Essential Question: What is a probability distribution for a discrete random variable, and how can it be displayed?

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1 COMMON CORE N 3 Locker LESSON Distributions Common Core Math Standards The student is expected to: COMMON CORE S-IC.A. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. Also A-APR.C.5(+), S-CP.B.7 Mathematical Practices COMMON CORE 3.1 MP.7 Using Structure Language Objective Work with a partner to fill in a chart describing empirical, theoretical, and cumulative probability. Name Class Date 3.1 Distributions Essential Question: What is a probability distribution for a discrete random variable, and how can it be displayed? Explore Using Simulation to Obtain an Empirical Distribution 3 1 Resource Locker A random variable is a variable whose value is determined by the outcome of a probability experiment. For example, when you roll a number cube, you can use the random variable X to represent the number you roll. The possible values of X are 1,, 3,, 5, and. A probability distribution is a data distribution that gives the probabilities of the values of a random variable. A probability distribution can be represented by a histogram in which the values of the random variable that is, the possible outcomes are on the horizontal axis, and probabilities are on the vertical axis. The probability distribution for rolling a number cube is shown. Notice that it is a uniform distribution. When the values of a random variable are discrete, as is the case for rolling a number cube, a histogram for the probability distribution typically shows bars that each have a width of 1 and are centered on a value of the variable. The area of each bar therefore equals the probability of the corresponding outcome, and the combined areas of the bars are the sum of the probabilities, which is Result of rolling a number cube ENGAGE Essential Question: What is a probability distribution for a discrete random variable, and how can it be displayed? A probability distribution shows the probability (either empirical or theoretical) associated with each possible value of the random variable. The distribution can be displayed using a histogram with bars that are centered on the possible values and having a height equal to the probability for that value. A cumulative probability is the probability that a random variable is less than or equal to a given value. You can find cumulative probabilities from a histogram by adding the areas of the bars for all outcomes less than or equal to the given value. Suppose you flip a coin 5 times in a row. Use a simulation to determine the probability distribution for the number of times the coin lands heads up. When you flip a coin, the possible outcomes are heads and tails. Use a graphing calculator to generate the integers and 1 randomly, associating each with tails and each 1 with heads. To do the simulation, press MATH and then select the probability (PRB) menu. Choose 5:randInt and enter a, a comma, a 1, and a closing parenthesis. Now press ENTER 5 times to generate a group of s and 1s. This simulates one trial (that is, one set of 5 coin flips). Carry out trials and record your results in the table. Answers will vary. Possible answer: Trial 1 3 Number of heads 1 3 PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how a census can provide information that can be used to calculate a probability. Then preview the Lesson Performance Task. Module Lesson 1 Name Class Date 3.1 Distributions Essential Question: What is a probability distribution for a discrete random variable, and how can it be displayed? S-IC.A. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. Also A-APR.C.5(+), S-CP.B.7 Explore Using Simulation to Obtain an Empirical Distribution 1 Resource A random variable is a variable whose value is determined by the outcome of a probability experiment. For example, when you roll a number cube, you can use the random variable X to represent the number you roll. The possible values of X are 1,, 3,, 5, and. A probability distribution is a data distribution that gives the probabilities of the values of a random variable. A probability distribution can be represented by a histogram in which the values of the random variable that is, the possible outcomes are on the horizontal axis, and probabilities are on the vertical axis. The probability distribution for rolling a number cube is shown. Notice that it is a uniform distribution When the values of a random variable are discrete, as is the case for rolling a number cube, a histogram for the probability distribution typically shows bars that each have a width of 1 and are centered on a value of the variable. The area of each bar therefore equals the probability of the corresponding outcome, and the combined areas of the bars are the sum of the probabilities, which is 1. A cumulative probability is the probability that a random variable is less than or equal to a given value. You can find cumulative probabilities from a histogram by adding the areas of the bars for all outcomes less than or equal to the given value. Suppose you flip a coin 5 times in a row. Use a simulation to determine the probability distribution for the number of times the coin lands heads up. When you flip a coin, the possible outcomes are heads and tails. Use a graphing calculator to generate the integers and 1 randomly, associating each with tails and each 1 with heads. To do the simulation, press MATH and then select the probability (PRB) menu. Choose 5:randInt and enter a, a comma, a 1, and a closing parenthesis. Now press ENTER 5 times to generate a group of s and 1s. This simulates one trial (that is, one set of 5 coin flips). Carry out trials and record your results in the table. Result of rolling a number cube Answers will vary. Possible answer: Trial Number of heads Module Lesson 1 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition Lesson 3.1

