{"id":5155,"date":"2025-07-14T07:46:23","date_gmt":"2025-07-14T06:46:23","guid":{"rendered":"https:\/\/pcool.dyndns.org\/?page_id=5155"},"modified":"2025-07-14T13:56:21","modified_gmt":"2025-07-14T12:56:21","slug":"probability-and-odds","status":"publish","type":"page","link":"https:\/\/pcool.dyndns.org\/index.php\/probability-and-odds\/","title":{"rendered":"Probability and Odds"},"content":{"rendered":"\n<p>Risk, probability and odds can be confusing. <\/p>\n\n\n\n<p>Risk has a broader, less specific and often refers to an expected loss. Risk can be absolute (no context) or relative (in context, for example sex). Risks don&#8217;t necessarily add up to one.<\/p>\n\n\n\n<p><a href=\"https:\/\/pcool.dyndns.org\/index.php\/summary-probability\/\" data-type=\"page\" data-id=\"993\">Probability<\/a> is more specific and always has a value between 0 and 1. The probability of an event is:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\(Probability = \\frac{EventHappening}{EventHappening + EventNotHappening} \\)<script src=\"https:\/\/pcool.dyndns.org\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/pcool.dyndns.org\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<p>Probabilities for and against add up to one; <strong>the probabilities for and against are each other&#8217;s complement.<\/strong><\/p>\n\n\n\n<p>The odds are for over against:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\(Odds = \\frac{EventHappening}{EventNotHappening} \\)<\/div>\n\n\n\n<p>The odds ratio is the ratio of the odds for and against. <strong>The odds ratio for and against are each other reciprocal.<\/strong><\/p>\n\n\n\n<p>Of course, it is possible to &#8216;convert&#8217; odds to probability. <\/p>\n\n\n\n<p>To convert odds to probability:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\(Odds = \\frac{a}{b}: Probability = \\frac{a}{a+b} \\)<\/div>\n\n\n\n<p>Similarly for probability:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\(Probability = p: Odds = \\frac{p}{1-p} \\)<\/div>\n\n\n\n<p>To illustrate on an example from the Titanic:<\/p>\n\n\n\n<table id=\"tablepress-47\" class=\"tablepress tablepress-id-47\">\n<thead>\n<tr class=\"row-1\">\n\t<td class=\"column-1\"><\/td><th class=\"column-2\">Alive<\/th><th class=\"column-3\">Dead<\/th><th class=\"column-4\">Total<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">Female<\/td><td class=\"column-2\">308<\/td><td class=\"column-3\">154<\/td><td class=\"column-4\">462<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">Male<\/td><td class=\"column-2\">142<\/td><td class=\"column-3\">709<\/td><td class=\"column-4\">851<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">Total<\/td><td class=\"column-2\">450<\/td><td class=\"column-3\">863<\/td><td class=\"column-4\">1313<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-47 from cache -->\n\n\n<p>Risk<\/p>\n\n\n\n<table id=\"tablepress-48\" class=\"tablepress tablepress-id-48\">\n<thead>\n<tr class=\"row-1\">\n\t<td class=\"column-1\"><\/td><th class=\"column-2\">Numerator<\/th><th class=\"column-3\">Denominator<\/th><th class=\"column-4\">Value<\/th><th class=\"column-5\">Comment<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">Overall Risk Death<\/td><td class=\"column-2\">863<\/td><td class=\"column-3\">1313<\/td><td class=\"column-4\">0.657<\/td><td class=\"column-5\"><\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">Risk Female Death<\/td><td class=\"column-2\">154<\/td><td class=\"column-3\">462<\/td><td class=\"column-4\">0.333<\/td><td class=\"column-5\">Not Complement of below<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">Risk Male Death<\/td><td class=\"column-2\">709<\/td><td class=\"column-3\">851<\/td><td class=\"column-4\">0.833<\/td><td class=\"column-5\">Not Complement of above<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">Relative Risk Male Death<\/td><td class=\"column-2\">0.833<\/td><td class=\"column-3\">0.333<\/td><td class=\"column-4\">2.5<\/td><td class=\"column-5\"><\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\"><\/td><td class=\"column-3\"><\/td><td class=\"column-4\"><\/td><td class=\"column-5\"><\/td>\n<\/tr>\n<tr class=\"row-7\">\n\t<td class=\"column-1\">Probability Female Death<\/td><td class=\"column-2\">154<\/td><td class=\"column-3\">863<\/td><td class=\"column-4\">0.178<\/td><td class=\"column-5\">Complement of below<\/td>\n<\/tr>\n<tr class=\"row-8\">\n\t<td class=\"column-1\">Probability Male Death<\/td><td class=\"column-2\">709<\/td><td class=\"column-3\">863<\/td><td class=\"column-4\">0.822<\/td><td class=\"column-5\">Complement of above<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-48 from cache -->\n\n\n<p>And for the odds:<\/p>\n\n\n\n<table id=\"tablepress-49\" class=\"tablepress tablepress-id-49\">\n<thead>\n<tr class=\"row-1\">\n\t<td class=\"column-1\"><\/td><th class=\"column-2\">Numerator<\/th><th class=\"column-3\">Denominator<\/th><th class=\"column-4\">Value<\/th><th class=\"column-5\">Comment<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">Female Odds<\/td><td class=\"column-2\">154<\/td><td class=\"column-3\">308<\/td><td class=\"column-4\">1\/2<\/td><td class=\"column-5\"><\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">Male Odds<\/td><td class=\"column-2\">709<\/td><td class=\"column-3\">142<\/td><td class=\"column-4\">5\/1<\/td><td class=\"column-5\"><\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">Odds ratio Males<\/td><td class=\"column-2\">5<\/td><td class=\"column-3\">0.5<\/td><td class=\"column-4\">10<\/td><td class=\"column-5\">Reciprocal of below<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">Odds ratio Females<\/td><td class=\"column-2\">0.5<\/td><td class=\"column-3\">5<\/td><td class=\"column-4\">0.1<\/td><td class=\"column-5\">Reciprocal of above<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-49 from cache -->","protected":false},"excerpt":{"rendered":"<p>Risk, probability and odds can be confusing. Risk has a broader, less specific and often refers to an expected loss. Risk can be absolute (no context) or relative (in context, for example sex). Risks don&#8217;t necessarily add up to one. Probability is more specific and always has a value between 0 and 1. The probability [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-5155","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages\/5155","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/comments?post=5155"}],"version-history":[{"count":3,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages\/5155\/revisions"}],"predecessor-version":[{"id":5220,"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/pages\/5155\/revisions\/5220"}],"wp:attachment":[{"href":"https:\/\/pcool.dyndns.org\/index.php\/wp-json\/wp\/v2\/media?parent=5155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}