Question 1<\/h3>\r\nWould this outcome make you suspicious that there is something wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 2<\/h3>\r\nSuppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex] X = [\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex] X [\/latex]?\r\n\r\n<\/div>\r\nThis variable\u00a0[latex] X [\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.\r\n\r\nA probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.\r\n\r\nQuestion 3<\/h3>\r\nWe want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?\r\n\r\n<\/div>\r\n\r\nQuestion 4<\/h3>\r\nEveryone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.\r\n\r\n \t- Which value do you expect to occur most often?<\/li>\r\n \t
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\r\n \t
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\r\n<\/ol>\r\n<\/div>\r\n\r\n
Question 5<\/h3>\r\nSketch the class dotplot.\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nLet [latex] X = [\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/17175219\/8A-InClass-2-300x168.png\")
\r\n\r\nNote: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number\u00a0 of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n 0.1608<\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n 0.0322<\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n 0.0032<\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n 0.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nWhat is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nCalculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nIf you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 11<\/h3>\r\nIt\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n<\/div>\r\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/17175359\/8A-InClass-3-300x172.png\")
\r\n\r\nSo far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.\r\n\r\nHowever, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.\r\n\r\nWhen we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.\r\n\r\nFor example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.\r\n\r\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/17175458\/8A-InClass-4-300x116.png\")
\r\n\r\nIn some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex] X = [\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex] X [\/latex].\r\n\r\nQuestion 12<\/h3>\r\nMark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.\r\n\r\n<\/div>\r\n![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/17175639\/8A-InClass-5.png\")
\r\n\r\nQuestion 13<\/h3>\r\nIs it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
You and your family are playing a board game that has a spinner with 6 equally-sized\u00a0\u00a0sections of different colors: red, orange,\u00a0\u00a0yellow, green, blue, and purple. In the first 5 spins, the spinner lands on the purple section\u00a0\u00a04 times.<\/p>\n
![\"\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/08\/17174130\/8A-InClass-1.png\")
\nCredit: iStock\/SolStock<\/p>\n
\nQuestion 1<\/h3>\n
Would this outcome make you suspicious that there is something wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
Suppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex]X =[\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex]X[\/latex]?<\/p>\n<\/div>\n
This variable\u00a0[latex]X[\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.<\/p>\n
A probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.<\/p>\n
\nQuestion 3<\/h3>\n
We want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
Everyone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.<\/p>\n
\n- Which value do you expect to occur most often?<\/li>\n
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\n
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\n<\/ol>\n<\/div>\n\n
Question 5<\/h3>\n
Sketch the class dotplot.<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
Let [latex]X =[\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n <\/td>\n<\/tr>\n \n1<\/td>\n <\/td>\n<\/tr>\n \n2<\/td>\n <\/td>\n<\/tr>\n \n3<\/td>\n <\/td>\n<\/tr>\n \n4<\/td>\n <\/td>\n<\/tr>\n \n5<\/td>\n <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\nQuestion 7<\/h3>\n
What could we do to obtain more accurate empirical estimates of the probability?<\/p>\n<\/div>\n
Probability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)<\/p>\n
<\/p>\n
Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number\u00a0 of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n 0.4019<\/td>\n<\/tr>\n \n1<\/td>\n 0.4019<\/td>\n<\/tr>\n \n2<\/td>\n 0.1608<\/td>\n<\/tr>\n \n3<\/td>\n 0.0322<\/td>\n<\/tr>\n \n4<\/td>\n 0.0032<\/td>\n<\/tr>\n \n5<\/td>\n 0.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\nQuestion 8<\/h3>\n
What is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Calculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
If you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 11<\/h3>\n
It\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n<\/div>\n
<\/p>\n
So far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.<\/p>\n
However, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.<\/p>\n
When we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.<\/p>\n