2 Report your results to your teacher in order to combine everyone s results. Use the combined class data to complete the table. To find the relative frequency for an outcome, divide the frequency of the outcome by the total number of trials in the class and round to the nearest hundredth. Answers will vary. Possible answer (based on trials): Number of heads Frequency Relative frequency Enter the outcomes ( through 5) into your calculator as list L 1. Enter the relative frequencies as list L. Make a histogram by turning on a statistics plot, selecting the histogram option, and using L 1 for Xlist and L for Freq as shown. Then set the viewing window as shown. Finally, press GRAPH to obtain a histogram like the one shown. Describe the shape of the probability distribution. The distribution is mounded and has tails that are roughly symmetric, so the distribution is approximately normal. Reflect 1. Discussion If you flipped a coin 5 times and got 5 heads, would this cause you to question whether the coin is fair? Why or why not? Answers may vary. Possible answer: The probability of getting 5 heads in 5 flips with a fair coin is very low, so I might question whether the coin is fair. Module Lesson 1 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.7, which asks students to "look for and make use of structure." Students will use histograms to display and analyze probability distributions. They will recognize the difference between a theoretically generated probability distribution and an experimentally generated probability distribution. They will be able to explain why theoretical and experimental probability distributions are not necessarily the same, and when they should expect them to be more alike (which happens as the number of trials in the experiment increases). EXPLORE Using Simulation to Obtain an Empirical Distribution INTEGRATE TECHNOLOGY Students have the option of completing the Explore activity either in the book or online. QUESTIONING STRATEGIES How do the values on the x-axis of a probability distribution compare to those on the x-axis for a data distribution histogram? For a data distribution histogram, the x-axis shows data values (or groups of data values). For a probability distribution, the x-axis shows possible outcomes (that is, the values of the random variable). How do the values on the y-axis of a probability distribution compare to those on the x-axis for a data distribution histogram? For a data distribution histogram, the y-axis shows the frequency of each data value (or group of data values). For a probability distribution, the y-axis shows the probability of each outcome. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Ask students to discuss how generating random numbers models flipping a coin. Students should observe that the random numbers are divided into two groups of equal size, and since the numbers are generated randomly, you are as likely to get a number in the first group as you are to get a number in the second group. This is the same situation as flipping a coin you are as likely to get heads as you are to get tails on each flip. Distributions 1118

