{"id":3463,"date":"2023-10-11T17:26:12","date_gmt":"2023-10-11T17:26:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3463"},"modified":"2025-02-11T04:36:39","modified_gmt":"2025-02-11T04:36:39","slug":"additional-concepts-in-probability-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/additional-concepts-in-probability-cheat-sheet\/","title":{"raw":"Additional Concepts in Probability: Cheat Sheet","rendered":"Additional Concepts in Probability: Cheat Sheet"},"content":{"raw":"<h4 style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+18_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a pdf of this page here.<\/a><\/h4>\r\n<h2>Essential Concepts<\/h2>\r\n<ul>\r\n\t<li>A chance experiment involves making observations in situations where there is uncertainty about which of two or more possible outcomes will result.<\/li>\r\n\t<li>The probability of an event is a numeric measure of how likely the event is to happen.<\/li>\r\n\t<li>A probability is always a number from [latex]0[\/latex] to [latex]1[\/latex], inclusive (which means that [latex]0[\/latex] and [latex]1[\/latex] are included). Probability may be written as a percentage, from [latex]0\\%[\/latex] to [latex]100\\%[\/latex], inclusive.<\/li>\r\n\t<li>\r\n<div>A set is a collection of distinct objects, called elements of the set. The elements of a set can be either numbers or objects that have a particular characteristic in common. For example, we can define the set [latex] A = \\{...\\} [\/latex] where the elements of the set are listed inside the curly bracket.<\/div>\r\n<\/li>\r\n\t<li>A Venn diagram is a visual representation that illustrates the outcomes of a chance experiment.\u00a0A Venn diagram represents each set by a circle\/oval, usually drawn inside of a containing box representing the sample space. Overlapping areas indicate elements common to both sets.<\/li>\r\n\t<li>The complement of event [latex]A[\/latex] is denoted [latex]A'[\/latex] (read \"A prime\") or [latex]A^c[\/latex] (read \"A complement\"). The complement of an event [latex]A[\/latex] consists of all outcomes that are NOT in [latex]A[\/latex].<\/li>\r\n\t<li>The intersection of two events, [latex]A[\/latex] and [latex]B[\/latex], is denoted by [latex]A \\cap B[\/latex]. [latex]A[\/latex] and<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> [latex]B[\/latex] ([latex]A \\cap B[\/latex]) means that the outcomes are in\u00a0<strong>both<\/strong> event [latex]A[\/latex] and even [latex]B[\/latex].<\/span><\/li>\r\n\t<li>The union of two events, [latex]A[\/latex] or [latex]B[\/latex], is denoted by [latex]A \\cup B[\/latex]. The union of two events consists of the set of all elements in the collection of both events. The outcomes in the event [latex]A[\/latex] or [latex]B[\/latex] are the outcomes that are in event [latex]A[\/latex], in event [latex]B[\/latex], or in both event [latex]A[\/latex] and event [latex]B[\/latex]. In term of probability, [latex]P(A \\text{ or }B)= P(A \\cup B)=[\/latex] the relative frequency of either event [latex]A[\/latex] or [latex]B[\/latex] (or both) with respect to the sample space.<\/li>\r\n\t<li>Mutually exclusive<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> describes two or more events that cannot happen simultaneously. <\/span>If event [latex]A[\/latex] and event [latex]B[\/latex] are mutually exclusive, then [latex]A \\cap B[\/latex] is an empty set and [latex]P(A \\cap B) = 0[\/latex].<\/li>\r\n\t<li>The conditional probability\u00a0of [latex]A[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>given [latex]B[\/latex], denoted as [latex]P(A\\text{ given }B) = P(A|B)[\/latex], represents the probability of event [latex]A[\/latex] occurring given that event [latex]B[\/latex] has occurred.