing the coin? How many outcomes are there in the sample space of rolling the die and tossing the coin? Is there another way to decide how many outcomes are in the sample space? Let s see if this works for another situation. A fast food restaurant offers 5 sandwiches and 3 sides. How many different meals of a sandwich and side can you order? If our theory holds true, how could we find the number of outcomes in the sample space? 11

12 Make a table to see if this is correct. Were we correct? Probabilities of Independent Events The probability of independent events is, denoted by. Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6 Head Tail Fill in the table again and then use the table to find the following probabilities: 1. P(rolling a 3) = 2. P(Tails) = 3. P(rolling a 3 AND getting tails) = 4. P(rolling an even) = 5. P(heads) = 6. P(rolling an even AND getting heads) = What do you notice about the answers to 3 and 6? 12

13 Multiplication Rule of Probability The probability of two independent events occurring can be found by the following formula: Examples: 1. At City High School, 30% of students have part-time jobs and 25% of students are on the honor roll. What is the probability that a student chosen at random has a part-time job and is on the honor roll? Write your answer in context. 2. The following table represents data collected from a grade 12 class in DEF High School. Suppose 1 student was chosen at random from the grade 12 class. (a) What is the probability that the student is female? (b) What is the probability that the student is going to university? Now suppose 2 people both randomly chose 1 student from the grade 12 class. Assume that it's possible for them to choose the same student. (c) What is the probability that the first person chooses a student who is female and the second person chooses a student who is going to university? 3. Suppose a card is chosen at random from a deck of cards, replaced, and then a second card is chosen. Would these events be independent? How do we know? What is the probability that both cards are 7s? 13

14 Probabilities of Depended Events Determine whether the events are independent or dependent: 1. Selecting a marble from a container and selecting a jack from a deck of cards. 2. Rolling a number less than 4 on a die and rolling a number that is even on a second die. 3. Choosing a jack from a deck of cards and choosing another jack, without replacement. 4. Winning a hockey game and scoring a goal. We cannot use the multiplication rule for finding probabilities of dependent events because the one event affects the probability of the other event occurring. Instead, we need to think about how the occurrence of one event will effect the sample space of the second event to determine the probability of the second event occurring. Then we can multiply the new probabilities. Examples: 1. Suppose a card is chosen at random from a deck, the card is NOT replaced, and then a second card is chosen from the same deck. What is the probability that both will be 7s? This is similar the earlier example, but these events are dependent? How do we know? How does the first event affect the sample space of the second event? Now find the probability that both cards will be 7s. 2. A box contains 5 red marbles and 5 purple marbles. What is the probability of drawing 2 purple marbles and 1 red marble in succession without replacement? 3. In Example 2, what is the probability of first drawing all 5 red marbles in succession and then drawing all 5 purple marbles in succession without replacement? 14

15 Name CCM2 Unit 6 Lesson 2 Homework Independent and Dependent Events 1. Determine which of the following are examples of independent or dependent events. a. Rolling a 5 on one die and rolling a 5 on a second die. b. Choosing a cookie from the cookie jar and choosing a jack from a deck of cards. c. Selecting a book from the library and selecting a book that is a mystery novel. d. Going to the beach and bringing an umbrella. e. Getting gasoline for your car and getting diesel fuel for your car. f. Choosing an 8 from a deck of cards, replacing it, and choosing a face card. g. Choosing a jack from a deck of cards and choosing another jack, without replacement. h. Being lunchtime and eating a sandwich. 2. A coin and a die are tossed. Calculate the probability of getting tails and a In Tania's homeroom class, 9% of the students were born in March and 40% of the students have a blood type of O+. What is the probability of a student chosen at random from Tania's homeroom class being born in March and having a blood type of O+? 4. If a baseball player gets a hit in 31% of his at-bats, what it the probability that the baseball player will get a hit in 5 at-bats in a row? 5. What is the probability of tossing 2 coins one after the other and getting 1 head and 1 tail? 6. 2 cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is the probability that they both will be clubs? 7. 2 cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is the probability that they both will be face cards? 8. If the probability of receiving at least 1 piece of mail on any particular day is 22%, what is the probability of not receiving any mail for 3 days in a row? 9. Johnathan is rolling 2 dice and needs to roll an 11 to win the game he is playing. What is the probability that Johnathan wins the game? 15