3 INTEGRATE TECHNOLOGY Discuss with students the use of the calculator s random number generator to simulate the trials in the experiment. Lead students to observe that generating the data in this way guarantees that the results reflect those that would be produced by a fair coin. Point out that this may be preferable to using an actual coin, on the chance that the coin or tossing process is not fair. AVOID COMMON ERRORS Students may, in error, use the frequencies of each outcome to draw their histograms. Reinforce that since the histogram represents a probability distribution, the histogram must be drawn using the relative frequencies, which are the probabilities of occurrence for each possible value of the random variable. CONNECT VOCABULARY Ask students if they know what accumulate means. to increase in amount gradually Then explain that cumulative is an adjective that means increasing in amount by successive (one after another) additions. EXPLAIN 1 Displaying and Analyzing a Theoretical Distribution Image Credits: Eyebyte/ Alamy Images Explain 1 Displaying and Analyzing a Theoretical Distribution Recall that a binomial experiment involves repeated trials where each trial has only two outcomes: success or failure. The probability of success on each trial is p, and the probability of failure on each trial is q = 1 - p. The binomial probability of r successes in n trials is given by P (X = r) = n C r p r q n-r. Example 1 Calculate all the theoretical probabilities for the given binomial experiment. Then draw a histogram of the probability distribution, observe its shape, and use it to find the specified probabilities. A binomial experiment consists of flipping a fair coin for 5 trials where getting heads is considered a success. Find the probability of getting 3 or more heads and the probability of getting at least 1 head. To calculate the probabilities, set n equal to 5 and let r range from to 5 r in n C r p q n-r. Since the coin is fair, p = and q =. Number of heads Theoretical probability Create a histogram. 5_ 1 _ 1 _ The distribution is mounded and has symmetric tails, so it is a normal distribution. The probability of getting 3 or more heads is: P (X 3) = P (X = 3) + P (X = ) + P (X = 5) = _ 1 + 5_ + = _ =.5 The probability of getting at least 1 head is: P (X 1) = 1 - P (X = ) = 1 - = _ 31. 5_ Number of Heads INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 You may want to review Pascal s Triangle and its relationship to binomial probability. Students can use the appropriate values from the triangle to help calculate the probabilities. Module Lesson 1 COLLABORATIVE LEARNING Whole Class Activity Have students work as a class to design and perform an experiment that simulates the tossing of a coin that is not fair. Have them decide the degree of bias of the coin, and use this measure to design the simulation. Have each student perform a certain number of trials of the experiment. Then combine the results to form one set of data for the class. Have students determine the probability distribution and construct the related histogram. Have them analyze the results, and use them to critique the design of their simulation. 111 Lesson 3.1

4 B A binomial experiment consists of flipping a biased coin for 5 trials where getting heads is considered a success. The coin lands heads up 75% of the time. Find the probability of getting 3 or more heads and the probability of getting at least 1 head. To calculate the probabilities, set n equal to 5 and let r range from to 5 in n C r p r q n-r. Since the coin is biased such that it lands heads up 75% of the time, p = 3 and q =. Number of heads Theoretical probability Create a histogram Number of Heads left The distribution is mounded and has a tail to the left, so the distribution is skewed. The probability of getting 3 or more heads is: P (X 3) = P (X = 3) + P (X = ) + P (X = 5) = _ + _ + _ = _ 1 The probability of getting at least 1 head is: P (X 1) = 1 - P (X = ) = 1 - _ = _ 1.8. QUESTIONING STRATEGIES How would you describe the shape of the histogram? It is mound-shaped and symmetric, with high probabilities in the middle tapering to low probabilities at the ends. What does the shape of the histogram indicate? You are least likely to get the outcomes represented by the bars on the far ends, slightly more likely to get the outcomes represented by the bars as you move towards the middle, and most likely to get the outcomes represented by the bars in the middle. How does the shape of the histogram in Part A of the Example compare to that from the Explore? How do you explain this difference? The histogram in the Example is perfectly symmetrical, while the one from the Explore is not. This is because the histogram in the Explore was based on experimental results, which only approximate the theoretical results. Module 3 11 Lesson 1 DIFFERENTIATE INSTRUCTION Kinesthetic Experience Some students may better understand the process and the meaning of the results of a coin-flipping experiment if they perform the experiment using an actual coin rather than the random number generator on a calculator. Have students work in pairs, flipping the coin and recording the results. Have each pair complete the given number of trials, and then combine their results with those of other pairs to create a table of the data and the related probabilities. Have each student draw a histogram for the probability distribution. Students might also compare their results with those obtained from the random number generator. Distributions 11

