{"id":700,"date":"2025-07-14T21:13:54","date_gmt":"2025-07-14T21:13:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=700"},"modified":"2026-01-15T20:59:09","modified_gmt":"2026-01-15T20:59:09","slug":"module-12-probability-and-counting-theory-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-12-probability-and-counting-theory-cheat-sheet\/","title":{"raw":"Probability and Counting Theory: Cheat Sheet","rendered":"Probability and Counting Theory: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Counting Principles<\/h3>\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Addition Principle<\/li>\r\n<\/ul>\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">If one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways, and both events cannot occur simultaneously, then there are [latex]A + B[\/latex] total ways for the first event OR the second event to occur<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiplication Principle (Fundamental Counting Principle)\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li>If one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways after the first event has occurred, then the two events can occur in [latex]A \\cdot B[\/latex] ways<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h4 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Permutations<\/h4>\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The number of ways to arrange [latex]n[\/latex] distinct objects in a specific order is [latex]n![\/latex] (n factorial)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Formula for selecting [latex]r[\/latex] objects from [latex]n[\/latex] distinct objects in order: [latex]_nP_r=P(n,r) = \\frac{n!}{(n-r)!}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Permutations of Non-Distinct Objects<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">When some objects are identical, many arrangements are duplicates<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]\\frac{n!}{r_1! \\cdot r_2! \\cdot \\ldots \\cdot r_k!}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]n[\/latex] is the total number of objects<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]r_1, r_2, \\ldots, r_k[\/latex] represent the number of each type of identical object<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Divide by factorials of repeated items to eliminate duplicate arrangements<\/li>\r\n<\/ul>\r\n<h4 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Combinations<\/h4>\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Formula for selecting [latex]r[\/latex] objects from [latex]n[\/latex] distinct objects: [latex]_nC_r=\\binom{n}{r}=C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Use when order doesn't matter (e.g., selecting committee members, choosing pizza toppings)<\/li>\r\n<\/ul>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Probability<\/h3>\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Probability is a number between [latex]0[\/latex] and [latex]1[\/latex] (or [latex]0%[\/latex] and [latex]100%[\/latex]) describing the likelihood of an event<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The sample space [latex]S[\/latex] is the set of all possible outcomes<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">An event [latex]E[\/latex] is any subset of the sample space<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">A probability model lists all possible outcomes and their associated probabilities<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The sum of all probabilities in a probability model must equal [latex]1[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Probability with Equally Likely Outcomes<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]P(E) = \\frac{\\text{number of elements in } E}{\\text{number of elements in } S} = \\frac{n(E)}{n(S)}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Union of Two Events<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The union [latex]E \\cup F[\/latex] represents the event that [latex]E[\/latex] OR [latex]F[\/latex] occurs (or both)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]P(E \\cup F) = P(E) + P(F) - P(E \\cap F)[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Mutually Exclusive Events<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Events that cannot occur at the same time (no outcomes in common)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">For mutually exclusive events: [latex]P(E \\cap F) = 0[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Complement of an Event<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The complement [latex]E'[\/latex] is the set of all outcomes in the sample space that are not in [latex]E[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]P(E') = 1 - P(E)[\/latex]<\/li>\r\n<\/ul>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Binomial Theorem Applications<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding a Single Term in a Binomial Expansion<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The [latex]r+1[\/latex]th term of [latex](x+y)^n[\/latex] is: [latex]\\binom{n}{r}x^{n-r}y^r[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">To find a specific term without fully expanding:\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Identify [latex]n[\/latex] (the exponent)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Determine which term you want: [latex]r+1[\/latex]th term<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solve for [latex]r[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Substitute into the formula<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The binomial coefficient [latex]\\binom{n}{r}[\/latex] is the same as [latex]C(n,r)[\/latex]<\/li>\r\n<\/ul>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\r\n<table style=\"width: 93.3421%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>Permutations of [latex]n[\/latex] distinct objects taken [latex]r[\/latex] at a time<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]P(n,r) = \\frac{n!}{(n-r)!}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>Permutations of [latex]n[\/latex] non-distinct objects<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]\\frac{n!}{r_1! \\cdot r_2! \\cdot \\ldots \\cdot r_k!}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>Combinations of [latex]n[\/latex] distinct objects taken [latex]r[\/latex] at a time<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>Probability of the union of two events<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]P(E \\cup F) = P(E) + P(F) - P(E \\cap F)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>Probability of the union of mutually exclusive events<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]P(E \\cup F) = P(E) + P(F)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>Probability of the complement<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]P(E') = 1 - P(E)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.7946%;\"><strong>The [latex](r+1)[\/latex]th term of a binomial expansion<\/strong><\/td>\r\n<td style=\"width: 58.9056%;\">[latex]\\binom{n}{r}x^{n-r}y^r[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Glossary<\/h2>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Addition Principle<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">States that if one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways, and both events cannot occur at the same time, then there are [latex]A + B[\/latex] ways for the first event OR the second event to occur.