{"id":362,"date":"2025-12-22T08:53:25","date_gmt":"2025-12-22T08:53:25","guid":{"rendered":"https:\/\/blog.samuelgill.net\/?p=362"},"modified":"2025-12-22T08:53:25","modified_gmt":"2025-12-22T08:53:25","slug":"probability-probably-a-good-idea","status":"publish","type":"post","link":"https:\/\/blog.samuelgill.net\/index.php\/2025\/12\/22\/probability-probably-a-good-idea\/","title":{"rendered":"Probability &#8211; Probably a good Idea"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">The concept of probability is ubiquitous in our modern world, but how does it work, and how do we construct these tools from the ground up?<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The Probability Space<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The underlying tool we will use for probability is our probability space, this consists of 3 main items<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-deb220df7c93f9fc29c492a09b1d863b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>, the sample space, which is the collection of outcomes.<\/li>\n\n\n\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3c2a3bd25cd2107bf2f7f77081555147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;&#32;&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#101;&#113;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#80;&#125;&#40;&#32;&#92;&#79;&#109;&#101;&#103;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"\/>, the event space, which is the collection of possible events, which are collections of elements of the sample space.<\/li>\n\n\n\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-61ba77d7a7503ae46199311c9933f370_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#58;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;&#32;&#92;&#116;&#111;&#32;&#91;&#48;&#44;&#49;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -5px;\"\/>, the probability map, which gives the probability of a given event.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\"><strong>Example<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s take the example of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-d64371cdedcf640615545644fbabcfca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> sided dice roll&#8230;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-deb220df7c93f9fc29c492a09b1d863b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/> would be the possible outcomes, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-64263dabc49fa59bfe19aabc4ade8d73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;&#32;&#61;&#32;&#32;&#92;&#123;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#53;&#44;&#54;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -5px;\"\/><\/li>\n\n\n\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-c4eab677339e9b830360978dfba91f7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"15\" style=\"vertical-align: -1px;\"\/> would be all possible combinations of elements of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-deb220df7c93f9fc29c492a09b1d863b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>. This is because not only are there events such as the die lands on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-9f383212662555a816a0ad59c8874dbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> (Which would be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-675f607de2324992efcaf0447e3fe4a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#52;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -5px;\"\/>), but there are also the events such as the die lands on an even number (Which would be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3a931cb438c7eca290ca10a654c75d87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#50;&#44;&#52;&#44;&#54;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"\/>).<\/li>\n\n\n\n<li>Provided the die is fair, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-90c96d818e9aee3b76003836187b4236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: 0px;\"\/> would simply be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-00d01368defee6af09e7978db521cb61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#41;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#117;&#109;&#125;&#40;&#65;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"132\" style=\"vertical-align: -6px;\"\/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-5881cd7e567823bfedc7da8f99535640_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#117;&#109;&#125;&#40;&#65;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"\/> is the number of elements in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-84d731b5fec7151bcb789a9e4caa6866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-medium-font-size wp-block-paragraph\"><strong>Axioms of Probability<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These then obey certain axioms of probability. First, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-c4eab677339e9b830360978dfba91f7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"15\" style=\"vertical-align: -1px;\"\/> must be a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-d15968dd984c20d5c6a125874a81f5eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>-algebra on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-deb220df7c93f9fc29c492a09b1d863b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/>. This is a type of structure that follows these 3 main properties.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-63595baa672abf5711c21188b0ad8688_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#109;&#112;&#116;&#121;&#115;&#101;&#116;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -1px;\"\/><\/li>\n\n\n\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-61786f6a7d5da3874dec85078c776886_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;&#32;&#92;&#105;&#109;&#112;&#108;&#105;&#101;&#115;&#32;&#92;&#79;&#109;&#101;&#103;&#97;&#32;&#92;&#115;&#101;&#116;&#109;&#105;&#110;&#117;&#115;&#32;&#65;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"177\" style=\"vertical-align: -5px;\"\/><\/li>\n\n\n\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-84e178fd0e0636e5114e3e3094f128a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#123;&#105;&#125;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;&#32;&#92;&#105;&#109;&#112;&#108;&#105;&#101;&#115;&#32;&#65;&#95;&#49;&#32;&#92;&#99;&#117;&#112;&#32;&#65;&#95;&#50;&#32;&#92;&#99;&#117;&#112;&#32;&#65;&#95;&#51;&#32;&#92;&#108;&#100;&#111;&#116;&#115;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"266\" style=\"vertical-align: -3px;\"\/><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Or in other words<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The event of nothing is an event<\/li>\n\n\n\n<li>If some event can happen, then it not happening is also an event that can happen.