For example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.<\/p>\n
<\/p>\n
In some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex]X =[\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex]X[\/latex].<\/p>\n
\nQuestion 12<\/h3>\n
Mark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.<\/p>\n<\/div>\n
<\/p>\n
\nQuestion 13<\/h3>\n
Is it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.<\/p>\n<\/div>\n
<\/p>\n
<\/p>\n","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5006","chapter","type-chapter","status-publish","hentry"],"part":4997,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions"}],"predecessor-version":[{"id":5012,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions\/5012"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4997"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 2<\/h3>\r\nSuppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex] X = [\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex] X [\/latex]?\r\n\r\n<\/div>\r\nThis variable\u00a0[latex] X [\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.\r\n\r\nA probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.\r\n\r\nQuestion 3<\/h3>\r\nWe want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?\r\n\r\n<\/div>\r\n\r\nQuestion 4<\/h3>\r\nEveryone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.\r\n\r\n \t- Which value do you expect to occur most often?<\/li>\r\n \t
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\r\n \t
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\r\n<\/ol>\r\n<\/div>\r\n\r\n
Question 5<\/h3>\r\nSketch the class dotplot.\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nLet [latex] X = [\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n\r\n\r\nNote: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number\u00a0 of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n 0.1608<\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n 0.0322<\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n 0.0032<\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n 0.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nWhat is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nCalculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nIf you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 11<\/h3>\r\nIt\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n<\/div>\r\n\r\n\r\nSo far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.\r\n\r\nHowever, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.\r\n\r\nWhen we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.\r\n\r\nFor example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.\r\n\r\n\r\n\r\nIn some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex] X = [\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex] X [\/latex].\r\n\r\nQuestion 12<\/h3>\r\nMark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.\r\n\r\n<\/div>\r\n\r\n\r\nQuestion 13<\/h3>\r\nIs it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
You and your family are playing a board game that has a spinner with 6 equally-sized\u00a0\u00a0sections of different colors: red, orange,\u00a0\u00a0yellow, green, blue, and purple. In the first 5 spins, the spinner lands on the purple section\u00a0\u00a04 times.<\/p>\n
\nCredit: iStock\/SolStock<\/p>\n
\nQuestion 1<\/h3>\n
Would this outcome make you suspicious that there is something wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
Suppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex]X =[\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex]X[\/latex]?<\/p>\n<\/div>\n
This variable\u00a0[latex]X[\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.<\/p>\n
A probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.<\/p>\n
\nQuestion 3<\/h3>\n
We want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
Everyone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.<\/p>\n
\n- Which value do you expect to occur most often?<\/li>\n
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\n
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\n<\/ol>\n<\/div>\n\n
Question 5<\/h3>\n
Sketch the class dotplot.<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
Let [latex]X =[\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n <\/td>\n<\/tr>\n \n1<\/td>\n <\/td>\n<\/tr>\n \n2<\/td>\n <\/td>\n<\/tr>\n \n3<\/td>\n <\/td>\n<\/tr>\n \n4<\/td>\n <\/td>\n<\/tr>\n \n5<\/td>\n <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\nQuestion 7<\/h3>\n
What could we do to obtain more accurate empirical estimates of the probability?<\/p>\n<\/div>\n
Probability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)<\/p>\n
<\/p>\n
Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number\u00a0 of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n 0.4019<\/td>\n<\/tr>\n \n1<\/td>\n 0.4019<\/td>\n<\/tr>\n \n2<\/td>\n 0.1608<\/td>\n<\/tr>\n \n3<\/td>\n 0.0322<\/td>\n<\/tr>\n \n4<\/td>\n 0.0032<\/td>\n<\/tr>\n \n5<\/td>\n 0.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\nQuestion 8<\/h3>\n
What is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Calculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
