{"id":252,"date":"2017-04-15T03:20:12","date_gmt":"2017-04-15T03:20:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/two-way-tables-2-of-5\/"},"modified":"2022-08-01T16:04:42","modified_gmt":"2022-08-01T16:04:42","slug":"two-way-tables-2-of-5","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/chapter\/two-way-tables-2-of-5\/","title":{"raw":"Two-Way Tables (2 of 5)","rendered":"Two-Way Tables (2 of 5)"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning OUTCOMES<\/h3>\r\n<ul>\r\n \t<li>Calculate marginal, joint, and conditional percentages and interpret them as probability estimates.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the previous section, we used the information in a two-table to examine the relationship between two categorical variables. Our goal was to answer the big question: <em>Are the variables related?<\/em>\r\n\r\nIn this section, we continue to work with two-way tables, but we ask a different set of questions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<h2>Community College Enrollment<\/h2>\r\nThe following table summarizes the full-time enrollment at a community college located in a West Coast city. There are a total of 12,000 full-time students enrolled at the college. The two categorical variables here are <em>gender<\/em> and <em>program<\/em>. The programs include academic and vocational programs at the college. Assume that a student can enroll in only one program.\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\" style=\"height: 30px;\">\r\n<td style=\"height: 30px;\"><\/td>\r\n<td style=\"height: 30px;\"><strong>Arts-Sci<\/strong><\/td>\r\n<td style=\"height: 30px;\"><strong>Bus-Econ<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15.3906px;\">\r\n<td style=\"height: 15.3906px;\" align=\"center\"><strong>Female<\/strong><\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">4,660<\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">435<\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">494<\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">421<\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">105<\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">83<\/td>\r\n<td style=\"height: 15.3906px;\" align=\"center\">6,198<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" align=\"center\"><strong>Male<\/strong><\/td>\r\n<td style=\"height: 15px;\" align=\"center\">4,334<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">490<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">564<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">223<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">97<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">94<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">5,802<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 30px;\">\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\">8,994<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">925<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">1,058<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">644<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">202<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">177<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">12,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s consider a few preliminary questions to get familiar with this new data set.\r\n<ul style=\"list-style-type: none;\">\r\n \t<li>1. <em>What proportion of the total number of students are male students?<\/em><\/li>\r\n<\/ul>\r\n<strong>Answer:<\/strong>\r\n<p style=\"text-align: center;\">[latex]\\frac{\\mathrm{number\\; of\\; male\\; students}}{\\mathrm{total\\; number\\; of\\; students}}=\\frac{\\mathrm{5,802}}{\\mathrm{12,000}}=\\mathrm{0.4835}(\\mathrm{or\\; 48.35\\%})[\/latex]<\/p>\r\n\r\n<ul style=\"list-style-type: none;\">\r\n \t<li>2. <em>What proportion of the total number of students are Bus-Econ students?<\/em><\/li>\r\n<\/ul>\r\n<strong>Answer:<\/strong>\r\n<p style=\"text-align: center;\">[latex]\\frac{\\mathrm{number\\; of\\; Bus-Econ\\; students}}{\\mathrm{total\\; number\\; of\\; students}}=\\frac{925}{\\mathrm{12,000}}=\\mathrm{0.077}(\\mathrm{or\\; 7.7\\%})[\/latex]<\/p>\r\nNote that to calculate this proportion, we used two numbers in the margin that relate to just one of the categorical variables (program). This calculation is therefore called a <strong>marginal proportion<\/strong>.