16 10. Thomas bought a bag of jelly beans that contained 10 red jelly beans, 15 blue jelly beans, and 12 green jelly beans. What is the probability of Thomas reaching into the bag and pulling out a blue or green jelly bean and then reaching in again and pulling out a red jelly bean? Assume that the first jelly bean is not replaced. 11. For question 10, what if the order was reversed? In other words, what is the probability of Thomas reaching into the bag and pulling out a red jelly bean and then reaching in again and pulling out a blue or green jelly bean without replacement? 12. What is the probability of drawing 2 face cards one after the other from a standard deck of cards without replacement? 13. There are 3 quarters, 7 dimes, 13 nickels, and 27 pennies in Jonah's piggy bank. If Jonah chooses 2 of the coins at random one after the other, what is the probability that the first coin chosen is a nickel and the second coin chosen is a quarter? Assume that the first coin is not replaced. 14. For question 13, what is the probability that neither of the 2 coins that Jonah chooses are dimes? Assume that the first coin is not replaced. 15. Jenny bought a half-dozen doughnuts, and she plans to randomly select 1 doughnut each morning and eat it for breakfast until all the doughnuts are gone. If there are 3 glazed, 1 jelly, and 2 plain doughnuts, what is the probability that the last doughnut Jenny eats is a jelly doughnut? 16. Steve will draw 2 cards one after the other from a standard deck of cards without replacement. What is the probability that his 2 cards will consist of a heart and a diamond? Source: 16

17 Day 2 Guided Notes: Mutually Exclusive and Inclusive events Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or you roll a 2? Can these both occur at the same time? Why or why not? Mutually Exclusive Events: The probability of two mutually exclusive events occurring at the same time, P(A and B), is To find the probability of one of two mutually exclusive events occurring, use the following formula: Examples: 1. If you randomly chose one of the integers 1 10, what is the probability of choosing either an odd number or an even number? Are these mutually exclusive events? Why or why not? Complete the following statement: P(odd or even) = P( ) + P( ) Now fill in with numbers: P(odd or even) = + = Does this answer make sense? 2. Two fair dice are rolled. What is the probability of getting a sum less than 7 or a sum equal to 10? Are these events mutually exclusive? Sometimes using a table of outcomes is useful. Complete the following table using the sums of two dice: P(getting a sum less than 7 OR sum of 10) = This means 17

18 Mutually Inclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number or a number less than 4? Can these both occur at the same time? If so, when? Mutually Inclusive Events: Probability of the Union of Two Events: The Addition Rule: *** *** Examples: 1. What is the probability of choosing a card from a deck of cards that is a club or a ten? P(choosing a club or a ten) = 2. What is the probability of choosing a number from 1 to 10 that is less than 5 or odd? 3. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the first 10 letters of the alphabet on it or randomly choosing a tile with a vowel on it? 4. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the last 5 letters of the alphabet on it or randomly choosing a tile with a vowel on it? Source: 18