If you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 11<\/h3>\n
It\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n<\/div>\n
<\/p>\n
So far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.<\/p>\n
However, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.<\/p>\n
When we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.<\/p>\n
For example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.<\/p>\n
<\/p>\n
In some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex]X =[\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex]X[\/latex].<\/p>\n
\nQuestion 12<\/h3>\n
Mark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.<\/p>\n<\/div>\n
<\/p>\n
\nQuestion 13<\/h3>\n
Is it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.<\/p>\n<\/div>\n
<\/p>\n
<\/p>\n","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5006","chapter","type-chapter","status-publish","hentry"],"part":4997,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions"}],"predecessor-version":[{"id":5012,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions\/5012"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4997"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 3<\/h3>\r\nWe want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?\r\n\r\n<\/div>\r\n\r\nQuestion 4<\/h3>\r\nEveryone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.\r\n\r\n \t- Which value do you expect to occur most often?<\/li>\r\n \t
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\r\n \t
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\r\n<\/ol>\r\n<\/div>\r\n\r\n
Question 5<\/h3>\r\nSketch the class dotplot.\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nLet [latex] X = [\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n\r\n\r\nNote: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number\u00a0 of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n 0.1608<\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n 0.0322<\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n 0.0032<\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n 0.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nWhat is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nCalculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nIf you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 11<\/h3>\r\nIt\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n<\/div>\r\n\r\n\r\nSo far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.\r\n\r\nHowever, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.\r\n\r\nWhen we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.\r\n\r\nFor example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.\r\n\r\n\r\n\r\nIn some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex] X = [\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex] X [\/latex].\r\n\r\nQuestion 12<\/h3>\r\nMark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.\r\n\r\n<\/div>\r\n\r\n\r\nQuestion 13<\/h3>\r\nIs it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
You and your family are playing a board game that has a spinner with 6 equally-sized\u00a0\u00a0sections of different colors: red, orange,\u00a0\u00a0yellow, green, blue, and purple. In the first 5 spins, the spinner lands on the purple section\u00a0\u00a04 times.<\/p>\n
\nCredit: iStock\/SolStock<\/p>\n
\nQuestion 1<\/h3>\n
Would this outcome make you suspicious that there is something wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
Suppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex]X =[\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex]X[\/latex]?<\/p>\n<\/div>\n
This variable\u00a0[latex]X[\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.<\/p>\n
A probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.<\/p>\n
\nQuestion 3<\/h3>\n
We want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
Everyone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.<\/p>\n
\n- Which value do you expect to occur most often?<\/li>\n
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\n
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\n<\/ol>\n<\/div>\n\n
Question 5<\/h3>\n
Sketch the class dotplot.<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
Let [latex]X =[\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n <\/td>\n<\/tr>\n \n1<\/td>\n <\/td>\n<\/tr>\n \n2<\/td>\n <\/td>\n<\/tr>\n \n3<\/td>\n <\/td>\n<\/tr>\n \n4<\/td>\n <\/td>\n<\/tr>\n \n5<\/td>\n <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\nQuestion 7<\/h3>\n
What could we do to obtain more accurate empirical estimates of the probability?<\/p>\n<\/div>\n
Probability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)<\/p>\n
<\/p>\n
Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number\u00a0 of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n 0.4019<\/td>\n<\/tr>\n \n1<\/td>\n 0.4019<\/td>\n<\/tr>\n \n2<\/td>\n 0.1608<\/td>\n<\/tr>\n \n3<\/td>\n 0.0322<\/td>\n<\/tr>\n \n4<\/td>\n 0.0032<\/td>\n<\/tr>\n \n5<\/td>\n 0.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\nQuestion 8<\/h3>\n
What is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Calculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