\r\n\r\nNote: This proportion does not help us determine if gender is related to program because it involves only one of the variables.\r\n\r\nNow consider the following question:\r\n<ul style=\"list-style-type: none;\">\r\n \t<li><em>If we choose one student at random from among all 12,000 students at the college, how likely is it that this student will be in the Bus-Econ program?<\/em><\/li>\r\n<\/ul>\r\nFrom our previous calculation, we know that only about 8% (7.7%) of the students at the college are in the Bus-Econ program. That's a fairly low number, so it is not very likely that our random student will be a Bus-Econ student.\r\n\r\nOne way to state our conclusion is to say:\r\n<ul style=\"list-style-type: none;\">\r\n \t<li>There is about an <strong>8% chance<\/strong> of picking a Bus-Econ major.<\/li>\r\n<\/ul>\r\nThis means that if we selected 100 students at random, we would expect on average that 8 of them would be in the Bus-Econ program.\r\n\r\nHere is another way to state this conclusion:\r\n<ul style=\"list-style-type: none;\">\r\n \t<li>There is about an <strong>0.08 probability<\/strong> of picking a Bus-Econ major.<\/li>\r\n<\/ul>\r\nBecause this probability is exactly the same as the marginal proportion we calculated earlier, we call it a <strong>marginal probability<\/strong>.\r\n\r\n<\/div>\r\n<strong>Note:<\/strong> <em><strong>P<\/strong><\/em> <strong>for Probability<\/strong>\r\n\r\nIt is customary to use the capital letter <em>P <\/em>to stand for probability. So instead of writing \u201cThe probability that a student is in Bus-Econ program equals 0.08,\u201d we can write <em>P<\/em>(student is in Bus-Econ) = 0.08.\r\n\r\nThe following table is used for the next Try It and Did I Get This? activities.\r\n<table style=\"width: 495px;\">\r\n<tbody>\r\n<tr class=\"oli_table\" style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 59px;\"><\/td>\r\n<td style=\"height: 30px; width: 38px;\"><strong>Arts-Sci<\/strong><\/td>\r\n<td style=\"height: 30px; width: 38px;\"><strong>Bus-Econ<\/strong><\/td>\r\n<td style=\"height: 30px; width: 36px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\r\n<td style=\"height: 30px; width: 57px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\r\n<td style=\"height: 30px; width: 68px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\r\n<td style=\"height: 30px; width: 65px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\r\n<td style=\"height: 30px; width: 46px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15.6875px;\">\r\n<td style=\"height: 15.6875px; width: 59px;\" align=\"center\"><strong>Female<\/strong><\/td>\r\n<td style=\"height: 15.6875px; width: 38px;\" align=\"center\">4,660<\/td>\r\n<td style=\"height: 15.6875px; width: 38px;\" align=\"center\">435<\/td>\r\n<td style=\"height: 15.6875px; width: 36px;\" align=\"center\">494<\/td>\r\n<td style=\"height: 15.6875px; width: 57px;\" align=\"center\">421<\/td>\r\n<td style=\"height: 15.6875px; width: 68px;\" align=\"center\">105<\/td>\r\n<td style=\"height: 15.6875px; width: 65px;\" align=\"center\">83<\/td>\r\n<td style=\"height: 15.6875px; width: 46px;\" align=\"center\">6,198<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 59px;\" align=\"center\"><strong>Male<\/strong><\/td>\r\n<td style=\"height: 15px; width: 38px;\" align=\"center\">4,334<\/td>\r\n<td style=\"height: 15px; width: 38px;\" align=\"center\">490<\/td>\r\n<td style=\"height: 15px; width: 36px;\" align=\"center\">564<\/td>\r\n<td style=\"height: 15px; width: 57px;\" align=\"center\">223<\/td>\r\n<td style=\"height: 15px; width: 68px;\" align=\"center\">97<\/td>\r\n<td style=\"height: 15px; width: 65px;\" align=\"center\">94<\/td>\r\n<td style=\"height: 15px; width: 46px;\" align=\"center\">5,802<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 30px;\">\r\n<td style=\"height: 30px; width: 59px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\r\n<td style=\"height: 30px; width: 38px;\" align=\"center\">8,994<\/td>\r\n<td