{"id":157,"date":"2022-05-18T16:37:11","date_gmt":"2022-05-18T16:37:11","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/probability-topics\/"},"modified":"2022-11-09T16:07:49","modified_gmt":"2022-11-09T16:07:49","slug":"probability-topics","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/probability-topics\/","title":{"raw":"Activity 4.7: Probability Topics","rendered":"Activity 4.7: Probability Topics"},"content":{"raw":"&nbsp;\r\n<div id=\"fs-id1172778073391\" class=\"statistics lab\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\r\n<div data-type=\"title\">Probability Topics<\/div>\r\nClass time:\r\n<p id=\"element-279\">Names:<\/p>\r\n<p id=\"fs-idp7760432\"><span data-type=\"title\">Student Learning Outcomes<\/span><\/p>\r\n\r\n<ul>\r\n \t<li>The student will use theoretical and empirical methods to estimate probabilities.<\/li>\r\n \t<li>The student will appraise the differences between the two estimates.<\/li>\r\n \t<li>The student will demonstrate an understanding of long-term relative frequencies.<\/li>\r\n<\/ul>\r\n<p id=\"element-410\"><span data-type=\"title\">Do the Experiment<\/span> Count out 40 mixed-color M&amp;Ms\u00ae which is approximately one small bag\u2019s worth. Record the number of each color in <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl009\">(Figure)<\/a>. Use the information from this table to complete <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl010\">(Figure)<\/a>. Next, put the M&amp;Ms in a cup. The experiment is to pick two M&amp;Ms, one at a time. Do <strong>not<\/strong> look at them as you pick them. The first time through, replace the first M&amp;M before picking the second one. Record the results in the \u201cWith Replacement\u201d column of <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl011\">(Figure)<\/a>. Do this 24 times. The second time through, after picking the first M&amp;M, do <strong>not<\/strong> replace it before picking the second one. Then, pick the second one. Record the results in the \u201cWithout Replacement\u201d column section of <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl012\">(Figure)<\/a>. After you record the pick, put <strong>both<\/strong> M&amp;Ms back. Do this a total of 24 times, also. Use the data from <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl012\">(Figure)<\/a> to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do <strong>not<\/strong> multiply out any fractions.<\/p>\r\n\r\n<table summary=\"Partially filled theoretical data table. The first column lists the color (6 rows) and the blank second column lists the quantity values.\"><caption><span data-type=\"title\">Population<\/span><\/caption>\r\n<thead>\r\n<tr>\r\n<th>Color<\/th>\r\n<th>Quantity<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Yellow (<em data-effect=\"italics\">Y<\/em>)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Green (<em data-effect=\"italics\">G<\/em>)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Blue (<em data-effect=\"italics\">BL<\/em>)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Brown (<em data-effect=\"italics\">B<\/em>)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Orange (<em data-effect=\"italics\">O<\/em>)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Red (<em data-effect=\"italics\">R<\/em>)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"M05_ch03-tbl010\" summary=\"\"><caption><span data-type=\"title\">Theoretical Probabilities<\/span><\/caption>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>With Replacement<\/th>\r\n<th>Without Replacement<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(2 reds)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub><em data-effect=\"italics\">B<\/em><sub>2<\/sub> OR <em data-effect=\"italics\">B<\/em><sub>1<\/sub><em data-effect=\"italics\">R<\/em><sub>2<\/sub>)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub> AND <em data-effect=\"italics\">G<\/em><sub>2<\/sub>)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">G<\/em><sub>2<\/sub>|<em data-effect=\"italics\">R<\/em><sub>1<\/sub>)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(no yellows)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(doubles)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(no doubles)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-idp15354112\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\r\n<div data-type=\"title\">Note<\/div>\r\n<p id=\"fs-idp27436528\"><em data-effect=\"italics\">G<\/em><sub>2<\/sub> = green on second pick; <em data-effect=\"italics\">R<\/em><sub>1<\/sub> = red on first pick; <em data-effect=\"italics\">B<\/em><sub>1<\/sub> = brown on first pick; <em data-effect=\"italics\">B<\/em><sub>2<\/sub> = brown on second pick; doubles = both picks are the same colour.<\/p>\r\n\r\n<\/div>\r\n<table summary=\"Blank empirical results table with the first column designated for with replacement and the second column listed for without replacement. 24 empty cells.