If you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 11<\/h3>\n
It\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n<\/div>\n
<\/p>\n
So far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.<\/p>\n
However, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.<\/p>\n
When we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.<\/p>\n
For example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.<\/p>\n
<\/p>\n
In some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex]X =[\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex]X[\/latex].<\/p>\n
\nQuestion 12<\/h3>\n
Mark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.<\/p>\n<\/div>\n
<\/p>\n
\nQuestion 13<\/h3>\n
Is it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.<\/p>\n<\/div>\n
<\/p>\n
<\/p>\n","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5006","chapter","type-chapter","status-publish","hentry"],"part":4997,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions"}],"predecessor-version":[{"id":5012,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions\/5012"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4997"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 4<\/h3>\r\nEveryone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.\r\n\r\n \t- Which value do you expect to occur most often?<\/li>\r\n \t
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\r\n \t
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\r\n<\/ol>\r\n<\/div>\r\n\r\n
Question 5<\/h3>\r\nSketch the class dotplot.\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nLet [latex] X = [\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n\r\n\r\nNote: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number\u00a0 of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n 0.1608<\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n 0.0322<\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n 0.0032<\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n 0.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nWhat is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nCalculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nIf you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 11<\/h3>\r\nIt\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n<\/div>\r\n\r\n\r\nSo far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.\r\n\r\nHowever, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.\r\n\r\nWhen we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.\r\n\r\nFor example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.\r\n\r\n\r\n\r\nIn some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex] X = [\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex] X [\/latex].\r\n\r\nQuestion 12<\/h3>\r\nMark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.\r\n\r\n<\/div>\r\n\r\n\r\nQuestion 13<\/h3>\r\nIs it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
You and your family are playing a board game that has a spinner with 6 equally-sized\u00a0\u00a0sections of different colors: red, orange,\u00a0\u00a0yellow, green, blue, and purple. In the first 5 spins, the spinner lands on the purple section\u00a0\u00a04 times.<\/p>\n
\nCredit: iStock\/SolStock<\/p>\n
\nQuestion 1<\/h3>\n
Would this outcome make you suspicious that there is something wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
Suppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex]X =[\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex]X[\/latex]?<\/p>\n<\/div>\n
This variable\u00a0[latex]X[\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.<\/p>\n
A probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.<\/p>\n
\nQuestion 3<\/h3>\n
We want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
Everyone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.<\/p>\n
\n- Which value do you expect to occur most often?<\/li>\n
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\n
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\n<\/ol>\n<\/div>\n\n
Question 5<\/h3>\n
Sketch the class dotplot.<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
Let [latex]X =[\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n <\/td>\n<\/tr>\n \n1<\/td>\n <\/td>\n<\/tr>\n \n2<\/td>\n <\/td>\n<\/tr>\n \n3<\/td>\n <\/td>\n<\/tr>\n \n4<\/td>\n <\/td>\n<\/tr>\n \n5<\/td>\n <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\nQuestion 7<\/h3>\n
What could we do to obtain more accurate empirical estimates of the probability?<\/p>\n<\/div>\n
Probability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)<\/p>\n
<\/p>\n
Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number\u00a0 of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n 0.4019<\/td>\n<\/tr>\n \n1<\/td>\n 0.4019<\/td>\n<\/tr>\n \n2<\/td>\n 0.1608<\/td>\n<\/tr>\n \n3<\/td>\n 0.0322<\/td>\n<\/tr>\n \n4<\/td>\n 0.0032<\/td>\n<\/tr>\n \n5<\/td>\n 0.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\nQuestion 8<\/h3>\n
What is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Calculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
If you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 11<\/h3>\n