style=\"height: 30px; width: 38px;\" align=\"center\">925<\/td>\r\n<td style=\"height: 30px; width: 36px;\" align=\"center\">1,058<\/td>\r\n<td style=\"height: 30px; width: 57px;\" align=\"center\">644<\/td>\r\n<td style=\"height: 30px; width: 68px;\" align=\"center\">202<\/td>\r\n<td style=\"height: 30px; width: 65px;\" align=\"center\">177<\/td>\r\n<td style=\"height: 30px; width: 46px;\" align=\"center\">12,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nhttps:\/\/assess.lumenlearning.com\/practice\/1e0bf3af-ac06-4d84-b6d7-1a70a84d491f\r\n\r\nhttps:\/\/assess.lumenlearning.com\/practice\/4fd49588-6fa5-4346-af07-61b0f7274365\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<h2>Conditional Probability<\/h2>\r\nHere is the same community college enrollment data.\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\" style=\"height: 30px;\">\r\n<td style=\"height: 30px;\"><\/td>\r\n<td style=\"height: 30px;\"><strong>Arts-Sci<\/strong><\/td>\r\n<td style=\"height: 30px;\"><strong>Bus-Econ<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" align=\"center\"><strong>Female<\/strong><\/td>\r\n<td style=\"height: 15px;\" align=\"center\">4,660<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">435<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">494<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">421<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">105<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">83<\/td>\r\n<td style=\"height: 15px;\" align=\"center\">6,198<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 15.6719px;\">\r\n<td style=\"height: 15.6719px;\" align=\"center\"><strong>Male<\/strong><\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">4,334<\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">490<\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">564<\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">223<\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">97<\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">94<\/td>\r\n<td style=\"height: 15.6719px;\" align=\"center\">5,802<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\" style=\"height: 30px;\">\r\n<td style=\"height: 30px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\r\n<td style=\"height: 30px;\" align=\"center\">8,994<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">925<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">1,058<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">644<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">202<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">177<\/td>\r\n<td style=\"height: 30px;\" align=\"center\">12,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHere is our first question:\r\n<ul style=\"list-style-type: none;\">\r\n \t<li><em>If we select a female student at random, what is the probability that she is in the Health Sciences program?<\/em><\/li>\r\n<\/ul>\r\n<strong>Answer:<\/strong> Of the 6,198 female students at the college, <strong>421<\/strong> are enrolled in Health Sciences. (Find these numbers in the table.) The probability we are looking for is:\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{421}}{\\text{6,198}}\\approx \\text{0.07}[\/latex]<\/p>\r\nTherefore, the probability that a female student is in the Health Sciences program is approximately 0.07.\r\n\r\n<strong>Focus on Language<\/strong>\r\n\r\nWe need to pause here and be very careful about the language we use in describing this situation.\r\n\r\nNote that we <em>start<\/em> with a female student and <em>then<\/em> ask what is the probability that this female student is in the Health Sciences department.\r\n\r\nIn this case, our <em>starting point is that the student is a female<\/em>. This information sets the conditions for calculating the probability. Once the condition <em>(student is female)<\/em> is set, we focus on the female student population. In terms of the two-way table, it means that the only numbers we will be using are in the Female row: 421 and 6,198.\r\n\r\n<strong>What Is a Conditional Probability?