\u00a0<\/li>\r\n\t<li>A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of \u201cbranches\u201d that are labeled with either frequencies or probabilities. In a tree diagram, probabilities are multiplied along the branches to find the probability at the end of a branch due to the Multiplication Rule of Probability.<\/li>\r\n\t<li>Bayes' Theorem is a fundamental concept in probability theory that allows us to update our beliefs about an event when new evidence or information becomes available.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>bayes' theorem<\/strong><\/p>\r\n<p>[latex]P(A|B) = \\dfrac{P(A)\\times P(B|A)}{P(B)}= \\dfrac{P(A)\\times P(B|A)}{P(A)\\times P(B|A) + P(A')\\times P(B|A')}[\/latex]<\/p>\r\n<p>or<\/p>\r\n<p>[latex]P(A|B) = \\dfrac{P(\\text{the path of events A and B})}{\\text{ sum of } P(\\text{paths containing event B})}[\/latex].<\/p>\r\n<p><strong>probability of an event<\/strong><\/p>\r\n<p>[latex]P(\\text{event}) = \\dfrac{\\text{number of outcomes in event}}{\\text{number of all possible outcomes}}[\/latex]<\/p>\r\n<p><strong>conditional probability<\/strong><\/p>\r\n<p>[latex] P(A|B) = \\dfrac{P(A \\text{ and } B)}{P(B)} =\\dfrac{P(A \\cap B)}{P(B)}[\/latex]<\/p>\r\n<p><strong>complement of an event<\/strong><\/p>\r\n<p style=\"text-align: left;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex](not [latex]A[\/latex]) = [latex]1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">or<\/p>\r\n<p style=\"text-align: left;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex]([latex]A'[\/latex]) = [latex]1[\/latex]<\/p>\r\n<p><strong>independent events<\/strong><\/p>\r\n<p>[latex]P(A \\text{ and }B) = P(A) \\times P(B) [\/latex]<\/p>\r\n<p><strong>mutually exclusive events<\/strong><\/p>\r\n<p>[latex]P(A \\text{ or } B) = P(A \\cup B) = P(A)+P(B)[\/latex]<\/p>\r\n<h2>Glossary<\/h2>\r\n<p><strong>bayes' theorem<\/strong><\/p>\r\n<p>a fundamental concept in probability theory that allows us to update our beliefs about an event when new evidence or information becomes available<\/p>\r\n<p><strong>chance experiment<\/strong><\/p>\r\n<p>making observations in situations where there is uncertainty about which of two or more possible outcomes will result<\/p>\r\n<p><strong>complement<\/strong><\/p>\r\n<p>[latex]A'[\/latex] or [latex]A^c[\/latex], consists of all outcomes that are not\u00a0in [latex]A[\/latex]<\/p>\r\n<p><strong>conditional probability<\/strong><\/p>\r\n<p>calculated based on the assumption that one event has already occurred<\/p>\r\n<p><strong>event<\/strong><\/p>\r\n<p>an outcome or collection of outcomes for a chance experiment<\/p>\r\n<p><strong>intersection<\/strong><\/p>\r\n<p>all elements in event [latex]A[\/latex] also belong in event [latex]B[\/latex]<\/p>\r\n<p><strong>mutually exclusive<\/strong><\/p>\r\n<p>two or more events that cannot happen simultaneously<\/p>\r\n<p><strong>[latex]P(\\text{event})[\/latex]<\/strong><\/p>\r\n<p>probability of an event<\/p>\r\n<p><b>probability<\/b><\/p>\r\n<p>a numeric measure (the number of outcomes in the event divided by the number of possible outcomes in the sample space) of how likely the event is to happen<\/p>\r\n<p><strong>sample space<\/strong><\/p>\r\n<p>the list of all possible outcomes of a chance experiment<\/p>\r\n<p><strong>set<\/strong><\/p>\r\n<p>a collection of distinct objects, called elements of the set<\/p>\r\n<p><strong>tree diagram<\/strong><\/p>\r\n<p>a special type of graph used to determine the outcomes of an experiment<\/p>\r\n<p><strong>venn diagram<\/strong><\/p>\r\n<p>a diagram that represents each set by a circle\/oval, usually drawn inside of a containing box representing the sample space<\/p>\r\n<p><strong>union<\/strong><\/p>\r\n<p>the set of all elements in the collection of both events<\/p>","rendered":"<h4 style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+18_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a pdf of this page here.