<\/li>\n\n\n\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-19cae74a902d8652424f91731cc08468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\"\/> are a list of events that can happen, then the event that any of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-19cae74a902d8652424f91731cc08468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\"\/> occur is an event that can happen.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">And <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-90c96d818e9aee3b76003836187b4236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: 0px;\"\/> must follow the following axioms.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3969e10210730ccb1b3aafa94f98bf64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#92;&#79;&#109;&#101;&#103;&#97;&#41;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\"\/><\/li>\n\n\n\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-8a1b90c4746e60803ce9762d1a8b3d51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#41;&#32;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"\/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-1fc2884619a1fb89680dfc3deb74fdba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#111;&#114;&#97;&#108;&#108;&#32;&#65;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"60\" style=\"vertical-align: -1px;\"\/><\/li>\n\n\n\n<li>Given <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-b1a53becc2b62803958d12cea09c4ba4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#105;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -3px;\"\/> disjoint, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-e341aed7f9137dc5d73b09f9f77de2db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#92;&#98;&#105;&#103;&#99;&#117;&#112;&#32;&#65;&#95;&#105;&#41;&#32;&#61;&#32;&#92;&#115;&#117;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#95;&#105;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"\/><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Which means that,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The probability that any outcome will happen is 1, or in other words, at least one outcome must happen.<\/li>\n\n\n\n<li>The probability of any event must be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-2a0612c52be0b04f86be030ab704aa48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: -3px;\"\/>.<\/li>\n\n\n\n<li>Given any <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-19cae74a902d8652424f91731cc08468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -3px;\"\/> such that they don\u2019t intersect, meaning <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-745a371e130ef1d0d2c6330bdabac80d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#95;&#105;&#32;&#92;&#99;&#97;&#112;&#32;&#65;&#95;&#106;&#32;&#61;&#32;&#92;&#101;&#109;&#112;&#116;&#121;&#115;&#101;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"91\" style=\"vertical-align: -6px;\"\/> (which although not technically true, one can think of this as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-4c998550b3bd9bf28dfa7b49e2ac5540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#95;&#105;&#32;&#92;&#99;&#97;&#112;&#32;&#65;&#95;&#106;&#41;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"116\" style=\"vertical-align: -6px;\"\/>), then the probability of any of the events happening is the same as the sum of the probabilities of each event.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Bayes Theorem<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Now that we have defined our probability function, we can use it to dicover some facts about the probability of events. Note for all the future discussions we need to assume, unless stated otherwise, all probabilities will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-e5264b3ce4c3c82037123a5054e9eab8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"27\" style=\"vertical-align: -2px;\"\/>, this simplifies some of the upcoming work.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One important tool for this is going to be the symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-eba6296d2284ef32a4179959727b196e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"1\" style=\"vertical-align: -5px;\"\/>, which in words means &#8220;given&#8221;. Thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-a91ab789a7af7de90d9dd2f2801bb721_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#124;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> means the probability that the event <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-84d731b5fec7151bcb789a9e4caa6866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> occurs, given the event <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-4b01dac4b384fbc022d366988c55170b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> occured. We also need to define something called a partition, this is a colleciton of events <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-c70a65017bc874161a71cba6c35c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\"\/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-8b5455e0860dbf249097c1c9f73906da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;&#32;&#92;&#105;&#110;&#32;&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"37\" style=\"vertical-align: -1px;\"\/> such that all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-c70a65017bc874161a71cba6c35c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\"\/> are disjoint, and together <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3bd181a4d037212082b30ca6b85d5eed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#105;&#103;&#99;&#117;&#112;&#32;&#66;&#95;&#105;&#32;&#61;&#32;&#92;&#79;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"\/>. Or in words, this is a collection of events such that exactly one of them will occur. For example if we are rolling a die, the partition could be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-9e0d35d0053ac7d3b392e36e1d6560ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#95;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\"\/> being the number is odd and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-a5da1af0e126aadef53045f837505193_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#95;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\"\/> being the number is even.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The first fact that will be of use for us is that <br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>|<\/mi><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>\u2229<\/mo><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbb{P}(A|B) = \\frac{\\mathbb{P}(A \\cap B)}{\\mathbb{P}(B)}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This actually does not need a proof, as this is the definition of what <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-a91ab789a7af7de90d9dd2f2801bb721_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#124;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> is (In our current logic, the actual definition is far more complex and beyond the scope of this post).