<\/strong>\r\n\r\nThe probability we calculated earlier is an example of a <strong>conditional probability<\/strong>. In general, a conditional probability is one that is based on a given condition. Here the <em>given condition <\/em>is that the student is female.\r\n\r\nHere is the notation we use for a conditional probability:\r\n<ul>\r\n \t<li>Original question: <em>If we select a female student at random, what is the probability that she is in the Health Sciences program?<\/em><\/li>\r\n \t<li>Notation: <em>P<\/em>(student is in Health Sciences <strong>given that <\/strong>student is female).<\/li>\r\n \t<li>We also write this as <em>P<\/em>(Health Sciences <strong>given <\/strong>female).<\/li>\r\n<\/ul>\r\nAn even shorter way of writing this is to use a vertical bar <strong>|<\/strong> in place of <em>given:<\/em> <em>P<\/em>(Health Sciences | female).\r\n\r\n<\/div>\r\nThe following table is used for the next Try It and Did I Get This? activities.\r\n<table style=\"width: 755px;\">\r\n<tbody>\r\n<tr class=\"oli_table\">\r\n<td style=\"width: 100px;\"><\/td>\r\n<td style=\"width: 54px;\"><strong>Arts-Sci<\/strong><\/td>\r\n<td style=\"width: 65px;\"><strong>Bus-Econ<\/strong><\/td>\r\n<td style=\"width: 65px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\r\n<td style=\"width: 102px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\r\n<td style=\"width: 112px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\r\n<td style=\"width: 92px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\r\n<td style=\"width: 77px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td style=\"width: 100px;\" align=\"center\"><strong>Female<\/strong><\/td>\r\n<td style=\"width: 54px;\" align=\"center\">4,660<\/td>\r\n<td style=\"width: 65px;\" align=\"center\">435<\/td>\r\n<td style=\"width: 65px;\" align=\"center\">494<\/td>\r\n<td style=\"width: 102px;\" align=\"center\">421<\/td>\r\n<td style=\"width: 112px;\" align=\"center\">105<\/td>\r\n<td style=\"width: 92px;\" align=\"center\">83<\/td>\r\n<td style=\"width: 77px;\" align=\"center\">6,198<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td style=\"width: 100px;\" align=\"center\"><strong>Male<\/strong><\/td>\r\n<td style=\"width: 54px;\" align=\"center\">4,334<\/td>\r\n<td style=\"width: 65px;\" align=\"center\">490<\/td>\r\n<td style=\"width: 65px;\" align=\"center\">564<\/td>\r\n<td style=\"width: 102px;\" align=\"center\">223<\/td>\r\n<td style=\"width: 112px;\" align=\"center\">97<\/td>\r\n<td style=\"width: 92px;\" align=\"center\">94<\/td>\r\n<td style=\"width: 77px;\" align=\"center\">5,802<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td style=\"width: 100px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\r\n<td style=\"width: 54px;\" align=\"center\">8,994<\/td>\r\n<td style=\"width: 65px;\" align=\"center\">925<\/td>\r\n<td style=\"width: 65px;\" align=\"center\">1,058<\/td>\r\n<td style=\"width: 102px;\" align=\"center\">644<\/td>\r\n<td style=\"width: 112px;\" align=\"center\">202<\/td>\r\n<td style=\"width: 92px;\" align=\"center\">177<\/td>\r\n<td style=\"width: 77px;\" align=\"center\">12,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nhttps:\/\/assess.lumenlearning.com\/practice\/9eab88f9-333b-48ae-acdf-c01824976d0b\r\n\r\nhttps:\/\/assess.lumenlearning.com\/practice\/c8a3d18b-eb8f-443c-a8fb-415c5832547b\r\n\r\n<\/div>\r\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1bkwrnzogfnvOVSzsUbMDAg_birWypZ3SWqqZLQfEX9Q\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning OUTCOMES<\/h3>\n<ul>\n<li>Calculate marginal, joint, and conditional percentages and interpret them as probability estimates.<\/li>\n<\/ul>\n<\/div>\n<p>In the previous section, we used the information in a two-table to examine the relationship between two categorical variables. Our goal was to answer the big question: <em>Are the variables related?<\/em><\/p>\n<p>In this section, we continue to work with two-way tables, but we ask a different set of questions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<h2>Community College Enrollment<\/h2>\n<p>The following table summarizes the full-time enrollment at a community college located in a West Coast city. There are a total of 12,000 full-time students enrolled at the college. The two categorical variables here are <em>gender<\/em> and <em>program<\/em>. The programs include academic and vocational programs at the college. Assume that a student can enroll in only one program.