19.4 Mutually Exclusive and Overlapping Events

Transcription

1 S 1 A B Locker LESSON 19. Mutually Exclusive and Overlapping Events Common Core Math Standards The student is expected to: S-CP.A. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Also S-CP.7 Mathematical Practices MP.6 Precision Language Objective Give a partner an example of a mutually exclusive event. Explain how you know the events are mutually exclusive. Repeat with an example of overlapping events. Name Class Date 19. Mutually Exclusive and Overlapping Events Essential Question: How are probabilities affected when events are mutually exclusive or overlapping? S 1 A 3 B 5 Resource Locker Explore 1 Finding the Probability of Mutually Exclusive Events Two events are mutually exclusive events if they cannot both occur in the same trial of an experiment. For example, if you flip a coin it cannot land heads up and tails up in the same trial. Therefore, the events are mutually exclusive. A number dodecahedron has sides numbered 1 through. What is the probability that you roll the cube and the result is an even number or a 7? A Let A be the event that you roll an even number. Let B be the event that you roll a 7. Let S be the sample space. Complete the Venn diagram by writing all outcomes in the sample space in the appropriate region. ENGAGE Essential Question: How are probabilities affected when events are mutually exclusive or overlapping? The probability of mutually exclusive events is the sum of the individual probabilities, while the probability of overlapping events is the sum of the individual probabilities minus the probability that both events occur. B D Calculate P (A). P (A) = _ 6 = _ 1 2 Calculate P (A or B). n (S) = n (A or B) = n (A) + n (B) 9 = = 7 C E 11 Calculate P (B). P (B) = _ 1 Calculate P (A) + P (B). Compare the answer to Step D. P (A) + P (B) = _ 1 + _ 1 = _ 7 2 PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photograph. Ask students to estimate the probability that two people in the photo have the same birthday. Then preview the Lesson Performance Task. n (A or B) So, P (A or B) = _ = _ 7. n (S) P (A) + P (B) equals P (A or B). Module Lesson Name Class Date 19. Mutually Exclusive and Overlapping Events Essential Question: How are probabilities affected when events are mutually exclusive or overlapping? S-CP. For the full text of this standard, see the table starting on page CA2 of Volume 1. Also S-CP.B Resource Explore 1 Finding the Probability of Mutually Exclusive Events Two events are mutually exclusive events if they cannot both occur in the same trial of an experiment. For example, if you flip a coin it cannot land heads up and tails up in the same trial. Therefore, the events are mutually exclusive. A number dodecahedron has sides numbered 1 through. What is the probability that you roll the cube and the result is an even number or a 7? Let A be the event that you roll an even number. Let B be the event that you roll a 7. Let S be the sample space. Complete the Venn diagram by writing all outcomes in the sample space in the appropriate region. Calculate P (A). 6 1 P (A) = _ = _ 2 Calculate P (A or B). n (S) = n (A or B) = n (A) + n (B) = + = n (A or B) So, P (A or B) = = _. n (S) _ 7 Calculate P (B). 1 P (B) = _ Calculate P (A) + P (B). Compare the answer to Step D. P (A) + P (B) = _ + _ = 1 2 equals P (A) + P (B) P (A or B). 1 7 _ Module Lesson HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 985 Lesson 19.

