{"id":1284,"date":"2021-08-23T19:27:37","date_gmt":"2021-08-23T19:27:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1284"},"modified":"2023-12-05T09:09:49","modified_gmt":"2023-12-05T09:09:49","slug":"summary-binomial-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-binomial-distribution\/","title":{"raw":"Summary: Binomial Distribution","rendered":"Summary: Binomial Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Binomial experiments consisted of a fixed number of independent trials.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The probability of success and failure is the same in each trial of a binomial experiment.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>Bernoulli trials:<\/strong> an experiment with the following characteristics:\r\n<ol>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">There are only two possible outcomes called \u201csuccess\u201d and \u201cfailure\u201d for each trial.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The probability [latex]p[\/latex] of a success is the same for any trial (so the probability [latex]q = 1 \u2212 p[\/latex] of a failure is the same for any trial).<\/li>\r\n<\/ol>\r\n<strong>binomial experiment:\u00a0<\/strong>a statistical experiment that satisfies the following three conditions:\r\n<ol>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">There are a fixed number of trials, [latex]n[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">There are only two possible outcomes, called \"success\" and, \"failure,\" for each trial. The letter [latex]p[\/latex] denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The [latex]n[\/latex] trials are independent and are repeated using identical conditions.<\/li>\r\n<\/ol>\r\n<strong>binomial probability distribution:<\/strong> a discrete random variable [latex](RV)[\/latex] that arises from Bernoulli trials; there are a fixed number, [latex]n[\/latex], of independent trials. The notation is: [latex]X \\sim B(n, p)[\/latex]. The mean is [latex]\\mu = np[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{np(1-p)}[\/latex]. The probability of exactly [latex]x[\/latex] successes in [latex]n[\/latex] trials is\r\n<p style=\"text-align: center;\">[latex]P(X=x) =(_{x}^{n})p^{x}(q)^{n-x}, \\mathrm{where} (_{x}^{n}) = \\frac{n!}{x!(n-x)!}[\/latex]<\/p>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Binomial experiments consisted of a fixed number of independent trials.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The probability of success and failure is the same in each trial of a binomial experiment.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>Bernoulli trials:<\/strong> an experiment with the following characteristics:<\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">There are only two possible outcomes called \u201csuccess\u201d and \u201cfailure\u201d for each trial.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The probability [latex]p[\/latex] of a success is the same for any trial (so the probability [latex]q = 1 \u2212 p[\/latex] of a failure is the same for any trial).<\/li>\n<\/ol>\n<p><strong>binomial experiment:\u00a0<\/strong>a statistical experiment that satisfies the following three conditions:<\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">There are a fixed number of trials, [latex]n[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">There are only two possible outcomes, called &#8220;success&#8221; and, &#8220;failure,&#8221; for each trial. The letter [latex]p[\/latex] denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The [latex]n[\/latex] trials are independent and are repeated using identical conditions.<\/li>\n<\/ol>\n<p><strong>binomial probability distribution:<\/strong> a discrete random variable [latex](RV)[\/latex] that arises from Bernoulli trials; there are a fixed number, [latex]n[\/latex], of independent trials. The notation is: [latex]X \\sim B(n, p)[\/latex]. The mean is [latex]\\mu = np[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{np(1-p)}[\/latex]. The probability of exactly [latex]x[\/latex] successes in [latex]n[\/latex] trials is<\/p>\n<p style=\"text-align: center;\">[latex]P(X=x) =(_{x}^{n})p^{x}(q)^{n-x}, \\mathrm{where} (_{x}^{n}) = \\frac{n!}{x!(n-x)!}[\/latex]<\/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-1284\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-key-terms<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":169134,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1284","chapter","type-chapter","status-publish","hentry"],"part":240,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1284","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1284\/revisions"}],"predecessor-version":[{"id":4061,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1284\/revisions\/4061"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/240"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1284\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1284"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1284"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1284"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1284"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}