5 Reflect. Why are the probabilities in the histogram you made in the Explore different from the probabilities given in the histogram from Part A? The histogram in the Explore was based on experimental probabilities, so there is randomness in the results. 3. For which coin, the fair coin in Part A or the biased coin in Part B, is flipping a coin 5 times and getting 5 heads more likely to occur? Explain. The biased coin in Part B is more likely to produce 5 heads in 5 flips because _ P (X = 5) = for the fair coin, and P (X = 5) =.37 (about 7. times more likely) for the biased coin.. Discussion Can you definitively conclude whether a coin that results in repeated heads when flipped is fair or biased? What might make you favor one conclusion over the other? Answers may vary. Possible answer: Assume the coin is fair so long as a result is not very unlikely. Determining when a result crosses over into being very unlikely is debatable, however. 1 Your Turn Calculate all the theoretical probabilities for the given binomial experiment. Then draw a histogram of the probability distribution, observe its shape, and use it to find the specified probabilities. 5. A binomial experiment consists of flipping a biased coin for trials where getting heads is considered a success. The coin lands heads up % of the time. Find the probability of getting or more heads and the probability of getting fewer than heads Number of Heads Number of heads 1 3 Theoretical probability The distribution is slightly skewed right. P (X ) = P (X = ) + P (X = 3) + P (X = ) = =.58 P (X < ) = 1 - P (X = ) = =.7 Module Lesson 1 LANGUAGE SUPPORT Communicate Math Have students work in pairs to complete a table like the one below. Be sure that students understand the differences among the types and how the distributions would differ. Type of Description Example Empirical probability Theoretical probability Cumulative probability 111 Lesson 3.1

6 Your Turn. A binomial experiment consists of flipping a biased coin for trials where getting tails is considered a success. The coin lands heads up % of the time. Find the probability of getting or more tails and the probability of getting fewer than tails. Number of heads 1 3 Theoretical probability The distribution is slightly skewed left. P (X ) = P (X = ) + P (X = 3) + P (X = ) = =.58 P (X < ) = 1 - P (X = ) Elaborate = = Number of Tails 7. What is a random variable, and what makes a random variable discrete? A random variable is a variable whose value is determined by the outcome of a probability experiment. If the outcomes are distinct, such as whole numbers, the random variable is discrete. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to describe the types of problems that require the calculation of a cumulative probability. Have them brainstorm different realworld scenarios in which the determination of cumulative probabilities would play an important role in decision-making. INTEGRATE TECHNOLOGY The graphing calculator command Int(1*rand)+1 can be used to generate random numbers from 1 to 1. Alternatively, a spreadsheet can generate these random numbers when the command =INT(RAND()*1+1) is entered into a cell. 8. How can a histogram for a probability distribution be used to calculate a cumulative probability? The probability is the sum of the areas of the bars in the histogram. The bars correspond to the values of the random variable that are less than or equal to a given value.. Essential Question Check-In What is a probability distribution for a discrete random variable? A probability distribution shows the probability (either empirical or theoretical) associated with each possible value of the random variable. SUMMARIZE THE LESSON How do you construct a histogram for a probability distribution based on the data from a binomial experiment? Determine the different numbers of possible successes, and use them to label the x-axis. Label the y-axis from to 1 to show the probabilities. Then, divide the number of successes for each possible value of the random variable by the total number of outcomes to find the probability for each value, and draw a bar to the height that would indicate that probability. Module 3 11 Lesson 1 Distributions 11

7 EVALUATE Evaluate: Homework and Practice 1. A spinner has three equal sections, labeled 1,, and 3. You spin the spinner twice and find the sum of the two numbers the spinner lands on. 1 Online Homework Hints and Help Extra Practice a. Let X be a random variable that represents the sum of the two numbers. What are the possible values of X?, 3,, 5, and 3 ASSIGNMENT GUIDE b. Complete the table. Concepts and Skills Explore Using Simulation to Obtain an Empirical Distribution Practice Sum 3 5 c. Make a histogram of the probability distribution. 3 _ Example 1 Displaying and Analyzing a Theoretical Distribution Exercises 1 7 QUESTIONING STRATEGIES What can you conclude about a binomial experiment if the histogram for the probability distribution is skewed? The probabilities of the two outcomes were not equally likely. If a binomial experiment consists of two equally likely outcomes, what will the histogram of the theoretical probability distribution look like? It will be mound-shaped and symmetric, with high probabilities in the middle, tapering to low probabilities at the ends. AVOID COMMON ERRORS Some students may have trouble making the connection between the outcome of a trial of an experiment and the value of the random variable associated with that outcome. Help to clarify this by encouraging students to determine the possible values of the random variable and describe what each value represents, before performing the experiment. 113 Lesson 3.1 Exercise Sum d. What is the probability that the sum is not? How is this probability represented in the histogram? The probability is 8_. This is the sum of the areas of the bars for the outcomes 3,, 5, and. Module Lesson 1 Depth of Knowledge (D.O.K.) 1 Skills/Concepts MP. Reasoning Skills/Concepts MP. Precision 3 Skills/Concepts MP.5 Using Tools COMMON CORE Mathematical Practices Skills/Concepts MP.7 Using Structure 5 Skills/Concepts MP. Reasoning Skills/Concepts MP. Precision 7 Skills/Concepts MP.3 Logic

