{"id":222,"date":"2023-12-07T14:07:19","date_gmt":"2023-12-07T19:07:19","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=222"},"modified":"2024-06-12T14:41:39","modified_gmt":"2024-06-12T18:41:39","slug":"binomial-distribution","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/binomial-distribution\/","title":{"raw":"Binomial Properties &amp; Calculating the Probability of 'X' Successes","rendered":"Binomial Properties &amp; Calculating the Probability of &#8216;X&#8217; Successes"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUnderstand what it means for an experiment to be a Binomial experiment and calculate the probability of [latex]x[\/latex] successes occurring using the <a href=\"https:\/\/www.ncl.ac.uk\/webtemplate\/ask-assets\/external\/maths-resources\/statistics\/distributions\/binomial-distribution.html#:~:text=The%20probability%20mass%20function%20of,%E2%88%92%20p%20)%20n%20%E2%88%92%20x%20.\">Binomial probability mass function<\/a>\u00a0formula.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Three Properties of Binomial Distributions<\/h2>\r\nIn order for an experiment to be considered a binomial distribution, it must satisfy three properties:\r\n<ol>\r\n \t<li>There is a fixed number of trials, each with 2 outcomes.<\/li>\r\n \t<li>The 'trial' outcomes are statistically independent.<\/li>\r\n \t<li>The probability, [latex]p[\/latex], of a 'success' is constant from trial to trial.<\/li>\r\n<\/ol>\r\n<h2>Two Ways of Calculating the Probability of '<em>x<\/em>' Successes<\/h2>\r\nIf we want to calculate the probability of exactly [latex]x[\/latex] successes occurring, there are two ways:\r\n<ol>\r\n \t<li><span style=\"color: #333333\">Using the formula<\/span>: [latex]P(x) = {}_nC_x \\cdot p^{x}\\cdot (1 - p)^{n-x}[\/latex]<\/li>\r\n \t<li><span style=\"color: #333333\">Using Excel<\/span>: = <span style=\"background-color: #ffffff;color: #000000\"><a style=\"background-color: #ffffff;color: #000000\" href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a><\/span>([latex]x[\/latex], [latex]n[\/latex], [latex]p[\/latex], 0)<\/li>\r\n<\/ol>\r\n<h1>two Parameters of Binomial Distributions<\/h1>\r\nThere are only two <a href=\"https:\/\/study.com\/skill\/learn\/calculating-the-paramters-of-a-binomial-distribution-explanation.html#:~:text=How%20to%20Calculate%20the%20Parameters,Compute%20the%20mean%20by%20evaluating\">parameters<\/a> needed to completely 'determine' a binomial distribution:\r\n<ul>\r\n \t<li>[latex]n[\/latex]\u00a0 = the number of trials<\/li>\r\n \t<li>[latex]p[\/latex] = the probability of success for each event\/trial.<\/li>\r\n<\/ul>\r\n<h1>What are trials and successes?<\/h1>\r\n<h2>Trials<\/h2>\r\nA 'trial' can be just about anything:\r\n<ul>\r\n \t<li>The flip of a coin, in which case the 2 outcomes (heads or tails).<\/li>\r\n \t<li>A salesman calling on her clients, and making a sale or not.<\/li>\r\n \t<li>The roll of a die (where a certain number is rolled or not)<\/li>\r\n \t<li>In general, we call the 2 outcomes 'successes' and 'failures'<\/li>\r\n<\/ul>\r\n<h2>Successes<\/h2>\r\nA 'success' can be just about anything:\r\n<ul>\r\n \t<li>Getting heads when flipping a coin<\/li>\r\n \t<li>Rolling a 6 when rolling dice<\/li>\r\n \t<li>Making a sale<\/li>\r\n<\/ul>\r\n<h2>Be Consistent<\/h2>\r\nJust be sure - when talking about trials and successes related to binomial problems:\r\n<ul>\r\n \t<li>Be sure to be consistent<\/li>\r\n \t<li>If you define success as rolling a 6,<\/li>\r\n \t<li>Be sure to use the probability of rolling as 6 as the probability of success.<\/li>\r\n<\/ul>\r\n<h1>Exploring the properties of Binomial distributions (ExeRCISE)<\/h1>\r\nIn this first example we will <span style=\"font-size: 20.5333px\">review the 3 properties of binomial distributions. In the next example, we will calculate a probability related to the example given below.<\/span>\r\n<h2>Example 24.1.1<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: A salesman calls on 10 clients everyday. 30% of all her calls in the past resulted in sales.