It\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n<\/div>\n
<\/p>\n
So far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.<\/p>\n
However, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.<\/p>\n
When we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.<\/p>\n
For example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.<\/p>\n
<\/p>\n
In some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex]X =[\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex]X[\/latex].<\/p>\n
\nQuestion 12<\/h3>\n
Mark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.<\/p>\n<\/div>\n
<\/p>\n
\nQuestion 13<\/h3>\n
Is it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.<\/p>\n<\/div>\n
<\/p>\n
<\/p>\n","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5006","chapter","type-chapter","status-publish","hentry"],"part":4997,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions"}],"predecessor-version":[{"id":5012,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions\/5012"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4997"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 5<\/h3>\r\nSketch the class dotplot.\r\n\r\n<\/div>\r\n\r\nQuestion 6<\/h3>\r\nLet [latex] X = [\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n\r\n\r\nNote: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number\u00a0 of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n 0.1608<\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n 0.0322<\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n 0.0032<\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n 0.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nWhat is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nCalculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nIf you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 11<\/h3>\r\nIt\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n<\/div>\r\n\r\n\r\nSo far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.\r\n\r\nHowever, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.\r\n\r\nWhen we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.\r\n\r\nFor example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.\r\n\r\n\r\n\r\nIn some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex] X = [\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex] X [\/latex].\r\n\r\nQuestion 12<\/h3>\r\nMark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.\r\n\r\n<\/div>\r\n\r\n\r\nQuestion 13<\/h3>\r\nIs it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
You and your family are playing a board game that has a spinner with 6 equally-sized\u00a0\u00a0sections of different colors: red, orange,\u00a0\u00a0yellow, green, blue, and purple. In the first 5 spins, the spinner lands on the purple section\u00a0\u00a04 times.<\/p>\n
\nCredit: iStock\/SolStock<\/p>\n
\nQuestion 1<\/h3>\n
Would this outcome make you suspicious that there is something wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
Suppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex]X =[\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex]X[\/latex]?<\/p>\n<\/div>\n
This variable\u00a0[latex]X[\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.<\/p>\n
A probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.<\/p>\n
\nQuestion 3<\/h3>\n
We want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
Everyone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.<\/p>\n
\n- Which value do you expect to occur most often?<\/li>\n
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\n
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\n<\/ol>\n<\/div>\n\n
Question 5<\/h3>\n
Sketch the class dotplot.<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
Let [latex]X =[\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n <\/td>\n<\/tr>\n \n1<\/td>\n <\/td>\n<\/tr>\n \n2<\/td>\n <\/td>\n<\/tr>\n \n3<\/td>\n <\/td>\n<\/tr>\n \n4<\/td>\n <\/td>\n<\/tr>\n \n5<\/td>\n <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\nQuestion 7<\/h3>\n
What could we do to obtain more accurate empirical estimates of the probability?<\/p>\n<\/div>\n
Probability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)<\/p>\n
<\/p>\n
Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number\u00a0 of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n 0.4019<\/td>\n<\/tr>\n \n1<\/td>\n 0.4019<\/td>\n<\/tr>\n \n2<\/td>\n 0.1608<\/td>\n<\/tr>\n \n3<\/td>\n 0.0322<\/td>\n<\/tr>\n \n4<\/td>\n 0.0032<\/td>\n<\/tr>\n \n5<\/td>\n 0.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\nQuestion 8<\/h3>\n
What is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Calculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
If you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 11<\/h3>\n
It\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n<\/div>\n
<\/p>\n
So far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.<\/p>\n
However, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.<\/p>\n
When we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.<\/p>\n
For example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.<\/p>\n
<\/p>\n
