{"id":3216,"date":"2025-08-15T23:27:40","date_gmt":"2025-08-15T23:27:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3216"},"modified":"2025-10-07T21:53:51","modified_gmt":"2025-10-07T21:53:51","slug":"probability-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-apply-it\/","title":{"raw":"Probability: Apply It","rendered":"Probability: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Compute probabilities of equally likely outcomes.<\/li>\r\n \t<li>Compute probabilities of the union of two events.<\/li>\r\n \t<li>Use the complement rule to find probabilities.<\/li>\r\n \t<li>Compute probability using counting theory.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\r\n<div data-test-render-count=\"1\">\r\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\r\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Fantasy Quest Dice Game<\/h2>\r\n<p class=\"whitespace-normal break-words\">You're playing a fantasy quest game where players roll two six-sided dice to determine the success of their character's actions. Different total values on the dice lead to different outcomes in the game.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Game Rules:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Rolling a sum of 7 or 11 means <strong>Critical Success<\/strong> (your character performs an amazing feat)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rolling a sum of 2, 3, or 4 means <strong>Failure<\/strong> (your action doesn't work)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rolling a sum of 10, 11, or 12 means <strong>High Roll<\/strong> (you get bonus points)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Any other sum means a <strong>Standard Success<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">What is the probability of rolling a sum of 7 on two six-sided dice?<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]First, identify the sample space. When rolling two dice, there are [latex]6 \\times 6 = 36[\/latex] equally likely outcomes. Next, count the outcomes where the sum equals 7: (1,6) (2,5), (3,4), (4,3), (5,2), (6,1)[\/latex] There are 6 ways to roll a sum of 7. [latex]P(\\text{sum of 7}) = \\dfrac{\\text{number of favorable outcomes}}{\\text{total number of outcomes}} = \\dfrac{6}{36} = \\dfrac{1}{6}[\/latex] The probability of rolling a sum of 7 is [latex]\\dfrac{1}{6}[\/latex] or approximately 16.67%.[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313353[\/ohm_question]<\/section>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313354[\/ohm_question]<\/section><section class=\"textbox recall\" aria-label=\"Recall\">The Complement Rule states that [latex]P(E') = 1 - P(E)[\/latex], where [latex]E'[\/latex] represents all outcomes not in event [latex]E[\/latex]. The sum of all probabilities in a sample space must equal 1.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">In an advanced version of the game, you draw 3 cards from a special deck containing 5 Spell cards, 4 Weapon cards, and 3 Shield cards. What is the probability that you draw exactly 2 Spell cards and 1 Weapon card?<\/p>\r\n[reveal-answer q=\"571087\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"571087\"]\r\n\r\nWe need to count the number of ways to choose exactly 2 Spell cards and 1 Weapon card.\r\n<ul>\r\n \t<li>There are [latex]5[\/latex] Spell cards, so there are [latex]C(5,2)[\/latex] ways to choose 2 Spell cards.<\/li>\r\n \t<li>There are [latex]4[\/latex] Weapon cards, so there are [latex]C(4,1)[\/latex] ways to choose 1 Weapon card.<\/li>\r\n<\/ul>\r\nSince we're choosing both at the same time, we use the Multiplication Principle:\r\n\r\n[latex]C(5,2) \\cdot C(4,1) = 10 \\cdot 4 = 40[\/latex] ways.\r\n\r\nNow to find the probability we need to know the total number of outcomes for the denominator. The total number of cards is [latex]5 + 4 + 3 = 12[\/latex], so there are [latex]C(12,3)[\/latex] ways to choose any 3 cards:\r\n\r\n[latex]C(12,3) = 220[\/latex]\r\n\r\n[latex]P(\\text{2 Spells and 1 Weapon}) = \\dfrac{40}{220} = \\dfrac{2}{11}[\/latex].\r\n\r\nThe probability of drawing exactly 2 Spell cards and 1 Weapon card is [latex]\\dfrac{2}{11}[\/latex] or approximately 18.18%.[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Compute probabilities of equally likely outcomes.<\/li>\n<li>Compute probabilities of the union of two events.<\/li>\n<li>Use the complement rule to find probabilities.<\/li>\n<li>Compute probability using counting theory.