\r\n\r\n<span style=\"color: #003366\"><strong>Question:<\/strong> <\/span>Is this a Binomial experiment?\r\n\r\n<strong><span style=\"color: #003366\">You Try<\/span><\/strong>: Let us find out by going through all the 3 basic characteristics.\r\n\r\n[h5p id=\"56\"]\r\n\r\n[h5p id=\"57\"]\r\n\r\n[h5p id=\"58\"]\r\n\r\n[h5p id=\"59\"]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: Because all three properties of the binomial distribution are satisfied, this is indeed a binomial distribution.\r\n<h1>Calculating Probabilities Using the Binomial Formula (EXAMPLE)<\/h1>\r\nNow that we know the situation given in the previous example is a binomial experiment, let us revisit this example when practicing using the binomial formula.\r\n<h2>Example 24.1.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup: <\/strong><\/span>Let us suppose the salesman is still calling 10 clients per day and the probability that she will make a sale with each client is 0.3.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What is the probability that 4 of her 10 calls in a day will result in sales?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: If we look at this formula first, we see that there are only 3 variables, [latex]n[\/latex],[latex] p[\/latex], and [latex]x[\/latex]. Recall that [latex]C[\/latex] stands for Combination and has no numeric value. We also know that [latex]n=10[\/latex] and [latex]p=0.3[\/latex] are the parameters, and we want to find the probability that [latex]x=4[\/latex].\r\n\r\n\\[P(x=4) = {}_{10}C_4\\cdot 0.3^4\\cdot (1 - 0.3)^{10-4}={}_{10}C_4\\cdot 0.3^4\\cdot (0.7)^{6}\\]\r\n\r\nLet us work out each of the 3 factors individually:\r\n\r\n\\[{}_{10}C_4=\\frac{10!}{4!(10-4)!}=\\frac{10!}{4!}{6!} =210\\]\r\n\r\n\\[0.3^4 = 0.0081\\]\r\n\r\n\\[0.7^6=0.0117649\\]\r\n\r\nSo:\r\n\r\n\\[P(x=4)= 201\\times 0.0081 \\times 0.117649 = 0.200120949 \\]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is a 20% chance that she will make exactly 4 sales in a day.\r\n<h1>Calculating Probabilities Using the Binomial Formula (EXErcise)<\/h1>\r\nWe can do the same steps as the previous example to find [latex] P(x=0)[\/latex], [latex]P(x=1)[\/latex], [latex]P(x=2)[\/latex], ... , [latex]P(x=10)[\/latex]. The sum of these 11 probabilities must, of course, equal 1.\r\n<h2>Example 24.1.3<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: We will continue with example 24.1 where [latex] n[\/latex]=10 and [latex] p[\/latex]=0.3\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: Can you calculate the probabilities in the exercises below?\r\n\r\n<span style=\"color: #003366\"><strong>You try<\/strong><\/span>: Solve for the probabilities below:\r\n\r\n[h5p id=\"60\"]\r\n\r\n[h5p id=\"61\"]\r\n\r\n[h5p id=\"62\"]\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Binomial Properties &amp; Calculating the Probability of \u2018X\u2019 Successes<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"63\"]\r\n\r\n[h5p id=\"64\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Understand what it means for an experiment to be a Binomial experiment and calculate the probability of [latex]x[\/latex] successes occurring using the <a href=\"https:\/\/www.ncl.ac.uk\/webtemplate\/ask-assets\/external\/maths-resources\/statistics\/distributions\/binomial-distribution.html#:~:text=The%20probability%20mass%20function%20of,%E2%88%92%20p%20)%20n%20%E2%88%92%20x%20.\">Binomial probability mass function<\/a>\u00a0formula.<\/p>\n<\/div>\n<\/div>\n<h2>Three Properties of Binomial Distributions<\/h2>\n<p>In order for an experiment to be considered a binomial distribution, it must satisfy three properties:<\/p>\n<ol>\n<li>There is a fixed number of trials, each with 2 outcomes.<\/li>\n<li>The &#8216;trial&#8217; outcomes are statistically independent.<\/li>\n<li>The probability, [latex]p[\/latex], of a &#8216;success&#8217; is constant from trial to trial.