19 Name CCM2 Unit 6 Lesson 3 Homework Mutually Exclusive and Inclusive Events 1. 2 dice are tossed. What is the probability of obtaining a sum equal to 6? 2. 2 dice are tossed. What is the probability of obtaining a sum less than 6? 3. 2 dice are tossed. What is the probability of obtaining a sum of at least 6? 4. Thomas bought a bag of jelly beans that contained 10 red jelly beans, 15 blue jelly beans, and 12 green jelly beans. What is the probability of Thomas reaching into the bag and pulling out a blue or green jelly bean? 5. A card is chosen at random from a standard deck of cards. What is the probability that the card chosen is a heart or spade? Are these events mutually exclusive? 6. 3 coins are tossed simultaneously. What is the probability of getting 3 heads or 3 tails? Are these events mutually exclusive? 7. In question 6, what is the probability of getting 3 heads and 3 tails when tossing the 3 coins simultaneously? 8. Are randomly choosing a person who is left-handed and randomly choosing a person who is right-handed mutually exclusive events? Explain your answer. 9. Suppose 2 events are mutually exclusive events. If one of the events is randomly choosing a boy from the freshman class of a high school, what could the other event be? Explain your answer. 10. Consider a sample set as. Event is the multiples of 4, while event is the multiples of 5. What is the probability that a number chosen at random will be from both and? 11. For question 10, what is the probability that a number chosen at random will be from either or? 19

20 12. Jack is a student in Bluenose High School. He noticed that a lot of the students in his math class were also in his chemistry class. In fact, of the 60 students in his grade, 28 students were in his math class, 32 students were in his chemistry class, and 15 students were in both his math class and his chemistry class. He decided to calculate what the probability was of selecting a student at random who was either in his math class or his chemistry class, but not both. Draw a Venn diagram and help Jack with his calculation. 13. Brenda did a survey of the students in her classes about whether they liked to get a candy bar or a new math pencil as their reward for positive behavior. She asked all 71 students she taught, and 32 said they would like a candy bar, 25 said they wanted a new pencil, and 4 said they wanted both. If Brenda were to select a student at random from her classes, what is the probability that the student chosen would want: 1. a candy bar or a pencil? 2. neither a candy bar nor a pencil? 14. A card is chosen at random from a standard deck of cards. What is the probability that the card chosen is a heart or a face card? Are these events mutually inclusive? 15. What is the probability of choosing a number from 1 to 10 that is greater than 5 or even? 16. A bag contains 26 tiles with a letter on each, one tile for each letter of the alphabet. What is the probability of reaching into the bag and randomly choosing a tile with one of the letters in the word ENGLISH on it or randomly choosing a tile with a vowel on it? 17. Are randomly choosing a teacher and randomly choosing a father mutually inclusive events? Explain your answer. 18. Suppose 2 events are mutually inclusive events. If one of the events is passing a test, what could the other event be? Explain your answer. Source: 20

21 Day 3 Guided Notes: Conditional Probability Conditional Probability: - Examples of conditional probability: The conditional probability of A given B is expressed as The formula is: Examples of Conditional Probability: 1. You are playing a game of cards where the winner is determined by drawing two cards of the same suit. What is the probability of drawing clubs on the second draw if the first card drawn is a club? 2. A bag contains 6 blue marbles and 2 brown marbles. One marble is randomly drawn and discarded. Then a second marble is drawn. Find the probability that the second marble is brown given that the first marble drawn was blue. 3. In Mr. Jonas' homeroom, 70% of the students have brown hair, 25% have brown eyes, and 5% have both brown hair and brown eyes. A student is excused early to go to a doctor's appointment. If the student has brown hair, what is the probability that the student also has brown eyes? 21

23 3. A survey asked students which types of music they listen to? Out of 200 students, 75 indicated pop music and 45 indicated country music with 22 of these students indicating they listened to both. Use a Venn diagram to find the probability that a randomly selected student listens to pop music given that they listen country music. Using Conditional Probability to Determine if Events are Independent If two events are statistically independent of each other, then: Let s revisit some previous examples and decide if the events are independent. 1. You are playing a game of cards where the winner is determined by drawing two cards of the same suit without replacement. What is the probability of drawing clubs on the second draw if the first card drawn is a club? Are the two events independent? Let drawing the first club be event A and drawing the second club be event B. 2. You are playing a game of cards where the winner is determined by drawing tow cards of the same suit. Each player draws a card, looks at it, then replaces the card randomly in the deck. Then they draw a second card. What is the probability of drawing clubs on the second draw if the first card drawn is a club? Are the two events independent? 3. In Mr. Jonas' homeroom, 70% of the students have brown hair, 25% have brown eyes, and 5% have both brown hair and brown eyes. A student is excused early to go to a doctor's appointment. If the student has brown hair, what is the probability that the student also has brown eyes? Are event A, having brown hair, and event B, having brown eyes, independent? 4. Using the table from the ice cream shop problem, determine whether age and choice of ice cream are independent events. 23