<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\" style=\"height: 30px;\">\n<td style=\"height: 30px;\"><\/td>\n<td style=\"height: 30px;\"><strong>Arts-Sci<\/strong><\/td>\n<td style=\"height: 30px;\"><strong>Bus-Econ<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15.3906px;\">\n<td style=\"height: 15.3906px;\" align=\"center\"><strong>Female<\/strong><\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">4,660<\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">435<\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">494<\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">421<\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">105<\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">83<\/td>\n<td style=\"height: 15.3906px;\" align=\"center\">6,198<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px;\">\n<td style=\"height: 15px;\" align=\"center\"><strong>Male<\/strong><\/td>\n<td style=\"height: 15px;\" align=\"center\">4,334<\/td>\n<td style=\"height: 15px;\" align=\"center\">490<\/td>\n<td style=\"height: 15px;\" align=\"center\">564<\/td>\n<td style=\"height: 15px;\" align=\"center\">223<\/td>\n<td style=\"height: 15px;\" align=\"center\">97<\/td>\n<td style=\"height: 15px;\" align=\"center\">94<\/td>\n<td style=\"height: 15px;\" align=\"center\">5,802<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 30px;\">\n<td style=\"height: 30px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\">8,994<\/td>\n<td style=\"height: 30px;\" align=\"center\">925<\/td>\n<td style=\"height: 30px;\" align=\"center\">1,058<\/td>\n<td style=\"height: 30px;\" align=\"center\">644<\/td>\n<td style=\"height: 30px;\" align=\"center\">202<\/td>\n<td style=\"height: 30px;\" align=\"center\">177<\/td>\n<td style=\"height: 30px;\" align=\"center\">12,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let\u2019s consider a few preliminary questions to get familiar with this new data set.<\/p>\n<ul style=\"list-style-type: none;\">\n<li>1. <em>What proportion of the total number of students are male students?<\/em><\/li>\n<\/ul>\n<p><strong>Answer:<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\mathrm{number\\; of\\; male\\; students}}{\\mathrm{total\\; number\\; of\\; students}}=\\frac{\\mathrm{5,802}}{\\mathrm{12,000}}=\\mathrm{0.4835}(\\mathrm{or\\; 48.35\\%})[\/latex]<\/p>\n<ul style=\"list-style-type: none;\">\n<li>2. <em>What proportion of the total number of students are Bus-Econ students?<\/em><\/li>\n<\/ul>\n<p><strong>Answer:<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\mathrm{number\\; of\\; Bus-Econ\\; students}}{\\mathrm{total\\; number\\; of\\; students}}=\\frac{925}{\\mathrm{12,000}}=\\mathrm{0.077}(\\mathrm{or\\; 7.7\\%})[\/latex]<\/p>\n<p>Note that to calculate this proportion, we used two numbers in the margin that relate to just one of the categorical variables (program). This calculation is therefore called a <strong>marginal proportion<\/strong>.<\/p>\n<p>Note: This proportion does not help us determine if gender is related to program because it involves only one of the variables.<\/p>\n<p>Now consider the following question:<\/p>\n<ul style=\"list-style-type: none;\">\n<li><em>If we choose one student at random from among all 12,000 students at the college, how likely is it that this student will be in the Bus-Econ program?<\/em><\/li>\n<\/ul>\n<p>From our previous calculation, we know that only about 8% (7.7%) of the students at the college are in the Bus-Econ program. That&#8217;s a fairly low number, so it is not very likely that our random student will be a Bus-Econ student.<\/p>\n<p>One way to state our conclusion is to say:<\/p>\n<ul style=\"list-style-type: none;\">\n<li>There is about an <strong>8% chance<\/strong> of picking a Bus-Econ major.<\/li>\n<\/ul>\n<p>This means that if we selected 100 students at random, we would expect on average that 8 of them would be in the Bus-Econ program.<\/p>\n<p>Here is another way to state this conclusion:<\/p>\n<ul style=\"list-style-type: none;\">\n<li>There is about an <strong>0.08 probability<\/strong> of picking a Bus-Econ major.