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>combination<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A selection of objects where the order does not matter; the number of ways to choose objects without regard to arrangement.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>complement of an event<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The complement [latex]E'[\/latex] is the set of all outcomes in the sample space that are not in event [latex]E[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>event<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Any subset of the sample space; a collection of outcomes from an experiment.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>experiment<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An activity with an observable result.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>intersection of two events<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The event [latex]E \\cap F[\/latex] that occurs if both event [latex]E[\/latex] and event [latex]F[\/latex] occur simultaneously. Mathematically represents \"and.\"<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Multiplication Principle<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">States that if one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways after the first event has occurred, then the two events can occur in [latex]A \\cdot B[\/latex] ways. Also known as the Fundamental Counting Principle.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>mutually exclusive events<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Events that have no outcomes in common; events that cannot occur at the same time.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>permutation<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An ordering of objects where the order matters; the number of ways to arrange objects in a specific sequence.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>probability<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A number between [latex]0[\/latex] and [latex]1[\/latex] (or [latex]0%[\/latex] and [latex]100%[\/latex]) that describes the likelihood that an event will occur.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>probability model<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A mathematical description of an experiment listing all possible outcomes and their associated probabilities. The sum of all probabilities must equal [latex]1[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sample space<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The set of all possible outcomes of an experiment, denoted [latex]S[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>union of two events<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The event [latex]E \\cup F[\/latex] that occurs if either event [latex]E[\/latex] or event [latex]F[\/latex] (or both) occurs. Mathematically represents \"or.\"<\/p>","rendered":"<h2>Essential Concepts<\/h2>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Counting Principles<\/h3>\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Addition Principle<\/li>\n<\/ul>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li style=\"list-style-type: none;\">\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">If one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways, and both events cannot occur simultaneously, then there are [latex]A + B[\/latex] total ways for the first event OR the second event to occur<\/li>\n<\/ul>\n<\/li>\n<li>Multiplication Principle (Fundamental Counting Principle)\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li>If one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways after the first event has occurred, then the two events can occur in [latex]A \\cdot B[\/latex] ways<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h4 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Permutations<\/h4>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The number of ways to arrange [latex]n[\/latex] distinct objects in a specific order is [latex]n![\/latex] (n factorial)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Formula for selecting [latex]r[\/latex] objects from [latex]n[\/latex] distinct objects in order: [latex]_nP_r=P(n,r) = \\frac{n!}{(n-r)!}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Permutations of Non-Distinct Objects<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">When some objects are identical, many arrangements are duplicates<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]\\frac{n!}{r_1! \\cdot r_2! \\cdot \\ldots \\cdot r_k!}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]n[\/latex] is the total number of objects<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]r_1, r_2, \\ldots, r_k[\/latex] represent the number of each type of identical object<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Divide by factorials of repeated items to eliminate duplicate arrangements<\/li>\n<\/ul>\n<h4 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Combinations<\/h4>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Formula for selecting [latex]r[\/latex] objects from [latex]n[\/latex] distinct objects: [latex]_nC_r=\\binom{n}{r}=C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Use when order doesn&#8217;t matter (e.g., selecting committee members, choosing pizza toppings)<\/li>\n<\/ul>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Probability<\/h3>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Probability is a number between [latex]0[\/latex] and [latex]1[\/latex] (or [latex]0%[\/latex] and [latex]100%[\/latex]) describing the likelihood of an event<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The sample space [latex]S[\/latex] is the set of all possible outcomes<\/li>\n<li class=\"whitespace-normal break-words pl-2\">An event [latex]E[\/latex] is any subset of the sample space<\/li>\n<li class=\"whitespace-normal break-words pl-2\">A probability model lists all possible outcomes and their associated probabilities<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The sum of all probabilities in a probability model must equal [latex]1[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Probability with Equally Likely Outcomes<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]P(E) = \\frac{\\text{number of elements in } E}{\\text{number of elements in } S} = \\frac{n(E)}{n(S)}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Union of Two Events<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The union [latex]E \\cup F[\/latex] represents the event that [latex]E[\/latex] OR [latex]F[\/latex] occurs (or both)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]P(E \\cup F) = P(E) + P(F) - P(E \\cap F)[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Mutually Exclusive Events<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Events that cannot occur at the same time (no outcomes in common)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">For mutually exclusive events: [latex]P(E \\cap F) = 0[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Complement of an Event<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The complement [latex]E'[\/latex] is the set of all outcomes in the sample space that are not in [latex]E[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Formula: [latex]P(E') = 1 - P(E)[\/latex]<\/li>\n<\/ul>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Binomial Theorem Applications<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding a Single Term in a Binomial Expansion<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The [latex]r+1[\/latex]th term of [latex](x+y)^n[\/latex] is: [latex]\\binom{n}{r}x^{n-r}y^r[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">To find a specific term without fully expanding:\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Identify [latex]n[\/latex] (the exponent)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Determine which term you want: [latex]r+1[\/latex]th term<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Solve for [latex]r[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Substitute into the formula<\/li>\n<\/ol>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The binomial coefficient [latex]\\binom{n}{r}[\/latex] is the same as [latex]C(n,r)[\/latex]<\/li>\n<\/ul>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\n<table style=\"width: 93.3421%;\">\n<tbody>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>Permutations of [latex]n[\/latex] distinct objects taken [latex]r[\/latex] at a time<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]P(n,r) = \\frac{n!}{(n-r)!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>Permutations of [latex]n[\/latex] non-distinct objects<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]\\frac{n!}{r_1! \\cdot r_2! \\cdot \\ldots \\cdot r_k!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>Combinations of [latex]n[\/latex] distinct objects taken [latex]r[\/latex] at a time<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>Probability of the union of two events<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]P(E \\cup F) = P(E) + P(F) - P(E \\cap F)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>Probability of the union of mutually exclusive events<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]P(E \\cup F) = P(E) + P(F)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>Probability of the complement<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]P(E') = 1 - P(E)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.7946%;\"><strong>The [latex](r+1)[\/latex]th term of a binomial expansion<\/strong><\/td>\n<td style=\"width: 58.9056%;\">[latex]\\binom{n}{r}x^{n-r}y^r[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Glossary<\/h2>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Addition Principle<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">States that if one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways, and both events cannot occur at the same time, then there are [latex]A + B[\/latex] ways for the first event OR the second event to occur.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>combination<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A selection of objects where the order does not matter; the number of ways to choose objects without regard to arrangement.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>complement of an event<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The complement [latex]E'[\/latex] is the set of all outcomes in the sample space that are not in event [latex]E[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>event<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Any subset of the sample space; a collection of outcomes from an experiment.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>experiment<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An activity with an observable result.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>intersection of two events<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The event [latex]E \\cap F[\/latex] that occurs if both event [latex]E[\/latex] and event [latex]F[\/latex] occur simultaneously. Mathematically represents &#8220;and.&#8221;<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Multiplication Principle<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">States that if one event can occur in [latex]A[\/latex] ways and a second event can occur in [latex]B[\/latex] ways after the first event has occurred, then the two events can occur in [latex]A \\cdot B[\/latex] ways. Also known as the Fundamental Counting Principle.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>mutually exclusive events<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Events that have no outcomes in common; events that cannot occur at the same time.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>permutation<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An ordering of objects where the order matters; the number of ways to arrange objects in a specific sequence.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>probability<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A number between [latex]0[\/latex] and [latex]1[\/latex] (or [latex]0%[\/latex] and [latex]100%[\/latex]) that describes the likelihood that an event will occur.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>probability model<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A mathematical description of an experiment listing all possible outcomes and their associated probabilities. The sum of all probabilities must equal [latex]1[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sample space<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The set of all possible outcomes of an experiment, denoted [latex]S[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>union of two events<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The event [latex]E \\cup F[\/latex] that occurs if either event [latex]E[\/latex] or event [latex]F[\/latex] (or both) occurs. Mathematically represents &#8220;or.&#8221;<\/p>\n","protected":false},"author":67,"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":513,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/700"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/700\/revisions"}],"predecessor-version":[{"id":5397,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/700\/revisions\/5397"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/700\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=700"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=700"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=700"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=700"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}