<\/a><\/h4>\n<h2>Essential Concepts<\/h2>\n<ul>\n<li>A chance experiment involves making observations in situations where there is uncertainty about which of two or more possible outcomes will result.<\/li>\n<li>The probability of an event is a numeric measure of how likely the event is to happen.<\/li>\n<li>A probability is always a number from [latex]0[\/latex] to [latex]1[\/latex], inclusive (which means that [latex]0[\/latex] and [latex]1[\/latex] are included). Probability may be written as a percentage, from [latex]0\\%[\/latex] to [latex]100\\%[\/latex], inclusive.<\/li>\n<li>\n<div>A set is a collection of distinct objects, called elements of the set. The elements of a set can be either numbers or objects that have a particular characteristic in common. For example, we can define the set [latex]A = \\{...\\}[\/latex] where the elements of the set are listed inside the curly bracket.<\/div>\n<\/li>\n<li>A Venn diagram is a visual representation that illustrates the outcomes of a chance experiment.\u00a0A Venn diagram represents each set by a circle\/oval, usually drawn inside of a containing box representing the sample space. Overlapping areas indicate elements common to both sets.<\/li>\n<li>The complement of event [latex]A[\/latex] is denoted [latex]A'[\/latex] (read &#8220;A prime&#8221;) or [latex]A^c[\/latex] (read &#8220;A complement&#8221;). The complement of an event [latex]A[\/latex] consists of all outcomes that are NOT in [latex]A[\/latex].<\/li>\n<li>The intersection of two events, [latex]A[\/latex] and [latex]B[\/latex], is denoted by [latex]A \\cap B[\/latex]. [latex]A[\/latex] and<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> [latex]B[\/latex] ([latex]A \\cap B[\/latex]) means that the outcomes are in\u00a0<strong>both<\/strong> event [latex]A[\/latex] and even [latex]B[\/latex].<\/span><\/li>\n<li>The union of two events, [latex]A[\/latex] or [latex]B[\/latex], is denoted by [latex]A \\cup B[\/latex]. The union of two events consists of the set of all elements in the collection of both events. The outcomes in the event [latex]A[\/latex] or [latex]B[\/latex] are the outcomes that are in event [latex]A[\/latex], in event [latex]B[\/latex], or in both event [latex]A[\/latex] and event [latex]B[\/latex]. In term of probability, [latex]P(A \\text{ or }B)= P(A \\cup B)=[\/latex] the relative frequency of either event [latex]A[\/latex] or [latex]B[\/latex] (or both) with respect to the sample space.<\/li>\n<li>Mutually exclusive<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> describes two or more events that cannot happen simultaneously. <\/span>If event [latex]A[\/latex] and event [latex]B[\/latex] are mutually exclusive, then [latex]A \\cap B[\/latex] is an empty set and [latex]P(A \\cap B) = 0[\/latex].<\/li>\n<li>The conditional probability\u00a0of [latex]A[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>given [latex]B[\/latex], denoted as [latex]P(A\\text{ given }B) = P(A|B)[\/latex], represents the probability of event [latex]A[\/latex] occurring given that event [latex]B[\/latex] has occurred.\u00a0<\/li>\n<li>A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of \u201cbranches\u201d that are labeled with either frequencies or probabilities. In a tree diagram, probabilities are multiplied along the branches to find the probability at the end of a branch due to the Multiplication Rule of Probability.<\/li>\n<li>Bayes&#8217; Theorem is a fundamental concept in probability theory that allows us to update our beliefs about an event when new evidence or information becomes available.