<\/li>\n<\/ul>\n<\/section>\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\n<div data-test-render-count=\"1\">\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Fantasy Quest Dice Game<\/h2>\n<p class=\"whitespace-normal break-words\">You&#8217;re playing a fantasy quest game where players roll two six-sided dice to determine the success of their character&#8217;s actions. Different total values on the dice lead to different outcomes in the game.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Game Rules:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Rolling a sum of 7 or 11 means <strong>Critical Success<\/strong> (your character performs an amazing feat)<\/li>\n<li class=\"whitespace-normal break-words\">Rolling a sum of 2, 3, or 4 means <strong>Failure<\/strong> (your action doesn&#8217;t work)<\/li>\n<li class=\"whitespace-normal break-words\">Rolling a sum of 10, 11, or 12 means <strong>High Roll<\/strong> (you get bonus points)<\/li>\n<li class=\"whitespace-normal break-words\">Any other sum means a <strong>Standard Success<\/strong><\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">What is the probability of rolling a sum of 7 on two six-sided dice?<\/p>\n<p class=\"whitespace-normal break-words\">[latex]First, identify the sample space. When rolling two dice, there are [latex]6 \\times 6 = 36[\/latex] equally likely outcomes. Next, count the outcomes where the sum equals 7: (1,6) (2,5), (3,4), (4,3), (5,2), (6,1)[\/latex] There are 6 ways to roll a sum of 7. [latex]P(\\text{sum of 7}) = \\dfrac{\\text{number of favorable outcomes}}{\\text{total number of outcomes}} = \\dfrac{6}{36} = \\dfrac{1}{6}[\/latex] The probability of rolling a sum of 7 is [latex]\\dfrac{1}{6}[\/latex] or approximately 16.67%.[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313353\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313353&theme=lumen&iframe_resize_id=ohm313353&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313354\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313354&theme=lumen&iframe_resize_id=ohm313354&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">The Complement Rule states that [latex]P(E') = 1 - P(E)[\/latex], where [latex]E'[\/latex] represents all outcomes not in event [latex]E[\/latex]. The sum of all probabilities in a sample space must equal 1.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">In an advanced version of the game, you draw 3 cards from a special deck containing 5 Spell cards, 4 Weapon cards, and 3 Shield cards. What is the probability that you draw exactly 2 Spell cards and 1 Weapon card?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q571087\">Show Solution<\/button><\/p>\n<div id=\"q571087\" class=\"hidden-answer\" style=\"display: none\">\n<p>We need to count the number of ways to choose exactly 2 Spell cards and 1 Weapon card.<\/p>\n<ul>\n<li>There are [latex]5[\/latex] Spell cards, so there are [latex]C(5,2)[\/latex] ways to choose 2 Spell cards.<\/li>\n<li>There are [latex]4[\/latex] Weapon cards, so there are [latex]C(4,1)[\/latex] ways to choose 1 Weapon card.<\/li>\n<\/ul>\n<p>Since we&#8217;re choosing both at the same time, we use the Multiplication Principle:<\/p>\n<p>[latex]C(5,2) \\cdot C(4,1) = 10 \\cdot 4 = 40[\/latex] ways.<\/p>\n<p>Now to find the probability we need to know the total number of outcomes for the denominator. The total number of cards is [latex]5 + 4 + 3 = 12[\/latex], so there are [latex]C(12,3)[\/latex] ways to choose any 3 cards:<\/p>\n<p>[latex]C(12,3) = 220[\/latex]<\/p>\n<p>[latex]P(\\text{2 Spells and 1 Weapon}) = \\dfrac{40}{220} = \\dfrac{2}{11}[\/latex].<\/p>\n<p>The probability of drawing exactly 2 Spell cards and 1 Weapon card is [latex]\\dfrac{2}{11}[\/latex] or approximately 18.18%.<\/p><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":24,"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":"apply_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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3216"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3216\/revisions"}],"predecessor-version":[{"id":4546,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3216\/revisions\/4546"}],"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\/3216\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3216"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3216"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3216"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3216"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}