24 Name CCM2 Unit 6 Lesson 4 Homework 1 Conditional Probability 1. Compete the following table using sums from rolling two dice. Us e the table to answer questions fair dice are rolled. What is the probability that the sum is even given that the first die that is rolled is a 2? 3. 2 fair dice are rolled. What is the probability that the sum is even given that the first die rolled is a 5? 4. 2 fair dice are rolled. What is the probability that the sum is odd given that the first die rolled is a 5? 5. Steve and Scott are playing a game of cards with a standard deck of playing cards. Steve deals Scott a black king. What is the probability that Scott s second card will be a red card? 6. Sandra and Karen are playing a game of cards with a standard deck of playing cards. Sandra deals Karen a red seven. What is the probability that Karen s second card will be a black card? 7. Donna discusses with her parents the idea that she should get an allowance. She says that in her class, 55% of her classmates receive an allowance for doing chores, and 25% get an allowance for doing chores and are good to their parents. Her mom asks Donna what the probability is that a classmate will be good to his or her parents given that he or she receives an allowance for doing chores. What should Donna's answer be? 8. At a local high school, the probability that a student speaks English and French is 15%. The probability that a student speaks French is 45%. What is the probability that a student speaks English, given that the student speaks French? 9. On a game show, there are 16 questions: 8 easy, 5 medium-hard, and 3 hard. If contestants are given questions randomly, what is the probability that the first two contestants will get easy questions? 10. On the game show above, what is the probability that the first contestant will get an easy question and the second contestant will get a hard question? 24

25 11. Figure 2.2 shows the counts of earned degrees for several colleges on the East Coast. The level of degree and the gender of the degree recipient were tracked. Row & Column totals are included. a. What is the probability that a randomly selected degree recipient is a female? b. What is the probability that a randomly chosen degree recipient is a man? c. What is the probability that a randomly selected degree recipient is a woman, given that they received a Master's Degree? d. For a randomly selected degree recipient, what is P(Bachelor's Degree Male)? 12. Animals on the endangered species list are given in the table below by type of animal and whether it is domestic or foreign to the United States. Complete the table and answer the following questions. Mammals Birds Reptiles Amphibians Total United States Foreign Total An endangered animal is selected at random. What is the probability that it is: a. a bird found in the United States? b. foreign or a mammal? c. a bird given that it is found in the United States? d. a bird given that it is foreign? Source: 25

26 Day 4 Guided Notes: Permutations and Combinations Fundamental Counting Principle: Example: A student is to roll a die and flip a coin. How many possible outcomes will there be? Example: For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? Permutation: Example: Find the number of ways to arrange the letters ABC: To find the number of Permutations of n items chosen r at a time, you can use the formula for finding P(n,r) or np r : Example: A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? 26

27 You can use your calculator to find permutations: Example: From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? Combination: To find the number of Combinations of n items chosen r at a time, C(n,r) or nc r, you can use the formula: Example: To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? You can use your calculator to find combinations: 27

28 Example: A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? Example: A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Example: The 25-member senior class council is selecting officers for president, vice president and secretary. Emily would like to be president, David would like to be vice president, and Jenna would like to be secretary. If the offices are filled at random, beginning with president, what is the probability that they are selected for these offices? Example: The 25-member senior class council is selecting members for the prom committee. Stephen, Marcus and Sabrina want would like to be on this committee. If the members are selected at random, what is the probability that all three are selected for this committee? 28