2 Reflect 1. Discussion How would you describe mutually exclusive events to another student in your own words? How could you use a Venn diagram to assist in your explanation? Possible answer: Mutually exclusive events are events that have no common outcomes, meaning that any outcome that belongs to one of the events cannot also belong to the other event. The Venn diagram could assist in this explanation by visually showing that the events have no outcomes in common because their intersection would be empty. 2. Look back over the steps. What can you conjecture about the probability of the union of events that are mutually exclusive? The probability of the union of events that are mutually exclusive is equal to the sum of the probabilities of the events. Explore 2 Finding the Probability of Overlapping Events The process used in the previous Explore can be generalized to give the formula for the probability of mutually exclusive events. Mutually Exclusive Events If A and B are mutually exclusive events, then P (A or B) = P (A) + P (B). Two events are overlapping events (or inclusive events) if they have one or more outcomes in common. What is the probability that you roll a number dodecahedron and the result is an even number or a number greater than 7? A Let A be the event that you roll an even number. Let B be the event that you roll a number greater than 7. Let S be the sample space. Complete the Venn diagram by writing all outcomes in the sample space in the appropriate region. S A B 9 11 Module Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.6, which calls for students to communicate with precision. In this lesson, students must decide if events are mutually exclusive or overlapping in order to apply the correct version of the Addition Rule. They may examine a variety of representations, including set notation, Venn diagrams, language analysis, and two-way tables, to make and support their decisions. 7 EXPLORE 1 Finding the Probability of Mutually Exclusive Events INTEGRATE TECHNOLOGY Students have the option of doing the Explore activity either in the book or online. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Ask students to give examples of mutually exclusive events using outcomes from different probability experiments, such as a spinner, a number cube, or a deck of cards. Discuss with students how they can be sure the events they describe are mutually exclusive. QUESTIONING STRATEGIES Explain why, in the case of mutually exclusive events, the probability of either event is equal to the sum of the probabilities of each event. The events have no outcomes in common so the probability will be the total of the individual probabilities. Why is a Venn diagram useful in finding the probability? The Venn diagram provides a picture of the sets and their outcomes that shows the sets do not overlap. EXPLORE 2 Finding the Probability of Overlapping Events INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Ask students to give examples of overlapping events using outcomes from different probability experiments, such as a spinner, a number cube, or a deck of cards. Discuss with students how they can be sure the events they describe are overlapping. Mutually Exclusive and Overlapping Events 986

3 INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Review the set notation used for P (A or B) and P (A and B), as well as how they are represented on Venn diagrams for both mutually exclusive and overlapping events. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss why P (A) + P (B) - P (A and B) can be used to find any probability P (A or B). For mutually exclusive events, P (A and B) = 0. Calculate P (A). Calculate P (B). P (A) = _ 6 1 = _ P (B) = _ 5 2 Calculate P (A and B). P (A and B) = _ 3 1 = _ Now, use P (A), P (B), and P (A and B) to calculate P (A or B). P (A) = P (B) = 5 P (A and B) = 2 P (A) + P (B) - P (A and B) = = 2 Reflect 1_ 1_ Use the Venn diagram to find P (A or B). P (A or B) = _ 8 2 = _ 3 3. Why must you subtract P (A and B) from P (A) + P (B) to determine P (A or B)? P (A and B) must be subtracted from P (A) + P (B) to determine P (A or B) because the outcomes in the event A and B are counted twice. Therefore, the outcomes in the intersection must be subtracted from the total. 1_ 1_ 2_ 3 QUESTIONING STRATEGIES How is P (A or B) different from P (A and B)? Or refers to the union of the events; and refers to the intersection, or overlap, of the events. How can a Venn diagram help you remember the Addition Rule for overlapping events? A Venn diagram shows where the events overlap. This can help me remember to subtract the intersection so it is not counted twice.. Look back over the steps. What can you conjecture about the probability of the union of two events that are overlapping? The probability of the union of two events that are overlapping is equal to the sum of the probabilities of the two separate events minus the probability of both events. Explain 1 Finding a Probability From a Two-Way Table of Data The previous Explore leads to the following rule. The Addition Rule P (A or B) = P (A) + P (B) - P (A and B) Example 1 Use the given two-way tables to determine the probabilities. P (senior or girl) Freshman Sophomore Junior Senior TOTAL Boy Girl Total To determine P (senior or girl), first calculate P (senior), P (girl), and P (senior and girl). Module Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work with a partner to describe a situation in which the probability of two events is mutually exclusive. Have them formulate and answer a question about the probability. Repeat with inclusive events. Have students share their problems with the class. 987 Lesson 19.

4 P (senior) = _ = _ 1 ; P (girl) = _ 808 = _ Use the addition rule to determine P (senior or girl). P (senior or girl) = P (senior) + P (girl) - P (senior and girl) = 1_ + _ _ = _ Therefore, the probability that a student is a senior or a girl is _ B P ( (domestic or late) c) Late On Time Total Domestic Flights International Flights Total P (senior and girl) = _ = _ c To determine P ((domestic or late) ), first calculate P (domestic or late). P (domestic) = _ 0 = _ 2 ; P (late) = _ 18 = _ 1 ; P (domestic and late) = _ = _ Use the addition rule to determine P (domestic or late). P (domestic or late) = P (domestic) + P (late) - P (domestic and late) = _ 2 + _ 1 - _ 1 = _ c Therefore, P ((domestic or late) ) = 1 - P (domestic or late) = 1 - _ 7 10 = _ 3 10 Image Credits: Elena Elisseeva/Cutcaster EXPLAIN 1 Finding a Probability From a Two-Way Table of Data AVOID COMMON ERRORS Students may not use the correct total for the denominator of the probability ratio when they use a two-way table to find a probability. They may use a total from a row or column. As needed, have students extend the table to include a total that shows the sum of the columns is equal to the sum of the rows. This is the total needed for the denominator. QUESTIONING STRATEGIES How do you know the events in the table are inclusive? Possible answer: The overall total of the rows and columns is the same. How do you identify the value that overlaps to apply the Addition Rule? The overlap value is in the cell where the column and row intersect. Module Lesson DIFFERENTIATE INSTRUCTION Multiple Representations Discuss alternate ways to visualize the data in the problems presented. Have students make two-way tables, Venn diagrams, or tree diagrams as alternate ways to view the data and understand the relationships. Discuss the advantages of each representation. Mutually Exclusive and Overlapping Events 988