8 . You roll two number cubes at the same time. Let X be a random variable that represents the absolute value of the difference of the numbers rolled. a. What are the possible values of X?, 1,, 3,, and 5 b. Complete the table. Absolute difference c. Is this probability distribution symmetric? Why or why not? No; the probability distribution is skewed right because in a histogram of the distribution, the tallest bar would occur at 1, with bars to the right decreasing in height. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Ask students to consider what must be true about the sum of the probabilities in a probability distribution. The sum must be 1. Ask them explain why this is so. Then encourage them to use this fact to check their work. d. Find the probability of getting a difference greater than 3. P (X > 3) = P (X = ) + P (X = 5) = + = = 1 3. A trick coin is designed to land heads up with a probability of 8%. You flip the coin 7 times. a. Complete the table? Number of heads Theoretical probability 1 _ 1 _ 8 78,15 78, _ 3 8 _ 1,5 _ 8,7 _,38 78,15 78,15 78,15 78,15 78,15 78,15 b. Make a histogram of the probability distribution Number of Heads c. What is the probability of getting or 7 heads? 8, P (X = ) + P (X = 7) = _ 7 78, 15 + _, 38 = _ 5, 5 78, 15 78, d. What is the probability of getting or more heads? P (X ) = P (X = ) + P (X = 5) + P (X = ) + P (X = 7) = 8 78, , 5 78,15 + 8, 7, , 15 78, 15 _ 75, 5 = 78, 15.7 Module 3 11 Lesson 1 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Strategic Thinking MP.3 Logic Distributions 11

9 PEER-TO-PEER DISCUSSION Ask students to discuss with a partner how the histograms for the probability distributions of two binomial experiments would differ if the probability of success in Experiment A was % and the probability of success in Experiment B was %. The histogram for Experiment A would be greatly skewed to the right, while the histogram for Experiment B would be slightly skewed to the left. e. Which is greater, the probability of getting an even number of heads or the probability of getting an odd number of heads? The probability of getting an even number of heads is P (X = ) + P (X = ) + _ 1 P (X = ) + P (X = ) = 78, , , , 7 37, = 78, 15 78, 15, f. Suppose you flip a coin 7 times and get 7 heads. Based on what you know now, would you question whether the coin is fair? Why or why not? Answers may vary. Possible answer: The probability of getting 7 heads in 7 flips of a fair coin is ( ) 7 = _ 1 18 _, 38 while the probability of getting an odd number of heads is P (X = 1) + _ 8 P (X = 3) + P (X = 5) + P (X = 7) = 78, , , 5 78, 15 +, 38, 15 = 78, 15 78, 15. So, the probability of getting an odd number of heads is greater..8. The probability of getting 7 heads in 7 flips of the trick coin is.1. Since getting 7 heads on the trick 78, 15 coin is more than times as likely to occur as getting 7 heads on the fair coin, I would question whether the coin is fair.. You flip a coin times in a row. The histogram shows the theoretical probability distribution for this situation. 1 3 Number of Heads a. What is the probability of getting 3 or more heads? P (X 3) = P (X = 3) + P (X = ) = + 1 = 5 =.315 b. What is the probability of getting at most heads? P (X ) = 1 - P (X 3) = =.875 c. How do you know that the coin is fair? The coin is fair because the distribution shown in the histogram is perfectly symmetric. If the probability of getting heads were not, then the distribution would be skewed. Module Lesson Lesson 3.1