In some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex]X =[\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex]X[\/latex].<\/p>\n
\nQuestion 12<\/h3>\n
Mark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.<\/p>\n<\/div>\n
<\/p>\n
\nQuestion 13<\/h3>\n
Is it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.<\/p>\n<\/div>\n
<\/p>\n
<\/p>\n","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5006","chapter","type-chapter","status-publish","hentry"],"part":4997,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions"}],"predecessor-version":[{"id":5012,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions\/5012"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4997"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Question 6<\/h3>\r\nLet [latex] X = [\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n <\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n\r\nQuestion 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n\r\n\r\nNote: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n\r\n\r\n\r\n[latex] X = [\/latex] Number\u00a0 of Purples<\/td>\r\n Probability<\/td>\r\n<\/tr>\r\n \r\n0<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n1<\/td>\r\n 0.4019<\/td>\r\n<\/tr>\r\n \r\n2<\/td>\r\n 0.1608<\/td>\r\n<\/tr>\r\n \r\n3<\/td>\r\n 0.0322<\/td>\r\n<\/tr>\r\n \r\n4<\/td>\r\n 0.0032<\/td>\r\n<\/tr>\r\n \r\n5<\/td>\r\n 0.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\nQuestion 8<\/h3>\r\nWhat is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.\r\n\r\n<\/div>\r\n\r\nQuestion 9<\/h3>\r\nCalculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.\r\n\r\n<\/div>\r\n\r\nQuestion 10<\/h3>\r\nIf you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?\r\n\r\n<\/div>\r\n\r\nQuestion 11<\/h3>\r\nIt\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.\r\n\r\n<\/div>\r\n\r\n\r\nSo far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.\r\n\r\nHowever, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.\r\n\r\nWhen we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.\r\n\r\nFor example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.\r\n\r\n\r\n\r\nIn some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex] X = [\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex] X [\/latex].\r\n\r\nQuestion 12<\/h3>\r\nMark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.\r\n\r\n<\/div>\r\n\r\n\r\nQuestion 13<\/h3>\r\nIs it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.\r\n\r\n<\/div>\r\n \r\n\r\n ","rendered":"
You and your family are playing a board game that has a spinner with 6 equally-sized\u00a0\u00a0sections of different colors: red, orange,\u00a0\u00a0yellow, green, blue, and purple. In the first 5 spins, the spinner lands on the purple section\u00a0\u00a04 times.<\/p>\n
\nCredit: iStock\/SolStock<\/p>\n
\nQuestion 1<\/h3>\n
Would this outcome make you suspicious that there is something wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 2<\/h3>\n
Suppose the spinner is fair, meaning that the arrow is equally likely to land on each\u00a0 of the 6 sections. You spin it 5 times and count how many times it lands on purple.\u00a0 Let [latex]X =[\/latex] the number of times the spinner lands on purple. What are the possible\u00a0 values for [latex]X[\/latex]?<\/p>\n<\/div>\n
This variable\u00a0[latex]X[\/latex] is classified as a discrete variable because it takes a fixed set of\u00a0 possible numerical values and it is not possible to get any value in between.<\/p>\n
A probability distribution includes all possible values of a random variable and the\u00a0 probabilities associated with those values. Probability distributions can be constructed\u00a0 empirically using simulation.<\/p>\n
\nQuestion 3<\/h3>\n
We want to model how often a fair spinner would land on purple, but we don\u2019t have any spinners. How else could we model this situation?<\/p>\n<\/div>\n
\nQuestion 4<\/h3>\n
Everyone in the class will simulate 5 \u201cspins\u201d and record how many times they get \u201cpurple.\u201d Then they will add their values to the class dotplot.<\/p>\n
\n- Which value do you expect to occur most often?<\/li>\n
- What shape do you expect the graph to have: roughly symmetric or skewed?<\/li>\n
- Do you think anyone in the class will get the color purple 5 times? If not, what do you think the largest number will be?<\/li>\n<\/ol>\n<\/div>\n\n
Question 5<\/h3>\n
Sketch the class dotplot.<\/p>\n<\/div>\n
\nQuestion 6<\/h3>\n
Let [latex]X =[\/latex] the number of times the spinner lands on purple. Based on the results of the simulation, estimate the probability of each possible value of this random variable. Record your answers in the following table.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n <\/td>\n<\/tr>\n \n1<\/td>\n <\/td>\n<\/tr>\n \n2<\/td>\n <\/td>\n<\/tr>\n \n3<\/td>\n <\/td>\n<\/tr>\n \n4<\/td>\n <\/td>\n<\/tr>\n \n5<\/td>\n <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n\nQuestion 7<\/h3>\n
What could we do to obtain more accurate empirical estimates of the probability?<\/p>\n<\/div>\n
Probability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)<\/p>\n
<\/p>\n
Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n