29 Name CCM2 Unit 6 Lesson 5 Homework Permutations and Combinations For 1-5, find the number of permutations How many ways can you plant a rose bush, a lavender bush and a hydrangea bush in a row? 5. How many ways can you pick a president, a vice president, a secretary and a treasurer out of 28 people for student council? For 6-10, find the probabilities. 6. What is the probability that a randomly generated arrangement of the letters A,E,L, Q and U will result in spelling the word EQUAL? 7. What is the probability that a randomly generated 3-letter arrangement of the letters in the word SPIN ends with the letter N? 8. A bag contains eight chips numbered 1 through 8. Two chips are drawn randomly from the bag and laid down in the order they were drawn. What is the probability that the 2-digit number formed is divisible by 3? 9. A prepaid telephone calling card comes with a randomly selected 4-digit PIN, using the digits 1 through 9 without repeating any digits. What is the probability that the PIN for a card chosen at random does not contain the number 7? 10. Janine makes a playlist of 8 songs and has her computer randomly shuffle them. If one song is by Little Bow Wow, what is the probability that this song will play first? For 11-13, calculate the number of combinations: For 14-18, a town lottery requires players to choose three different numbers from the numbers 1 through How many different combinations are there? 15. What is the probability that a player s numbers match all three numbers chosen by the computer? 16. What is the probability that two of a player s numbers match the numbers chosen by the computer? 17. What is the probability that one of a player s numbers matches the numbers chosen by the computer? 29

30 18. What is the probability that none of a player s numbers match the numbers chosen by the computer? 19. Looking at the odds that you came up with in question 14, devise a sensible payout plan for the lottery in other words, how big should the prizes be for players who match 1, 2, or all 3 numbers? Assume that tickets cost \$1. Don t forget to take into account the following: a. The town uses the lottery to raise money for schools and sports clubs. b. Selling tickets costs the town a certain amount of money. c. If payouts are too low, nobody will play! Source: 30

31 Day 5 Investigation: Theoretical vs. Experimental Probability Part 1: Theoretical Probability Probability is the chance or likelihood of an event occurring. We will study two types of probability, theoretical and experimental. Theoretical Probability: the probability of an event is the ratio or the number of favorable outcomes to the total possible outcomes. P(Event) = Number or favorable outcomes Total possible outcomes Sample Space: The set of all possible outcomes. For example, the sample space of tossing a coin is {Heads, Tails} because these are the only two possible outcomes. Theoretical probability is based on the set of all possible outcomes, or the sample space. 1. List the sample space for rolling a six-sided die (remember you are listing a set, so you should use brackets {} ): Find the following probabilities: P(2) P(3 or 6) P(odd) P(not a 4) P(1,2,3,4,5, or 6) P(8) 2. List the sample space for tossing two coins: Find the following probabilities: P(two heads) P(one head and one tail) P(head, then tail) P(all tails) P(no tails) 31

32 3. Complete the sample space for tossing two six-sided dice: {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2, ),,,, (3,1),,,,,,,,,,,,,,,,,,,,,,, } Find the following probabilities: P(a 1 and a 4) P(a 1, then a 4) P(sum of 8) P(sum of 12) P(doubles) P(sum of 15) 4. When would you expect the probability of an event occurring to be 1, or 100%? Describe an event whose probability of occurring is When would you expect the probability of an event occurring to be 0, or 0%? Describe an event whose probability of occurring is 0. Part 2: Experimental Probability Experimental Probability: the ratio of the number of times the event occurs to the total number of trials. P(Event) = Number or times the event occurs Total number of trials 1. Do you think that theoretical and experimental probabilities will be the same for a certain event occurring? Explain your answer. 32

33 2. Roll a six-sided die and record the number on the die. Repeat this 9 more times Number on Die Tally Frequency Total 10 Based on your data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) How do these compare to the theoretical probabilities in Part 1? Why do you think they are the same or different? 3. Record your data on the board (number on die and frequency only). Compare your data with other groups in your class. Explain what you observe about your data compared to the other groups. Try to make at least two observations. 4. Combine the frequencies of all the groups in your class with your data and complete the following table: Number on Die Total Frequency Based on the whole class data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) How do these compare to your group s probabilities? How do these compare to the theoretical probabilities from Part 1? 33