10 5. A spinner has equal sections that are labeled 1,, 3, and. You spin the spinner twice and find the sum of the numbers it lands on. Let X be a random variable that represents the sum of the numbers. a. Complete the table. Sum Frequency b. Make a histogram of the probability distribution Sum c. What is the probability of getting a sum of or more? P (X ) = P (X = ) + P (X = 7) + P (X = 8) = 3 + _ + 1 = =.375 d. Without actually calculating any probabilities, determine the relationship between P (X > 5) and P (X < 5). Explain your reasoning. 3 P (X > 5) = P (X < 5) because the histogram of the probability distribution is symmetric with respect to x = COMMUNICATING MATH Test for understanding by having students describe what their histograms indicate about the results of a given experiment, and what overall conclusions they can draw. COLLABORATIVE LEARNING Have students work in small groups to make a poster showing the difference between a frequency histogram and a histogram for a probability distribution. Have them generate or gather a set of data that they can use to construct both displays. Have them compare the two types of histograms, pointing out how they differ and how they are related. Module 3 11 Lesson 1 Distributions 11

11 . You roll number cubes at the same time. Let X be a random variable that represents the sum of the numbers rolled. a. Complete the table to show the sums that are possible. In the table, the row heads are the numbers that are possible on one number cube, and the column heads are the numbers that are possible on the other number cube b. Complete the second row of the table to show the number of ways that you can get each sum. Then find the probability of each sum to complete the third row. Sum Frequency c. Make a histogram of the probability distribution Sum d. What is the probability that you roll a sum of 5 or less? P (X 5) = P (X = ) + P (X = 3) + P (X = ) + P (X = 5) = = 1.78 e. What is the probability that you roll a sum of 1 four times in a row? If this happened, would you question whether the number cubes are fair? The probability of rolling a sum of 1 four times in a row is ( ) 1 1 = _.. This probability is so low that it calls 1, 7, into question the fairness of the number cubes. Module Lesson Lesson 3.1

12 7. A fair coin is flipped times. Match each specified probability on the left with its value on the right. (A value on the right may apply to more than one specified probability on the left.) A. The probability of getting at least heads B B. The probability of getting no more than 1 head C. The probability of getting 1 or heads D. The probability of getting no more than heads E. The probability of getting an even number of heads C A, D E, F 7 1 _ 11 F. The probability of getting an odd number of heads A. P (X ) = P (X = ) + P (X = 5) + P (X = ) = = _ = 11 B. P (X 1) = P (X = ) + P (X = 1) = 1 C. P (X = 1) + P (X = ) = H.O.T. Focus on Higher Order Thinking + 15 = 1 + = 7 D. P (X ) = P (X = ) + P (X = 1) + P (X = ) = = = _ 11 E. P (X = ) + P (X = ) + P (X = ) + P (X = ) = = F. P (X = 1) + P (X = 3) + P (X = 5) = + + = = = 8. Represent Real-World Situations About 1.% of the U.S. population that is 5 years old and over have a bachelor s degree only, and 1.5% have an advanced degree. a. Find the probability that of randomly selected people who are 5 years old or over, have at least a bachelor s degree. The probability that a person who is 5 years old or over has at least a bachelor s degree is So, the probability that of randomly selected people who are 5 years old or over have at least a bachelor s degree is about C (.3) (.7).55. b. Find the probability that of randomly selected people who are 5 years old or over, do not have even a bachelor s degree. The probability that a person who is 5 years old or over doesn t have at least a bachelor s degree is about =.7. So, the probability that of randomly selected people who are 5 years old or over don t have even a bachelor s degree is about C (.7) (.3).. Image Credits: Blend Images/Hill Street Studios/Corbis Module Lesson 1 Distributions 118