<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-medium-font-size\">Law of total probability<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The first consequence of our defintions above is the law of total probability (LTP), which states that given a partition <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-c70a65017bc874161a71cba6c35c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#95;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\"\/>, then<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><munder><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>i<\/mi><mo>\u2208<\/mo><mi>I<\/mi><\/mrow><\/munder><\/mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>|<\/mi><msub><mi>B<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>B<\/mi><mi>i<\/mi><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbb{P}(A) = \\sum_{i\\in I} \\mathbb{P}(A|B_i)\\mathbb{P}(B_i)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The proof of this fairly simple, but will be ommitted here (See if you can do it yourself!).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">What does this tell us? Essentially, this tells us that if we have an event, the probability that it occurs can be divided into some fractions of the probability of the partitions occuring, based on the chance that our event occurs given a partition occurs.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-medium-font-size\">Bayes Theorem.<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">You may have noticed that we can rearrange the formula for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-a91ab789a7af7de90d9dd2f2801bb721_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#124;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> and realised we can rearrange it to get a formula for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-8cca378089a6698838ef6136ebb54d21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#32;&#92;&#99;&#97;&#112;&#32;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"\/>, which is symmetric in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-84d731b5fec7151bcb789a9e4caa6866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-4b01dac4b384fbc022d366988c55170b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, thus <\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>|<\/mi><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mi>|<\/mi><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbb{P}(A|B)\\mathbb{P}(B) = \\mathbb{P}(B|A)\\mathbb{P}(A)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Hence,<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mi>|<\/mi><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mi>|<\/mi><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"> \\mathbb{P}(B|A) = \\frac{\\mathbb{P}(A|B)\\mathbb{P}(B)}{\\mathbb{P}(A)}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This is Bayes theorem (Some people use the law of total probability on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f11ef9eb4949a43e6899e2d2e19a3c64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#98;&#98;&#123;&#80;&#125;&#40;&#65;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"40\" style=\"vertical-align: -5px;\"\/> in their statement). Essentially it tells us that we can reverse conditional probabilities.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Random Variables<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The next key tool in the study of probability is that of the random variables. We can think of these as objects with a random value, such as the value on a die, the number of people in a room, or the height of a person.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-9837d3dd997b9179000fe080183bc37e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"\/> main types of such variables, these are continuous and discrete. Technically a random variable can be neither, but this only occurs in specific circumstances which wont be disscussed here.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-medium-font-size\">The basic construction<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A random variable is a map from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-deb220df7c93f9fc29c492a09b1d863b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"\/> to some output space, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f38787aadd30ad43ba6a40fcce89eb96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>. Very commonly <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f38787aadd30ad43ba6a40fcce89eb96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f97e7d39ba2e7e19be43ffe81a789132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"\/>, although this does not need to be the case, we shall assume <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f4afd4afbab2e284469bfdffbf6a8e91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"\/> from now on. We also require that a random variable satifies that the set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-2ec8331bf92cc5ccf6a28d9d927e8142_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#58;&#32;&#88;&#40;&#92;&#111;&#109;&#101;&#103;&#97;&#41;&#32;&#92;&#108;&#101;&#32;&#120;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"\/> is an event. This then lets us give a value to each possible event, for example if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-5ad1826ccb98e03282ae626ad33d351e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> was the year a person was born, then if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f1974df9a5fbef78f7d8b9edc42543f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> was the event that we chose Kurt G\u00f6del, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-71654c4976d4316c1fb71e29115d63ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#40;&#92;&#111;&#109;&#101;&#103;&#97;&#41;&#32;&#61;&#32;&#49;&#57;&#48;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"\/>. We could then say what is the probability that the age of a person is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-14d415d9c88a867c1ebce28d02f1458c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#48;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\"\/>, or more formally, what is the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-d194945ea8dc7e5d23607b24972524a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#92;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#58;&#32;&#88;&#40;&#92;&#111;&#109;&#101;&#103;&#97;&#41;&#32;&#61;&#32;&#49;&#57;&#48;&#54;&#92;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"169\" style=\"vertical-align: -5px;\"\/>, this is often shortened