\"><caption><span data-type=\"title\">Empirical Results<\/span><\/caption>\r\n<thead>\r\n<tr>\r\n<th>With Replacement<\/th>\r\n<th>Without Replacement<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<td>( __ , __ ) ( __ , __ )<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"M05_ch03-tbl012\" summary=\"\"><caption><span data-type=\"title\">Empirical Probabilities<\/span><\/caption>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>With Replacement<\/th>\r\n<th>Without Replacement<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(2 reds)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub><em data-effect=\"italics\">B<\/em><sub>2<\/sub> OR <em data-effect=\"italics\">B<\/em><sub>1<\/sub><em data-effect=\"italics\">R<\/em><sub>2<\/sub>)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub> AND <em data-effect=\"italics\">G<\/em><sub>2<\/sub>)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">G<\/em><sub>2<\/sub>|<em data-effect=\"italics\">R<\/em><sub>1<\/sub>)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(no yellows)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(doubles)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em data-effect=\"italics\">P<\/em>(no doubles)<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-idp8995504\"><span data-type=\"title\">Discussion Questions<\/span><\/p>\r\n\r\n<ol id=\"element-148\">\r\n \t<li>Why are the \u201cWith Replacement\u201d and \u201cWithout Replacement\u201d probabilities different?<\/li>\r\n \t<li>Convert <em data-effect=\"italics\">P<\/em>(no yellows) to decimal format for both Theoretical \u201cWith Replacement\u201d and for Empirical \u201cWith Replacement\u201d. Round to four decimal places.\r\n<ol id=\"sublist1\" type=\"a\">\r\n \t<li>Theoretical \u201cWith Replacement\u201d: <em data-effect=\"italics\">P<\/em>(no yellows) = _______<\/li>\r\n \t<li>Empirical \u201cWith Replacement\u201d: <em data-effect=\"italics\">P<\/em>(no yellows) = _______<\/li>\r\n \t<li>Are the decimal values \u201cclose\u201d? Did you expect them to be closer together or farther apart? Why?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>If you increased the number of times you picked two M&amp;Ms to 240 times, why would empirical probability values change?<\/li>\r\n \t<li>Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?<\/li>\r\n \t<li>Explain the differences in what <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">G<\/em><sub>1<\/sub> AND <em data-effect=\"italics\">R<\/em><sub>2<\/sub>) and <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub>|<em data-effect=\"italics\">G<\/em><sub>2<\/sub>) represent. Hint: Think about the sample space for each probability.<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<p>&nbsp;<\/p>\n<div id=\"fs-id1172778073391\" class=\"statistics lab\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\">Probability Topics<\/div>\n<p>Class time:<\/p>\n<p id=\"element-279\">Names:<\/p>\n<p id=\"fs-idp7760432\"><span data-type=\"title\">Student Learning Outcomes<\/span><\/p>\n<ul>\n<li>The student will use theoretical and empirical methods to estimate probabilities.<\/li>\n<li>The student will appraise the differences between the two estimates.<\/li>\n<li>The student will demonstrate an understanding of long-term relative frequencies.<\/li>\n<\/ul>\n<p id=\"element-410\"><span data-type=\"title\">Do the Experiment<\/span> Count out 40 mixed-color M&amp;Ms\u00ae which is approximately one small bag\u2019s worth. Record the number of each color in <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl009\">(Figure)<\/a>. Use the information from this table to complete <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl010\">(Figure)<\/a>. Next, put the M&amp;Ms in a cup. The experiment is to pick two M&amp;Ms, one at a time. Do <strong>not<\/strong> look at them as you pick them. The first time through, replace the first M&amp;M before picking the second one. Record the results in the \u201cWith Replacement\u201d column of <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl011\">(Figure)<\/a>. Do this 24 times. The second time through, after picking the first M&amp;M, do <strong>not<\/strong> replace it before picking the second one. Then, pick the second one. Record the results in the \u201cWithout Replacement\u201d column section of <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl012\">(Figure)<\/a>. After you record the pick, put <strong>both<\/strong> M&amp;Ms back. Do this a total of 24 times, also. Use the data from <a class=\"autogenerated-content\" href=\"#M05_ch03-tbl012\">(Figure)<\/a> to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do <strong>not<\/strong> multiply out any fractions.<\/p>\n<table summary=\"Partially filled theoretical data table. The first column lists the color (6 rows) and the blank second column lists the quantity values.