<\/li>\n<\/ol>\n<h2>Two Ways of Calculating the Probability of &#8216;<em>x<\/em>&#8216; Successes<\/h2>\n<p>If we want to calculate the probability of exactly [latex]x[\/latex] successes occurring, there are two ways:<\/p>\n<ol>\n<li><span style=\"color: #333333\">Using the formula<\/span>: [latex]P(x) = {}_nC_x \\cdot p^{x}\\cdot (1 - p)^{n-x}[\/latex]<\/li>\n<li><span style=\"color: #333333\">Using Excel<\/span>: = <span style=\"background-color: #ffffff;color: #000000\"><a style=\"background-color: #ffffff;color: #000000\" href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a><\/span>([latex]x[\/latex], [latex]n[\/latex], [latex]p[\/latex], 0)<\/li>\n<\/ol>\n<h1>two Parameters of Binomial Distributions<\/h1>\n<p>There are only two <a href=\"https:\/\/study.com\/skill\/learn\/calculating-the-paramters-of-a-binomial-distribution-explanation.html#:~:text=How%20to%20Calculate%20the%20Parameters,Compute%20the%20mean%20by%20evaluating\">parameters<\/a> needed to completely &#8216;determine&#8217; a binomial distribution:<\/p>\n<ul>\n<li>[latex]n[\/latex]\u00a0 = the number of trials<\/li>\n<li>[latex]p[\/latex] = the probability of success for each event\/trial.<\/li>\n<\/ul>\n<h1>What are trials and successes?<\/h1>\n<h2>Trials<\/h2>\n<p>A &#8216;trial&#8217; can be just about anything:<\/p>\n<ul>\n<li>The flip of a coin, in which case the 2 outcomes (heads or tails).<\/li>\n<li>A salesman calling on her clients, and making a sale or not.<\/li>\n<li>The roll of a die (where a certain number is rolled or not)<\/li>\n<li>In general, we call the 2 outcomes &#8216;successes&#8217; and &#8216;failures&#8217;<\/li>\n<\/ul>\n<h2>Successes<\/h2>\n<p>A &#8216;success&#8217; can be just about anything:<\/p>\n<ul>\n<li>Getting heads when flipping a coin<\/li>\n<li>Rolling a 6 when rolling dice<\/li>\n<li>Making a sale<\/li>\n<\/ul>\n<h2>Be Consistent<\/h2>\n<p>Just be sure &#8211; when talking about trials and successes related to binomial problems:<\/p>\n<ul>\n<li>Be sure to be consistent<\/li>\n<li>If you define success as rolling a 6,<\/li>\n<li>Be sure to use the probability of rolling as 6 as the probability of success.<\/li>\n<\/ul>\n<h1>Exploring the properties of Binomial distributions (ExeRCISE)<\/h1>\n<p>In this first example we will <span style=\"font-size: 20.5333px\">review the 3 properties of binomial distributions. In the next example, we will calculate a probability related to the example given below.<\/span><\/p>\n<h2>Example 24.1.1<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: A salesman calls on 10 clients everyday. 30% of all her calls in the past resulted in sales.<\/p>\n<p><span style=\"color: #003366\"><strong>Question:<\/strong> <\/span>Is this a Binomial experiment?<\/p>\n<p><strong><span style=\"color: #003366\">You Try<\/span><\/strong>: Let us find out by going through all the 3 basic characteristics.<\/p>\n<div id=\"h5p-56\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-56\" class=\"h5p-iframe\" data-content-id=\"56\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.1-1\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-57\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-57\" class=\"h5p-iframe\" data-content-id=\"57\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.1-2\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-58\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-58\" class=\"h5p-iframe\" data-content-id=\"58\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.1-3\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-59\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-59\" class=\"h5p-iframe\" data-content-id=\"59\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.1-4\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: Because all three properties of the binomial distribution are satisfied, this is indeed a binomial distribution.<\/p>\n<h1>Calculating Probabilities Using the Binomial Formula (EXAMPLE)<\/h1>\n<p>Now that we know the situation given in the previous example is a binomial experiment, let us revisit this example when practicing using the binomial formula.<\/p>\n<h2>Example 24.1.