34 What do you think would happen to the experimental probabilities if there were 200 trials? 500 trials? 1000 trials? 1,000,000 trials? 5. On your graphing calculator, go to APPS and open Prob Sim. Press any key and then select 2: Roll dice. Click Roll. Notice that there will be a bar on the graph at the right. What does this represent? Now push +1 nine more times. Push the right arrow to see the frequency of each number on the die. How many times did you get a 1? A 2? A 5? Now press the +1, +10, and +50 buttons until you have rolled 100 times. Based on the data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) Press the +50 button until you have rolled 1000 times. Based on the data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) Press the +50 button until you have rolled 5000 times. Based on the data, find the following experimental probabilities: P(2) P(3 or 6) P(odd) P(not a 4) What can you expect to happen to the experimental probabilities in the long run? In other words, as the number of trials increases, what happens to the experimental probabilities? Why can there be differences between experimental and theoretical probabilities in general? 34

35 Part 3: Which one do I use? So when do we use theoretical probability or experimental probability? Theoretical probability is always the best choice, when it can be calculated. But sometimes it is not possible to calculate theoretical probabilities because we cannot possible know all of the possible outcomes. In these cases, experimental probability is appropriate. For example, if we wanted to calculate the probability of a student in the class having green as his or her favorite color, we could not use theoretical probability. We would have to collect data on the favorite colors of each member of the class and use experimental probability. Determine whether theoretical or experimental probability would be appropriate for each of the following. Explain your reasoning: 1. What is the probability of someone tripping on the stairs today between first and second periods? 2. What is the probability of rolling a 3 on a six-sided die, then tossing a coin and getting a head? 3. What is the probability that a student will get 4 of 5 true false questions correct on a quiz? 4. What is the probability that a student is wearing exactly four buttons on his or her clothing today? 35

36 Probability Homework: Experimental vs. Theoretical Name 1) A baseball collector checked 350 cards in case on the shelf and found that 85 of them were damaged. Find the experimental probability of the cards being damaged. Show your work. 2) Jimmy rolls a number cube 30 times. He records that the number 6 was rolled 9 times. According to Jimmy's records, what is the experimental probability of rolling a 6? Show your work. 3) John, Phil, and Mike are going to a bowling match. Suppose the boys randomly sit in the 3 seats next to each other and one of the seats is next to an aisle. What is the probability that John will sit in the seat next to the aisle? 4) In Mrs. Johnson's class there are 12 boys and 16 girls. If Mrs. Johnson draws a name at random, what is the probability that the name will be that of a boy? 5) Antonia has 9 pairs of white socks and 7 pairs of black socks. Without looking, she pulls a black sock from the drawer. What is the probability that the next sock she pulls out will also be black? 36

37 6) Lenny tosses a nickel 50 times. It lands heads up 32 times and tails 18 times. What is the experimental probability that the nickel lands tails? 7) 8) A car manufacturer randomly selected 5,000 cars from their production line and found that 85 had some defects. If 100,000 cars are produced by this manufacturer, how many cars can be expected to have defects? (Source: The following advertisement appeared in the Sunday paper: Chew DentaGum! 4 out of 5 dentists surveyed agree that chewing DentaGum after eating reduces the risk of tooth decay! So enjoy a piece of delicious DentaGum and get fewer cavities! 10 dentists were surveyed. 9) According to the ad, what is the probability that a dentist chosen at random does not agree that chewing DentaGum after meals reduces the risk of tooth decay? 10) Is this probability theoretical or experimental? How do you know? 11) Do you think that the this advertisement is trying to influence the consumer to buy DentaGum? Why or why not? 12) What could be done to make this advertisement more believable? 37

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