\">\n<caption><span data-type=\"title\">Population<\/span><\/caption>\n<thead>\n<tr>\n<th>Color<\/th>\n<th>Quantity<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Yellow (<em data-effect=\"italics\">Y<\/em>)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Green (<em data-effect=\"italics\">G<\/em>)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Blue (<em data-effect=\"italics\">BL<\/em>)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Brown (<em data-effect=\"italics\">B<\/em>)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Orange (<em data-effect=\"italics\">O<\/em>)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Red (<em data-effect=\"italics\">R<\/em>)<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"M05_ch03-tbl010\" summary=\"\">\n<caption><span data-type=\"title\">Theoretical Probabilities<\/span><\/caption>\n<thead>\n<tr>\n<th><\/th>\n<th>With Replacement<\/th>\n<th>Without Replacement<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(2 reds)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub><em data-effect=\"italics\">B<\/em><sub>2<\/sub> OR <em data-effect=\"italics\">B<\/em><sub>1<\/sub><em data-effect=\"italics\">R<\/em><sub>2<\/sub>)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub> AND <em data-effect=\"italics\">G<\/em><sub>2<\/sub>)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">G<\/em><sub>2<\/sub>|<em data-effect=\"italics\">R<\/em><sub>1<\/sub>)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(no yellows)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(doubles)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(no doubles)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-idp15354112\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\">Note<\/div>\n<p id=\"fs-idp27436528\"><em data-effect=\"italics\">G<\/em><sub>2<\/sub> = green on second pick; <em data-effect=\"italics\">R<\/em><sub>1<\/sub> = red on first pick; <em data-effect=\"italics\">B<\/em><sub>1<\/sub> = brown on first pick; <em data-effect=\"italics\">B<\/em><sub>2<\/sub> = brown on second pick; doubles = both picks are the same colour.<\/p>\n<\/div>\n<table summary=\"Blank empirical results table with the first column designated for with replacement and the second column listed for without replacement. 24 empty cells.\">\n<caption><span data-type=\"title\">Empirical Results<\/span><\/caption>\n<thead>\n<tr>\n<th>With Replacement<\/th>\n<th>Without Replacement<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<tr>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<td>( __ , __ ) ( __ , __ )<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"M05_ch03-tbl012\" summary=\"\">\n<caption><span data-type=\"title\">Empirical Probabilities<\/span><\/caption>\n<thead>\n<tr>\n<th><\/th>\n<th>With Replacement<\/th>\n<th>Without Replacement<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(2 reds)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub><em data-effect=\"italics\">B<\/em><sub>2<\/sub> OR <em data-effect=\"italics\">B<\/em><sub>1<\/sub><em data-effect=\"italics\">R<\/em><sub>2<\/sub>)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub> AND <em data-effect=\"italics\">G<\/em><sub>2<\/sub>)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">G<\/em><sub>2<\/sub>|<em data-effect=\"italics\">R<\/em><sub>1<\/sub>)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(no yellows)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(doubles)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">P<\/em>(no doubles)<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-idp8995504\"><span data-type=\"title\">Discussion Questions<\/span><\/p>\n<ol id=\"element-148\">\n<li>Why are the \u201cWith Replacement\u201d and \u201cWithout Replacement\u201d probabilities different?<\/li>\n<li>Convert <em data-effect=\"italics\">P<\/em>(no yellows) to decimal format for both Theoretical \u201cWith Replacement\u201d and for Empirical \u201cWith Replacement\u201d. Round to four decimal places.\n<ol id=\"sublist1\" type=\"a\">\n<li>Theoretical \u201cWith Replacement\u201d: <em data-effect=\"italics\">P<\/em>(no yellows) = _______<\/li>\n<li>Empirical \u201cWith Replacement\u201d: <em data-effect=\"italics\">P<\/em>(no yellows) = _______<\/li>\n<li>Are the decimal values \u201cclose\u201d? Did you expect them to be closer together or farther apart? Why?<\/li>\n<\/ol>\n<\/li>\n<li>If you increased the number of times you picked two M&amp;Ms to 240 times, why would empirical probability values change?<\/li>\n<li>Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?<\/li>\n<li>Explain the differences in what <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">G<\/em><sub>1<\/sub> AND <em data-effect=\"italics\">R<\/em><sub>2<\/sub>) and <em data-effect=\"italics\">P<\/em>(<em data-effect=\"italics\">R<\/em><sub>1<\/sub>|<em data-effect=\"italics\">G<\/em><sub>2<\/sub>) represent. Hint: Think about the sample space for each probability.<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":32,"menu_order":25,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-157","chapter","type-chapter","status-publish","hentry"],"part":125,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/users\/32"}],"version-history":[{"count":3,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/157\/revisions"}],"predecessor-version":[{"id":659,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/157\/revisions\/659"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/parts\/125"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapters\/157\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/media?parent=157"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/pressbooks\/v2\/chapter-type?post=157"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/contributor?post=157"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/accintrostats\/wp-json\/wp\/v2\/license?post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}