\n\n\n\n[latex]X =[\/latex] Number\u00a0 of Purples<\/td>\n Probability<\/td>\n<\/tr>\n \n0<\/td>\n 0.4019<\/td>\n<\/tr>\n \n1<\/td>\n 0.4019<\/td>\n<\/tr>\n \n2<\/td>\n 0.1608<\/td>\n<\/tr>\n \n3<\/td>\n 0.0322<\/td>\n<\/tr>\n \n4<\/td>\n 0.0032<\/td>\n<\/tr>\n \n5<\/td>\n 0.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\nQuestion 8<\/h3>\n
What is the sum of all the probabilities in the probability distribution? Round to 3\u00a0 decimal places.<\/p>\n<\/div>\n
\nQuestion 9<\/h3>\n
Calculate the probability of landing on purple 4 or more times out of 5 spins if the\u00a0 spinner was really fair.<\/p>\n<\/div>\n
\nQuestion 10<\/h3>\n
If you were playing a board game and you landed on purple 4 or more times out of\u00a0 5 spins, would you be convinced that something was wrong with the spinner?<\/p>\n<\/div>\n
\nQuestion 11<\/h3>\n
It\u2019s difficult to judge whether or not the spinner is fair based on only 5 spins. With the Law of Large Numbers in mind, you decide to spin the spinner 100 times. How\u00a0 many times would the spinner have to land on purple for you to be convinced that the spinner was biased toward purple? Justify your answer based on the following\u00a0 probability distribution, which shows the outcomes we\u2019d expect if the spinner was\u00a0 fair. Note: Landing on purple is considered a \u201csuccess\u201d in this chance experiment.<\/p>\n<\/div>\n
<\/p>\n
So far, we have been using a discrete probability distribution that gives the probabilities\u00a0 for a fixed set of values. The spinner can land on purple 2 or 3 times, but 2.5 is\u00a0 impossible. It is not in the set of possible values.<\/p>\n
However, some variables are continuous, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one inch spurts but\u00a0 instead moves through the range of heights via immeasurably small increments. It is not\u00a0 possible to count all the possible heights that a person can be because even between\u00a0 64 inches and 65 inches, there are infinitely many possible heights.<\/p>\n
When we are using a discrete probability distribution, we calculate the probability for a\u00a0 range of values by adding up the probability of each outcome in the range. However,\u00a0 when we are using a continuous probability distribution, probabilities are represented\u00a0 as the area under a density curve. The total area under the curve is equal to 1.<\/p>\n
For example, suppose the following graph shows the distribution of heights (in inches)\u00a0 for women in a particular country. To find the probability that a randomly selected\u00a0 woman is between 60 and 66 inches tall, we would shade the area under the curve\u00a0 between these two values.<\/p>\n
<\/p>\n
In some situations, we may use a continuous probability distribution as an\u00a0 approximation even when the variable is technically discrete. Let\u2019s revisit the example of\u00a0 spinning a fair spinner 100 times, and let [latex]X =[\/latex] the number of times the spinner lands on\u00a0 purple. The following graph shows a continuous probability distribution used as an\u00a0 approximation to the discrete distribution of [latex]X[\/latex].<\/p>\n
\nQuestion 12<\/h3>\n
Mark the following graph to show the probability of landing on purple 20 or more\u00a0 times in 100 spins.<\/p>\n<\/div>\n
<\/p>\n
\nQuestion 13<\/h3>\n
Is it unlikely for a fair spinner to land on purple 20 or more times out of 100? Explain.<\/p>\n<\/div>\n
<\/p>\n
<\/p>\n","protected":false},"author":574340,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5006","chapter","type-chapter","status-publish","hentry"],"part":4997,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/574340"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions"}],"predecessor-version":[{"id":5012,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/revisions\/5012"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4997"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/5006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=5006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=5006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=5006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=5006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
| [latex] X = [\/latex] Number of Purples<\/td>\r\n | Probability<\/td>\r\n<\/tr>\r\n | ||||||||||||||||||||||||||||||||||||||||||
| 0<\/td>\r\n | <\/td>\r\n<\/tr>\r\n | ||||||||||||||||||||||||||||||||||||||||||
| 1<\/td>\r\n | <\/td>\r\n<\/tr>\r\n | ||||||||||||||||||||||||||||||||||||||||||
| 2<\/td>\r\n | <\/td>\r\n<\/tr>\r\n | ||||||||||||||||||||||||||||||||||||||||||
| 3<\/td>\r\n | <\/td>\r\n<\/tr>\r\n | ||||||||||||||||||||||||||||||||||||||||||
| 4<\/td>\r\n | <\/td>\r\n<\/tr>\r\n | ||||||||||||||||||||||||||||||||||||||||||
| 5<\/td>\r\n | <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n Question 7<\/h3>\r\nWhat could we do to obtain more accurate empirical estimates of the probability?\r\n\r\n<\/div>\r\nProbability distributions can also be constructed theoretically using probability rules. The\u00a0 graph and table below show the theoretical probability of each outcome. (In In-Class\u00a0 Activity 8.C, you will learn to calculate these values.)\r\n\r\n
|