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup: <\/strong><\/span>Let us suppose the salesman is still calling 10 clients per day and the probability that she will make a sale with each client is 0.3.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What is the probability that 4 of her 10 calls in a day will result in sales?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: If we look at this formula first, we see that there are only 3 variables, [latex]n[\/latex],[latex]p[\/latex], and [latex]x[\/latex]. Recall that [latex]C[\/latex] stands for Combination and has no numeric value. We also know that [latex]n=10[\/latex] and [latex]p=0.3[\/latex] are the parameters, and we want to find the probability that [latex]x=4[\/latex].<\/p>\n<p>\\[P(x=4) = {}_{10}C_4\\cdot 0.3^4\\cdot (1 &#8211; 0.3)^{10-4}={}_{10}C_4\\cdot 0.3^4\\cdot (0.7)^{6}\\]<\/p>\n<p>Let us work out each of the 3 factors individually:<\/p>\n<p>\\[{}_{10}C_4=\\frac{10!}{4!(10-4)!}=\\frac{10!}{4!}{6!} =210\\]<\/p>\n<p>\\[0.3^4 = 0.0081\\]<\/p>\n<p>\\[0.7^6=0.0117649\\]<\/p>\n<p>So:<\/p>\n<p>\\[P(x=4)= 201\\times 0.0081 \\times 0.117649 = 0.200120949 \\]<\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is a 20% chance that she will make exactly 4 sales in a day.<\/p>\n<h1>Calculating Probabilities Using the Binomial Formula (EXErcise)<\/h1>\n<p>We can do the same steps as the previous example to find [latex]P(x=0)[\/latex], [latex]P(x=1)[\/latex], [latex]P(x=2)[\/latex], &#8230; , [latex]P(x=10)[\/latex]. The sum of these 11 probabilities must, of course, equal 1.<\/p>\n<h2>Example 24.1.3<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: We will continue with example 24.1 where [latex]n[\/latex]=10 and [latex]p[\/latex]=0.3<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: Can you calculate the probabilities in the exercises below?<\/p>\n<p><span style=\"color: #003366\"><strong>You try<\/strong><\/span>: Solve for the probabilities below:<\/p>\n<div id=\"h5p-60\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-60\" class=\"h5p-iframe\" data-content-id=\"60\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.3 - P(x = 0)\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-61\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-61\" class=\"h5p-iframe\" data-content-id=\"61\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.3 - P(x = 1)\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-62\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-62\" class=\"h5p-iframe\" data-content-id=\"62\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 24.1.3 - P(x = 2)\"><\/iframe><\/div>\n<\/div>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Binomial Properties &amp; Calculating the Probability of \u2018X\u2019 Successes<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-63\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-63\" class=\"h5p-iframe\" data-content-id=\"63\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key Takeaways for Intro to Binomial Distributions Calculations\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-64\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-64\" class=\"h5p-iframe\" data-content-id=\"64\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key Takeaways for Intro to Binomial Distributions Calculations Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"author":883,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-222","chapter","type-chapter","status-publish","hentry"],"part":231,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/222","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/883"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/222\/revisions"}],"predecessor-version":[{"id":1952,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/222\/revisions\/1952"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/231"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/222\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=222"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=222"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=222"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}