5 ELABORATE AVOID COMMON ERRORS Students may forget to subtract the overlapping probability when finding the probability of overlapping events. Encourage students to draw Venn diagrams to help them remember how to apply the Addition Rule to solve probability problems. SUMMARIZE THE LESSON How do you find the probability of mutually exclusive events and overlapping events? For mutually exclusive events A and B, P (A or B) = P (A) + P (B). For overlapping events A and B, P (A or B) = P (A) + P (B) - P (A and B). Your Turn 5. Use the table to determine P (headache or no medicine). Took Medicine No Medicine TOTAL Headache No Headache TOTAL P (headache) 27 = 100 ; P (no medicine) = _ 0 Elaborate 100 = 2_ 5 ; P (headache and no medicine) = = 3 20 ; P (headache or no medicine) = P (headache) + P (no medicine) - P (headache and _ 27 no medicine) = = 13 therefore, the probability that a person has a headache or 25 takes no medicine is Give an example of mutually exclusive events and an example of overlapping events. Possible answer: If you roll a number cube, the event of rolling a 3 and the event of rolling an even number are mutually exclusive because you cannot obtain both outcomes at the same time. If you pull a card from a deck, the event of pulling an ace and the event of pulling a spade are overlapping because you can obtain both outcomes by pulling the ace of spades. 7. Essential Question Check-In How do you determine the probability of mutually exclusive events and overlapping events? To determine the probability of mutually exclusive events A and B, evaluate P (A or B) = P (A) + P (B). To determine the probability of overlapping events A and B, evaluate P (A or B) = P (A) + P (B) - P (A and B). Evaluate: Homework and Practice 1. A bag contains 3 blue marbles, 5 red marbles, and green marbles. You choose one without looking. What is the probability that it is red or green? Let S be the sample space, A be the event that you choose a red marble, and B be the Online Homework Hints and Help Extra Practice event that you choose a green marble. n (S) =, n (A or B) = n (A) + n (B) = 5 + = 9; n (A or B) P (A or B) = = 9 n (S) = 3_ ; the probability that you choose a red marble or a green marble is 3_. Module Lesson LANGUAGE SUPPORT Connect Vocabulary For some students, the phrase overlapping events may be unclear. Separate the class into groups. Have students work together to create lists of overlapping events other than those in the examples from the lesson. 989 Lesson 19.