<\/li>\n<\/ul>\n<p>Because this probability is exactly the same as the marginal proportion we calculated earlier, we call it a <strong>marginal probability<\/strong>.<\/p>\n<\/div>\n<p><strong>Note:<\/strong> <em><strong>P<\/strong><\/em> <strong>for Probability<\/strong><\/p>\n<p>It is customary to use the capital letter <em>P <\/em>to stand for probability. So instead of writing \u201cThe probability that a student is in Bus-Econ program equals 0.08,\u201d we can write <em>P<\/em>(student is in Bus-Econ) = 0.08.<\/p>\n<p>The following table is used for the next Try It and Did I Get This? activities.<\/p>\n<table style=\"width: 495px;\">\n<tbody>\n<tr class=\"oli_table\" style=\"height: 30px;\">\n<td style=\"height: 30px; width: 59px;\"><\/td>\n<td style=\"height: 30px; width: 38px;\"><strong>Arts-Sci<\/strong><\/td>\n<td style=\"height: 30px; width: 38px;\"><strong>Bus-Econ<\/strong><\/td>\n<td style=\"height: 30px; width: 36px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\n<td style=\"height: 30px; width: 57px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\n<td style=\"height: 30px; width: 68px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\n<td style=\"height: 30px; width: 65px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\n<td style=\"height: 30px; width: 46px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15.6875px;\">\n<td style=\"height: 15.6875px; width: 59px;\" align=\"center\"><strong>Female<\/strong><\/td>\n<td style=\"height: 15.6875px; width: 38px;\" align=\"center\">4,660<\/td>\n<td style=\"height: 15.6875px; width: 38px;\" align=\"center\">435<\/td>\n<td style=\"height: 15.6875px; width: 36px;\" align=\"center\">494<\/td>\n<td style=\"height: 15.6875px; width: 57px;\" align=\"center\">421<\/td>\n<td style=\"height: 15.6875px; width: 68px;\" align=\"center\">105<\/td>\n<td style=\"height: 15.6875px; width: 65px;\" align=\"center\">83<\/td>\n<td style=\"height: 15.6875px; width: 46px;\" align=\"center\">6,198<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px;\">\n<td style=\"height: 15px; width: 59px;\" align=\"center\"><strong>Male<\/strong><\/td>\n<td style=\"height: 15px; width: 38px;\" align=\"center\">4,334<\/td>\n<td style=\"height: 15px; width: 38px;\" align=\"center\">490<\/td>\n<td style=\"height: 15px; width: 36px;\" align=\"center\">564<\/td>\n<td style=\"height: 15px; width: 57px;\" align=\"center\">223<\/td>\n<td style=\"height: 15px; width: 68px;\" align=\"center\">97<\/td>\n<td style=\"height: 15px; width: 65px;\" align=\"center\">94<\/td>\n<td style=\"height: 15px; width: 46px;\" align=\"center\">5,802<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 30px;\">\n<td style=\"height: 30px; width: 59px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\n<td style=\"height: 30px; width: 38px;\" align=\"center\">8,994<\/td>\n<td style=\"height: 30px; width: 38px;\" align=\"center\">925<\/td>\n<td style=\"height: 30px; width: 36px;\" align=\"center\">1,058<\/td>\n<td style=\"height: 30px; width: 57px;\" align=\"center\">644<\/td>\n<td style=\"height: 30px; width: 68px;\" align=\"center\">202<\/td>\n<td style=\"height: 30px; width: 65px;\" align=\"center\">177<\/td>\n<td style=\"height: 30px; width: 46px;\" align=\"center\">12,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>\t<iframe id=\"assessment_practice_1e0bf3af-ac06-4d84-b6d7-1a70a84d491f\" class=\"resizable\" src=\"https:\/\/assess.lumenlearning.com\/practice\/1e0bf3af-ac06-4d84-b6d7-1a70a84d491f?iframe_resize_id=assessment_practice_id_1e0bf3af-ac06-4d84-b6d7-1a70a84d491f\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:300px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"assessment_practice_4fd49588-6fa5-4346-af07-61b0f7274365\" class=\"resizable\" src=\"https:\/\/assess.lumenlearning.com\/practice\/4fd49588-6fa5-4346-af07-61b0f7274365?iframe_resize_id=assessment_practice_id_4fd49588-6fa5-4346-af07-61b0f7274365\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:300px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<h2>Conditional Probability<\/h2>\n<p>Here is the same community college enrollment data.