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<p><strong>bayes&#8217; theorem<\/strong><\/p>\n<p>[latex]P(A|B) = \\dfrac{P(A)\\times P(B|A)}{P(B)}= \\dfrac{P(A)\\times P(B|A)}{P(A)\\times P(B|A) + P(A')\\times P(B|A')}[\/latex]<\/p>\n<p>or<\/p>\n<p>[latex]P(A|B) = \\dfrac{P(\\text{the path of events A and B})}{\\text{ sum of } P(\\text{paths containing event B})}[\/latex].<\/p>\n<p><strong>probability of an event<\/strong><\/p>\n<p>[latex]P(\\text{event}) = \\dfrac{\\text{number of outcomes in event}}{\\text{number of all possible outcomes}}[\/latex]<\/p>\n<p><strong>conditional probability<\/strong><\/p>\n<p>[latex]P(A|B) = \\dfrac{P(A \\text{ and } B)}{P(B)} =\\dfrac{P(A \\cap B)}{P(B)}[\/latex]<\/p>\n<p><strong>complement of an event<\/strong><\/p>\n<p style=\"text-align: left;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex](not [latex]A[\/latex]) = [latex]1[\/latex]<\/p>\n<p style=\"text-align: left;\">or<\/p>\n<p style=\"text-align: left;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex]([latex]A'[\/latex]) = [latex]1[\/latex]<\/p>\n<p><strong>independent events<\/strong><\/p>\n<p>[latex]P(A \\text{ and }B) = P(A) \\times P(B)[\/latex]<\/p>\n<p><strong>mutually exclusive events<\/strong><\/p>\n<p>[latex]P(A \\text{ or } B) = P(A \\cup B) = P(A)+P(B)[\/latex]<\/p>\n<h2>Glossary<\/h2>\n<p><strong>bayes&#8217; theorem<\/strong><\/p>\n<p>a fundamental concept in probability theory that allows us to update our beliefs about an event when new evidence or information becomes available<\/p>\n<p><strong>chance experiment<\/strong><\/p>\n<p>making observations in situations where there is uncertainty about which of two or more possible outcomes will result<\/p>\n<p><strong>complement<\/strong><\/p>\n<p>[latex]A'[\/latex] or [latex]A^c[\/latex], consists of all outcomes that are not\u00a0in [latex]A[\/latex]<\/p>\n<p><strong>conditional probability<\/strong><\/p>\n<p>calculated based on the assumption that one event has already occurred<\/p>\n<p><strong>event<\/strong><\/p>\n<p>an outcome or collection of outcomes for a chance experiment<\/p>\n<p><strong>intersection<\/strong><\/p>\n<p>all elements in event [latex]A[\/latex] also belong in event [latex]B[\/latex]<\/p>\n<p><strong>mutually exclusive<\/strong><\/p>\n<p>two or more events that cannot happen simultaneously<\/p>\n<p><strong>[latex]P(\\text{event})[\/latex]<\/strong><\/p>\n<p>probability of an event<\/p>\n<p><b>probability<\/b><\/p>\n<p>a numeric measure (the number of outcomes in the event divided by the number of possible outcomes in the sample space) of how likely the event is to happen<\/p>\n<p><strong>sample space<\/strong><\/p>\n<p>the list of all possible outcomes of a chance experiment<\/p>\n<p><strong>set<\/strong><\/p>\n<p>a collection of distinct objects, called elements of the set<\/p>\n<p><strong>tree diagram<\/strong><\/p>\n<p>a special type of graph used to determine the outcomes of an experiment<\/p>\n<p><strong>venn diagram<\/strong><\/p>\n<p>a diagram that represents each set by a circle\/oval, usually drawn inside of a containing box representing the sample space<\/p>\n<p><strong>union<\/strong><\/p>\n<p>the set of all elements in the collection of both events<\/p>\n","protected":false},"author":12,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2910,"module-header":"cheat_sheet","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\/3463"}],"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\/12"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3463\/revisions"}],"predecessor-version":[{"id":6274,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3463\/revisions\/6274"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2910"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3463\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3463"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3463"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3463"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}