6 2. A number icosahedron has 20 sides numbered 1 through 20. What is the probability that the result of a roll is a number less than or greater than 11? Let S be the sample space, A be the event that you roll a number less than, and B be the event that you roll a number greater than 11. EVALUATE n (S) = 20, n (A or B) = n (A) + n (B) = = = n (A or B) P (A or B) = n (S) 5 The probability that the result is a number less than or greater than 11 is 3_ = 3_ 3. A bag contains 26 tiles, each with a different letter of the alphabet written on it. You choose a tile without looking. What is the probability that you choose a vowel (a, e, i, o, or u) or a letter in the word GEOMETRY? Let S be the sample space, A be the event that you choose a vowel, and B be the event that you choose a letter in the word GEOMETRY. n (S) = 26, n (A or B) = n (A) + n (B) - n (A and B) = = 10 n (A or B) P (A or B) = = 10 n (S) 26 = 5 13 The probability that you choose a vowel or a letter in the word GEOMETRY is Persevere in Problem Solving You roll two number cubes at the same time. Each cube has sides numbered 1 through 6. What is the probability that the sum of the numbers rolled is even or greater than 9? (Hint: Create and fill out a probability chart.) Cube 2 Cube Let S be the sample space, A be the event that the sum of the numbers is even, and B be the event that the sum of the numbers is greater than 9. n (S) = 36, n (A or B) = n (A) + n (B) - n (A and B) = = 20 n (A or B) P (A or B) = = 20 n (S) 36 = 5_ 9 The probability that the sum of the numbers rolled is even or greater than 9 is 5_ 9. ASSIGNMENT GUIDE Concepts and Skills Explore 1 Finding the Probability of Mutually Exclusive Events Explore 2 Finding the Probability of Overlapping Events Example 1 Finding a Probability From a Two-Way Table of Data COMMUNICATING MATH Practice Exercises 1 2, Exercises 3 Exercises 5 10, Discuss the importance of filling in the total values for each row and column when the total values for a two-way table are not given. Module Lesson Exercise Depth of Knowledge (D.O.K.) Mathematical Practices Skills/Concepts MP.2 Reasoning 16 2 Skills/Concepts MP.3 Logic Skills/Concepts MP.1 Problem Solving 22 3 Strategic Thinking MP.3 Logic 23 3 Strategic Thinking MP.3 Logic 2 3 Strategic Thinking MP. Modeling 25 3 Strategic Thinking MP.3 Logic Mutually Exclusive and Overlapping Events 990

7 INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss with students why it is easier not to simplify the fractions until after they have used the Addition Rule to calculate probabilities. AVOID COMMON ERRORS Students may not recognize overlapping events or may forget to subtract the overlap. Suggest that students first decide if the events can overlap. If so, students should identify the probability of this event first. Discuss how students can use and to help them recognize whether events overlap. The table shows the data for car insurance quotes for 5 drivers made by an insurance company in one week. Teen Adult (20 or over) Total 0 accidents accident accidents 9 21 Total You randomly choose one of the drivers. Find the probability of each event. 5. The driver is an adult. 6. The driver is a teen with 0 or 1 accident. Total Adults Total Drivers = The driver is a teen. 8. The driver has 2+ accidents. Total Teens Total Drivers = 28 Drivers with 2+ accidents = 21 5 Total Drivers 5 9. The driver is a teen and has 2+ accidents. _ Teens with 2+ accidents 9 = Total Drivers 5 _ Teen with 0 or 1 accident = 19 Total Drivers 5 Image Credits: arek_ malang/shutterstock 10. The driver is a teen or a driver with 2+ accidents. Teen or Driver with 2+ accidents Total Drivers = = 0 5 = 8 25 Use the following information for Exercises The table shown shows the results of a customer satisfaction survey for a cellular service provider, by location of the customer. In the survey, customers were asked whether they would recommend a plan with the provider to a friend. Arlington Towson Parkville Total Yes No Total Module Lesson 991 Lesson 19.

8 One of the customers that was surveyed was chosen at random. Find the probability of each event. 11. The customer was from Towson and said No.. The customer was from Parkville. _ Towson and said no = 10 Total 150 = The customer was from Parkville or said Yes. Parkville = 7 Total The customer said Yes. 1. The customer was from Parkville and said Yes. _ Yes Total = = 58 Parkville and said Yes = 1 75 Total 150 _ Parkville or said Yes = 7 Total = = AVOID COMMON ERRORS Students might treat inclusive events as mutually exclusive and double count a probability. A total probability greater than 1 could indicate this error. Remind students to consider whether the events can occur simultaneously before they choose the form of the addition rule to use. 16. Explain why you cannot use the rule P (A or B) = P (A) + P (B) in Exercise 15. The events are not mutually exclusive because there are 1 customers who both live in Parkville and said yes, and they get counted twice when adding P (A) and P (B), so you must use the more general rule P (A or B) = P (A) + P (B) P (A and B). Use the following information for Exercises Roberto is the owner of a car dealership. He is assessing the success rate of his top three salespeople in order to offer one of them a promotion. Over two months, for each attempted sale, he records whether the salesperson made a successful sale or not. The results are shown in the chart. Successful Unsuccessful Total Becky 6 6 Raul 5 9 Darrell Total Roberto randomly chooses one of the attempted sales. 17. Find the probability that the sale was one of Becky s or Raul s successful sales. Let A be the set of Becky s successful sales attempts, let B be the set of Raul s successful sales attempts, and let S be the set of all sales attempts. n (S) = 36 n (A B) = n (A) + n (B) = 6 + = 10 n (A B) P (A B) = = 10 n (S) 36 = 5 18 The probability that the sale was one of Becky s or Raul s successful sales is Module Lesson Mutually Exclusive and Overlapping Events 992