<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\" style=\"height: 30px;\">\n<td style=\"height: 30px;\"><\/td>\n<td style=\"height: 30px;\"><strong>Arts-Sci<\/strong><\/td>\n<td style=\"height: 30px;\"><strong>Bus-Econ<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15px;\">\n<td style=\"height: 15px;\" align=\"center\"><strong>Female<\/strong><\/td>\n<td style=\"height: 15px;\" align=\"center\">4,660<\/td>\n<td style=\"height: 15px;\" align=\"center\">435<\/td>\n<td style=\"height: 15px;\" align=\"center\">494<\/td>\n<td style=\"height: 15px;\" align=\"center\">421<\/td>\n<td style=\"height: 15px;\" align=\"center\">105<\/td>\n<td style=\"height: 15px;\" align=\"center\">83<\/td>\n<td style=\"height: 15px;\" align=\"center\">6,198<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 15.6719px;\">\n<td style=\"height: 15.6719px;\" align=\"center\"><strong>Male<\/strong><\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">4,334<\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">490<\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">564<\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">223<\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">97<\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">94<\/td>\n<td style=\"height: 15.6719px;\" align=\"center\">5,802<\/td>\n<\/tr>\n<tr class=\"oli_table\" style=\"height: 30px;\">\n<td style=\"height: 30px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\n<td style=\"height: 30px;\" align=\"center\">8,994<\/td>\n<td style=\"height: 30px;\" align=\"center\">925<\/td>\n<td style=\"height: 30px;\" align=\"center\">1,058<\/td>\n<td style=\"height: 30px;\" align=\"center\">644<\/td>\n<td style=\"height: 30px;\" align=\"center\">202<\/td>\n<td style=\"height: 30px;\" align=\"center\">177<\/td>\n<td style=\"height: 30px;\" align=\"center\">12,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here is our first question:<\/p>\n<ul style=\"list-style-type: none;\">\n<li><em>If we select a female student at random, what is the probability that she is in the Health Sciences program?<\/em><\/li>\n<\/ul>\n<p><strong>Answer:<\/strong> Of the 6,198 female students at the college, <strong>421<\/strong> are enrolled in Health Sciences. (Find these numbers in the table.) The probability we are looking for is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{421}}{\\text{6,198}}\\approx \\text{0.07}[\/latex]<\/p>\n<p>Therefore, the probability that a female student is in the Health Sciences program is approximately 0.07.<\/p>\n<p><strong>Focus on Language<\/strong><\/p>\n<p>We need to pause here and be very careful about the language we use in describing this situation.<\/p>\n<p>Note that we <em>start<\/em> with a female student and <em>then<\/em> ask what is the probability that this female student is in the Health Sciences department.<\/p>\n<p>In this case, our <em>starting point is that the student is a female<\/em>. This information sets the conditions for calculating the probability. Once the condition <em>(student is female)<\/em> is set, we focus on the female student population. In terms of the two-way table, it means that the only numbers we will be using are in the Female row: 421 and 6,198.<\/p>\n<p><strong>What Is a Conditional Probability?<\/strong><\/p>\n<p>The probability we calculated earlier is an example of a <strong>conditional probability<\/strong>. In general, a conditional probability is one that is based on a given condition. Here the <em>given condition <\/em>is that the student is female.<\/p>\n<p>Here is the notation we use for a conditional probability:<\/p>\n<ul>\n<li>Original question: <em>If we select a female student at random, what is the probability that she is in the Health Sciences program?<\/em><\/li>\n<li>Notation: <em>P<\/em>(student is in Health Sciences <strong>given that <\/strong>student is female).<\/li>\n<li>We also write this as <em>P<\/em>(Health Sciences <strong>given <\/strong>female).<\/li>\n<\/ul>\n<p>An even shorter way of writing this is to use a vertical bar <strong>|<\/strong> in place of <em>given:<\/em> <em>P<\/em>(Health Sciences | female).