13 JOURNAL Have students describe why the histogram for the empirical probability distribution of an experiment could be different from the histogram for the theoretical probability distribution of the same experiment. c. Suppose all of randomly selected people have advanced degrees. Would you question the probability model? Explain. The probability that all have an advanced degree is (.15).13. This seems very unlikely. I might question whether the selection process was truly random, thinking instead that the sampling method was biased (for instance, it was a selfselected sample of people who read a professional journal, or it was random sample of a subset of the population, such as people living in a city where scientific research or high tech is a large part of the local economy).. Justify Reasoning Describe a way to get fair results from a coin that you suspect is biased. Explain how you know that the process is fair. Rather than flip the coin just once, flip the coin twice. If you flip the coin twice and the results match, then completely disregard the results. If you flip the coin twice and the results are different, then use the first result and disregard the second result. To see why the process is fair, let p be the probability of getting heads and 1 - p be the probability of getting tails. For a biased coin, p 1 - p. Assume that both probabilities are nonzero. Then the probability of getting two heads is p, the probability of getting two tails is (1 - p), and these probabilities are not equal if p and 1 - p are not equal. However, the probability of getting heads followed by tails is p (1 - p), the probability of getting tails followed by heads is (1 - p) p, and these probabilities are equal. So, by going with heads when heads appear first and going with tails when tails appear first, the two outcomes have equal probability, and the results are fair. 1. Construct Arguments Use the formula P (X = r) = n C r p r q n-r for a binomial experiment to explain why the probability distribution for the number of heads obtained when a fair coin is flipped n times is symmetric. Because the coin is fair, p = and q =, so the formula becomes P (X = r) = nc r p r q n - r = nc r ( = nc r (. Because ( ) r ( ) n - r is a constant for a given value of n, the only variable factor is nc r, which gives the number of ways that r (where r ranges from to n) of n flips of the coin can result in heads. But for every way there is to obtain r heads, there is a corresponding way to obtain n - r heads by simply switching heads and tails. (For instance, one way to obtain heads in 7 flips of a coin is HTTTHTT, and the corresponding way to obtain 5 heads in 7 flips is THHHTHH.) This means that nc r ( = nc n - r (, so P (X = r) = P (X = n - r), which shows ) n ) n that the probability distribution is symmetric. ) n ) n Module 3 11 Lesson 1 11 Lesson 3.1

14 Lesson Performance Task According to the U.S. Census, in 1 the number of people 18 years old or over in the U.S. was.1 million, and of those people, 1.5 million were married. a. Find the probability that of 1 randomly selected people 18 years old or over, are married. b. Consider a survey where all 1 of the people surveyed are married. What conclusion might you draw about that survey? a. The probability that a randomly selected person 18 years old or over is married is _ The probability that of 1 randomly selected people 18 years old.1 and over, are married is 1 C (.55) (.35) = 1 (.55) (.35).5. b. The probability that of 1 randomly selected people 18 years old and over, all 1 are married is 1 C 1 (.55) 1 (.35) = (.55) 1.3. This probability is low and might call into question whether the survey was actually random or representative of the general U.S. population. AVOID COMMON ERRORS Urge students solving without a calculator to check their substitutions in the expression nc xp x q n - x before computing. The sum of the exponents must be n and p + q = 1. The exponents in the expression should also match the factorials in the denominator of n C x. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 To calculate combinations, n C r, using a graphing calculator, press the value for n, and then the MATH key. Then press the right arrow key three times to scroll right to PRB. Press 3 to select the combination formula. Type the value for r and press ENTER. Module Lesson 1 EXTENSION ACTIVITY Have students find United States Census information about the voter turnout (the percent of voting-age population that voted) in the 1 Presidential election and in the most recent midyear election, a non-presidential election in which voting is for Congressional representatives. Then have them consider a sample small town with a number of people of voting age, find the number expected to vote in each election, and perform a simulation to estimate these numbers. Ask students to present and compare their findings. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. points: Student does not demonstrate understanding of the problem. Distributions 113