9 18. Find the probability that the sale was one of the unsuccessful sales or one of Raul s successful sales. Let A be the set of all unsuccessful sales attempts, let B be the set of Raul s successful sales attempts, and let S be the set of all sales attempts. n (S) = 36 n (A B) = n (A) + n (B) = + 20 = 2 n P (A B) = (A B) = 2 n (S) 36 = 2_ 3 The probability that the sale was one of the unsuccessful sales or one of Raul s successful sales is 2_ Find the probability that the sale was one of Darrell s unsuccessful sales or one of Raul s unsuccessful sales. Let A be the set of Darrell s unsuccessful sales attempts, let B be the set of Raul s unsuccessful sales attempts, and let S be the set of all sales attempts. n (S) = 36 n (A B) = n (A) + n (B) = = 1 n (A B) P (A B) = = 1 n (S) 36 = 7 18 The probability that the sale was one of Darrell s unsuccessful sales or one of Raul s unsuccessful sales is Find the probability that the sale was an unsuccessful sale or one of Becky s attempted sales. Let A be the set of all unsuccessful sales attempts, let B be the set of Becky s sales attempts, and let S be the set of all sales attempts. n (S) = 36 n (A B) = n (A) + n (B) - n (A B) = = 26 n (A B) P (A B) = = 26 n (S) 36 = The probability that the sale was an unsuccessful sale or one of Becky s attempted sales is Find the probability that the sale was a successful sale or one of Raul s attempted sales. Let A be the set of all successful sales attempts, let B be the set of Raul s sales attempts, and let S be the set of all sales attempts. n (S) = 36 n (A B) = n (A) + n (B) - n (A B) = = 21 n (A B) P (A B) = = 21 n (S) 36 = 7 The probability that the sale was a successful sale or one of Raul s attempted sales is 7. Module Lesson 993 Lesson 19.

10 22. You are going to draw one card at random from a standard deck of cards. A standard deck of cards has 13 cards (2, 3,, 5, 6, 7, 8, 9, 10, jack, queen, king, ace) in each of suits (hearts, clubs, diamonds, spades). The hearts and diamonds cards are red. The clubs and spades cards are black. Which of the following have a probability of less than 1? Choose all that apply. a. Drawing a card that is a spade and an ace b. Drawing a card that is a club or an ace c. Drawing a card that is a face card or a club d. Drawing a card that is black and a heart e. Drawing a red card and a number card from 2 9 n (spade ace) a. P (spade ace) = _ = 1 52 < 1 _ n (club ace) n (club) + n (ace) - n (club ace) b. P (club ace) = = = _ n (face club) n (face) + n (club) - n (face club) c. P (face club) = = = 52 = > 1_ n (black heart) d. P (black heart) = = 0 _ n (red 2-9) e. P (red 2-9) = = = 0 < 1_ 13 > 1_ 52 = = = = > 1_ H.O.T. Focus on Higher Order Thinking 23. Draw Conclusions A survey of 1108 employees at a software company finds that 621 employees take a bus to work and 5 employees take a train to work. Some employees take both a bus and a train, and 321 employees take only a train. To the nearest percent, find the probability that a randomly chosen employee takes a bus or a train to work. Explain. If 321 employees take only a train, and 5 total employees take a train, then = people take both a bus and a train to work. n (A B) = n (bus) + n (train) - n (bus and train) = = 92 n (S) = n (total surveyed) = 1108 n (A B) P (A B) = = _ 92 n (S) 1108 = % The probability that a randomly chosen employee takes a bus or a train to work is 85%. Image Credits: Peter Titmuss/Alamy Module Lesson Mutually Exclusive and Overlapping Events 99