<\/p>\n<\/div>\n<p>The following table is used for the next Try It and Did I Get This? activities.<\/p>\n<table style=\"width: 755px;\">\n<tbody>\n<tr class=\"oli_table\">\n<td style=\"width: 100px;\"><\/td>\n<td style=\"width: 54px;\"><strong>Arts-Sci<\/strong><\/td>\n<td style=\"width: 65px;\"><strong>Bus-Econ<\/strong><\/td>\n<td style=\"width: 65px;\" align=\"center\"><strong>Info Tech<\/strong><\/td>\n<td style=\"width: 102px;\" align=\"center\"><strong>Health Science<\/strong><\/td>\n<td style=\"width: 112px;\" align=\"center\"><strong>Graphics Design<\/strong><\/td>\n<td style=\"width: 92px;\" align=\"center\"><strong>Culinary Arts<\/strong><\/td>\n<td style=\"width: 77px;\" align=\"center\"><strong>Row Totals<\/strong><\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td style=\"width: 100px;\" align=\"center\"><strong>Female<\/strong><\/td>\n<td style=\"width: 54px;\" align=\"center\">4,660<\/td>\n<td style=\"width: 65px;\" align=\"center\">435<\/td>\n<td style=\"width: 65px;\" align=\"center\">494<\/td>\n<td style=\"width: 102px;\" align=\"center\">421<\/td>\n<td style=\"width: 112px;\" align=\"center\">105<\/td>\n<td style=\"width: 92px;\" align=\"center\">83<\/td>\n<td style=\"width: 77px;\" align=\"center\">6,198<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td style=\"width: 100px;\" align=\"center\"><strong>Male<\/strong><\/td>\n<td style=\"width: 54px;\" align=\"center\">4,334<\/td>\n<td style=\"width: 65px;\" align=\"center\">490<\/td>\n<td style=\"width: 65px;\" align=\"center\">564<\/td>\n<td style=\"width: 102px;\" align=\"center\">223<\/td>\n<td style=\"width: 112px;\" align=\"center\">97<\/td>\n<td style=\"width: 92px;\" align=\"center\">94<\/td>\n<td style=\"width: 77px;\" align=\"center\">5,802<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td style=\"width: 100px;\" align=\"center\"><strong>Column Totals<\/strong><\/td>\n<td style=\"width: 54px;\" align=\"center\">8,994<\/td>\n<td style=\"width: 65px;\" align=\"center\">925<\/td>\n<td style=\"width: 65px;\" align=\"center\">1,058<\/td>\n<td style=\"width: 102px;\" align=\"center\">644<\/td>\n<td style=\"width: 112px;\" align=\"center\">202<\/td>\n<td style=\"width: 92px;\" align=\"center\">177<\/td>\n<td style=\"width: 77px;\" align=\"center\">12,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>\t<iframe id=\"assessment_practice_9eab88f9-333b-48ae-acdf-c01824976d0b\" class=\"resizable\" src=\"https:\/\/assess.lumenlearning.com\/practice\/9eab88f9-333b-48ae-acdf-c01824976d0b?iframe_resize_id=assessment_practice_id_9eab88f9-333b-48ae-acdf-c01824976d0b\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:300px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"assessment_practice_c8a3d18b-eb8f-443c-a8fb-415c5832547b\" class=\"resizable\" src=\"https:\/\/assess.lumenlearning.com\/practice\/c8a3d18b-eb8f-443c-a8fb-415c5832547b?iframe_resize_id=assessment_practice_id_c8a3d18b-eb8f-443c-a8fb-415c5832547b\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:300px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1bkwrnzogfnvOVSzsUbMDAg_birWypZ3SWqqZLQfEX9Q\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-252\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":163,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"1069c50d-6e4a-4c0d-ba20-14c8af2cc9db, b98f7606-927e-441b-8623-4c9e637869ec","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-252","chapter","type-chapter","status-publish","hentry"],"part":245,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/252","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/252\/revisions"}],"predecessor-version":[{"id":2748,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/252\/revisions\/2748"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/parts\/245"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapters\/252\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/media?parent=252"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/pressbooks\/v2\/chapter-type?post=252"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/contributor?post=252"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-concepts-statistics\/wp-json\/wp\/v2\/license?post=252"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}