{"id":1033,"date":"2023-06-22T01:45:12","date_gmt":"2023-06-22T01:45:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/binomial-distribution-learn-it-2\/"},"modified":"2023-10-23T13:05:30","modified_gmt":"2023-10-23T13:05:30","slug":"binomial-distribution-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/binomial-distribution-learn-it-2\/","title":{"raw":"Binomial Distribution: Learn It 2","rendered":"Binomial Distribution: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use a binomial distribution to calculate probability<\/li>\r\n\t<li>Determine if a probability model meets the conditions for a binomial distribution<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Bernoulli Trial<\/h3>\r\n<section class=\"textbox example\">Flipping a coin is a classic example of a Bernoulli trial. When we flip a coin, there are two possible outcomes, heads or tails, and the different flips of the coin are independent. We can think of flipping tails as a \u201csuccess,\u201d and if the coin is fair, the probability of success is [latex]p = 0.5[\/latex] for every trial.Let's experiment flipping a coin [latex]3[\/latex] times and counting the number of tails obtained.Flipping a coin [latex]3[\/latex] times is a binomial experiment where [latex]n = 3 \\text{ and } p = 0.5[\/latex], and the random variable is the number of tails in [latex]3[\/latex] coin flips. Therefore, the distribution of [latex]X[\/latex] can be modeled using the binomial distribution.[reveal-answer q=\"225721\"]Outcomes of flipping a coin 3 times.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"225721\"]Recall that the outcomes of the experiment are as given in the following table:\r\n\r\n<div align=\"center\">\r\n<table style=\"height: 108px; width: 386px;\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 118.547px;\">\r\n<p style=\"text-align: center;\"><strong>Experimental <\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>Outcome<\/strong><\/p>\r\n<\/td>\r\n<td style=\"height: 12px; width: 243.766px;\">\r\n<p style=\"text-align: center;\"><strong>[latex]X[\/latex]<\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>Number of Tails in 3 Flips of a Coin<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>The probability of each value of [latex]X[\/latex] can be found by calculating its <strong>relative frequencies<\/strong>.<\/p>\r\n<p>Notice that since the trials here are <strong>independent<\/strong>, you can also find these probabilities by using the rule for independent events, [latex] P(A \\mbox{ and } B) = P(A) \\cdot P(B) [\/latex], in combination with the rule for finding OR probabilities.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1482[\/ohm2_question]<\/section>\r\n<p>Notice that the probabilities you found in part (a) and (b) are the same. This is due to the fact that the trials are independent!<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use a binomial distribution to calculate probability<\/li>\n<li>Determine if a probability model meets the conditions for a binomial distribution<\/li>\n<\/ul>\n<\/section>\n<h3>Bernoulli Trial<\/h3>\n<section class=\"textbox example\">Flipping a coin is a classic example of a Bernoulli trial. When we flip a coin, there are two possible outcomes, heads or tails, and the different flips of the coin are independent. We can think of flipping tails as a \u201csuccess,\u201d and if the coin is fair, the probability of success is [latex]p = 0.5[\/latex] for every trial.Let&#8217;s experiment flipping a coin [latex]3[\/latex] times and counting the number of tails obtained.Flipping a coin [latex]3[\/latex] times is a binomial experiment where [latex]n = 3 \\text{ and } p = 0.5[\/latex], and the random variable is the number of tails in [latex]3[\/latex] coin flips. Therefore, the distribution of [latex]X[\/latex] can be modeled using the binomial distribution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q225721\">Outcomes of flipping a coin 3 times.<\/button><\/p>\n<div id=\"q225721\" class=\"hidden-answer\" style=\"display: none\">Recall that the outcomes of the experiment are as given in the following table:<\/p>\n<div style=\"margin: auto;\">\n<table style=\"height: 108px; width: 386px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 118.547px;\">\n<p style=\"text-align: center;\"><strong>Experimental <\/strong><\/p>\n<p style=\"text-align: center;\"><strong>Outcome<\/strong><\/p>\n<\/td>\n<td style=\"height: 12px; width: 243.766px;\">\n<p style=\"text-align: center;\"><strong>[latex]X[\/latex]<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>Number of Tails in 3 Flips of a Coin<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>The probability of each value of [latex]X[\/latex] can be found by calculating its <strong>relative frequencies<\/strong>.<\/p>\n<p>Notice that since the trials here are <strong>independent<\/strong>, you can also find these probabilities by using the rule for independent events, [latex]P(A \\mbox{ and } B) = P(A) \\cdot P(B)[\/latex], in combination with the rule for finding OR probabilities.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1482\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1482&theme=lumen&iframe_resize_id=ohm1482&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Notice that the probabilities you found in part (a) and (b) are the same. This is due to the fact that the trials are independent!<\/p>\n","protected":false},"author":8,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2912,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1033"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1033\/revisions"}],"predecessor-version":[{"id":4080,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1033\/revisions\/4080"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2912"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1033\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1033"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1033"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1033"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}