11 JOURNAL Have students explain how the probabilities of mutually exclusive and inclusive events are really two versions of the same problem. 2. Communicate Mathematical Ideas Explain how to use a Venn diagram to find the probability of randomly choosing a multiple of 3 or a multiple of from the set of numbers from 1 to 25. Then find the probability. Let A be the set of multiples of 3 from 1 to 25 and B be the set of multiples of from 1 to 25. Create a Venn diagram representing the sets A and B A B 5 S A B Add the numbers of elements in A and B, and then subtract the number of elements in the overlap to get the numerator of the probability. The denominator is 25. n (A B) = = P (A B) = 25 The probability of randomly choosing a multiple of 3 or a multiple of from the set of numbers from 1 to 25 is Explain the Error Sanderson attempted to find the probability of randomly choosing a 10 or a diamond from a standard deck of playing cards. He used the following logic: Let S be the sample space, A be the event that the card is a 10, and B be the event that the card is a diamond. There are 52 cards in the deck, so n (S) = 52. There are four 10s in the deck, so n (A) =. There are 13 diamonds in the deck, so n (B) = 13. One 10 is a diamond, so n (A B) = 1. n (A B) P (A B) = _ n = (A) n (B) - n (A B) = _ 13-1 = 51 n (S) n (S) 52 _ 52 Describe and correct Sanderson s mistake. When finding n (A B), n (A) should be added to n (B), not multiplied. _ n (A) + n (B) - n (A B) P (A B) = = n (S) 52 = = 13 Module Lesson 995 Lesson 19.

12 Lesson Performance Task What is the smallest number of randomly chosen people that are needed in order for there to be a better than 50% probability that at least two of them will have the same birthday? The astonishing answer is 23. Follow these steps to find why. 1. Can a person have a birthday on two different days? Use the vocabulary of this lesson to explain your answer. Looking for the probability that two or more people in a group of 23 have matching birthdays is a challenge. Maybe there is one match but maybe there are five matches or seven or fourteen. A much easier way is to look for the probability that there are no matches in a group of 23. In other words, all 23 have different birthdays. Then use that number to find the answer. 2. There are 365 days in a non-leap year. a. Write an expression for the number of ways can you assign different birthdays to 23 people. (Hint: Think of the people as standing in a line, and you are going to assign a different number from 1 to 365 to each person.) b. Write an expression for the number ways can you assign any birthday to 23 people. (Hint: Now think about assigning any number from 1 to 365 to each of 23 people.) c. How can you use your answers to (a) and (b) to find the probability that no people in a group of 23 have the same birthday? Use a calculator to find the probability to the nearest ten-thousandth. d. What is the probability that at least two people in a group of 23 have the same birthday? Explain your reasoning. 1. No; The set of days of the year when you have a birthday and the set of days of the year when you do not have a birthday are mutually exclusive, not overlapping. 2. a. Because assigned numbers must be different, find the number of permutations of 365 numbers taken 23 at a time: 365 P 23 b. Because assigned numbers can be the same, the number of choices for each person is 365, so form a product where 365 appears as a factor 23 times: c. Divide 365 P 23 by and get to the nearest ten-thousandth. d. The complement of the event that at least two people in a group of 23 have the same birthday is the event that no people in a group of 23 have the same birthday, so P(at least two people in a group of 23 have the same birthday) = 1 - P(no people in a group of 23 have the same birthday) = or 50.73%. AVOID COMMON ERRORS Students hearing the birthday paradox for the first time often misunderstand it as claiming that in a group of 23 randomly chosen people, the chances are better than that one of them will have a specific given birthday, say July 23. Students are correct in thinking that that is highly unlikely. Stress that the paradox is that in a random group of 23, the chances are better than that two of them will have the same birthday. The actual date of the birthday, however, is unstated. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Tammy s birthday is July 23. How large must a group of randomly chosen people be in order for there to be a better than 50% chance that one of them will have the same birthday she has? ; _ % Module Lesson EXTENSION ACTIVITY Present another surprising probability paradox: (1) Mister Jones has two children. The oldest is a girl. 1_ What is the probability that the other child is a girl? Explain your reasoning. ; 2 possible outcomes for second child: {B, G}; 1 favorable 2 outcome: G (2) Mister Smith has two children. At least one of them is a girl. 1_ What is the probability that the other child is a girl? Explain your reasoning. 3 ; 3 possible combinations: {GG, BG, GB}; 1 favorable outcome: GG It may seem counterintuitive that specifying at least changes the probability that the second child will be a girl. To test this, have students flip two coins (H = girl, T = boy). They should ignore TT (boy, boy). If one is a heads, they should count the number of times the other is also a heads. Scoring Rubric 2 points: The student s answer is an accurate and complete execution of the task or tasks. 1 point: The student s answer contains attributes of an appropriate response but is flawed. 0 points: The student s answer contains no attributes of an appropriate response. Mutually Exclusive and Overlapping Events 996