{"id":1153,"date":"2021-08-19T18:30:14","date_gmt":"2021-08-19T18:30:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1153"},"modified":"2023-12-05T09:02:13","modified_gmt":"2023-12-05T09:02:13","slug":"summary-independent-and-mutually-exclusive-events","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-independent-and-mutually-exclusive-events\/","title":{"raw":"Summary: Independent and Mutually Exclusive Events","rendered":"Summary: Independent and Mutually Exclusive Events"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">For two events to be independent, the outcome of one event does not impact the outcome of a successive event. Tossing a fair coin or rolling a fair die are often considered independent events. Just because you rolled a 1 does not change the probability the next roll will be a 1.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Sampling with replacement is associated with independent events. Sampling without replacement is associated with dependent events.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">If two events are mutually exclusive, that means they cannot happen at the same time with a single outcome.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>dependent events: <\/strong>if two events are NOT independent, then we say that they are dependent\r\n\r\n<strong>independent events: <\/strong>the occurrence of one event has no effect on the probability of the occurrence of another event. Events [latex]A[\/latex] and [latex]B[\/latex] are independent if one of the following is true:\r\n<ol>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">[latex]P(A|B) = P(A)[\/latex]<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">[latex]P(B|A) = P(B)[\/latex]<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">[latex]P(A \\ \\mathrm{AND} \\ B) = P(A)P(B)[\/latex]<\/li>\r\n<\/ol>\r\n<strong>mutually exclusive:\u00a0<\/strong>two events are mutually exclusive if the probability that they both happen at the same time is zero. If events [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, then [latex]P(A \\ \\mathrm{AND} \\ B) = 0[\/latex].\r\n\r\n<strong>sampling with replacement:\u00a0<\/strong>if\u00a0each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once\r\n\r\n<strong>sampling without replacement:\u00a0<\/strong>when sampling is done without replacement, each member of a population may be chosen only once","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">For two events to be independent, the outcome of one event does not impact the outcome of a successive event. Tossing a fair coin or rolling a fair die are often considered independent events. Just because you rolled a 1 does not change the probability the next roll will be a 1.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Sampling with replacement is associated with independent events. Sampling without replacement is associated with dependent events.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">If two events are mutually exclusive, that means they cannot happen at the same time with a single outcome.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>dependent events: <\/strong>if two events are NOT independent, then we say that they are dependent<\/p>\n<p><strong>independent events: <\/strong>the occurrence of one event has no effect on the probability of the occurrence of another event. Events [latex]A[\/latex] and [latex]B[\/latex] are independent if one of the following is true:<\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">[latex]P(A|B) = P(A)[\/latex]<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">[latex]P(B|A) = P(B)[\/latex]<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">[latex]P(A \\ \\mathrm{AND} \\ B) = P(A)P(B)[\/latex]<\/li>\n<\/ol>\n<p><strong>mutually exclusive:\u00a0<\/strong>two events are mutually exclusive if the probability that they both happen at the same time is zero. If events [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, then [latex]P(A \\ \\mathrm{AND} \\ B) = 0[\/latex].<\/p>\n<p><strong>sampling with replacement:\u00a0<\/strong>if\u00a0each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once<\/p>\n<p><strong>sampling without replacement:\u00a0<\/strong>when sampling is done without replacement, each member of a population may be chosen only once<\/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-1153\">\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\/3-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/3-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":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/3-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-1153","chapter","type-chapter","status-publish","hentry"],"part":43,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1153","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":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1153\/revisions"}],"predecessor-version":[{"id":3456,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1153\/revisions\/3456"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/43"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1153\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1153"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1153"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1153"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}