to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-1cf673f60f2b8adc3622505933ac8a2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#88;&#32;&#61;&#32;&#49;&#57;&#48;&#54;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"\/>. This in words is then the probability that the sampled event is an event that gives the year <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-14d415d9c88a867c1ebce28d02f1458c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#48;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\"\/>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If X has finite or countably many outputs, then its discrete, and hence by the axioms above, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3aaf5082b213d2c2e78815b3c950a110_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#58;&#32;&#88;&#40;&#92;&#111;&#109;&#101;&#103;&#97;&#41;&#32;&#61;&#32;&#120;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"\/> is then an event. We can then define <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-e10e7640318676e67a842f1cc2606030_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#88;&#32;&#40;&#120;&#41;&#32;&#61;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#80;&#125;&#40;&#88;&#32;&#61;&#32;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -5px;\"\/>, we call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3b67504cf8811b3a6b0d99516cfb3bfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#95;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: -4px;\"\/> the probability mass function.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The defintion of a contionous random variable is more complex, as we can&#8217;t say that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-3aaf5082b213d2c2e78815b3c950a110_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#58;&#32;&#88;&#40;&#92;&#111;&#109;&#101;&#103;&#97;&#41;&#32;&#61;&#32;&#120;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"\/> is an event, so instead we say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-5ad1826ccb98e03282ae626ad33d351e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> is continuous if there exists a function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-5a3f70b1ea841fe1384a672f43898c58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#88;&#58;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;&#32;&#92;&#116;&#111;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"\/> such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-d0ef655a76615ed93ac4808d8002053e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#88;&#40;&#120;&#41;&#32;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"\/> for all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f96167d7dbba2ebc6caacfda335907c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f0951e86ee502cbd3620d81200d23883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#105;&#110;&#116;&#95;&#123;&#45;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#32;&#102;&#95;&#88;&#40;&#117;&#41;&#32;&#100;&#117;&#32;&#61;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"133\" style=\"vertical-align: -6px;\"\/>, and <br><\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>F<\/mi><mi>X<\/mi><\/msub><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u2119<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><msubsup><mo movablelimits=\"false\">\u222b<\/mo><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mi>x<\/mi><\/msubsup><msub><mi>f<\/mi><mi>X<\/mi><\/msub><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>u<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>u<\/mi><\/mrow><annotation encoding=\"application\/x-tex\"> F_X(x) = \\mathbb{P}(X \\le x) = \\int_{-\\infty}^{x} f_X(u) du<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Here we call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-eb6bcc1df553114384bec3c7464b42a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#95;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\"\/> the cummulative density function, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-6b76065113b915d461fa1b9e85765021_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#95;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\"\/> the probability density function.<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-medium-font-size\">Why?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">With this, we can define some useful properties, first we could want to consider the mean (or average) of a random variable, to do this we will define the expectation, this is expected value a random variable will give us, and is denoted <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-f156eff0201d116e8dd639fe3e539df7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#69;&#125;&#92;&#108;&#101;&#102;&#116;&#91;&#88;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -5px;\"\/>. For discrete random variables, this is calculated by<br><p class=\"ql-center-displayed-equation\" style=\"line-height: 38px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-90d508138a4781f0c2543d740041d406_l3.png\" height=\"38\" width=\"247\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#91;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#69;&#125;&#92;&#108;&#101;&#102;&#116;&#91;&#88;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#32;&#61;&#32;&#92;&#115;&#117;&#109;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#111;&#115;&#115;&#105;&#98;&#108;&#101;&#32;&#79;&#117;&#116;&#99;&#111;&#109;&#101;&#115;&#32;&#120;&#125;&#125;&#32;&#120;&#112;&#95;&#88;&#40;&#120;&#41;&#92;&#93;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Or in words, the expectation is the weighted average of all the outcomes, weighted by the probability of getting that outcome. Then for moving into the continous case, we replace sums with integrals to get<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>\ud835\udd3c<\/mi><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><mi>X<\/mi><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo>=<\/mo><msubsup><mo movablelimits=\"false\">\u222b<\/mo><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>\u221e<\/mi><\/mrow><mi>\u221e<\/mi><\/msubsup><mi>x<\/mi><msub><mi>f<\/mi><mi>X<\/mi><\/msub><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\mathbb{E}\\left[X\\right] = \\int_{-\\infty}^{\\infty} xf_X(x) dx<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Which in meaning is very similar to above, just using integrals.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Another property we could define is variance, here we will use the expectation to say the variance is the expected square (as so to make it positive) distance from the mean, and hence is calculated<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\ud835\udd3c<\/mi><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><msup><mrow><mo fence=\"true\" form=\"prefix\">(<\/mo><mi>X<\/mi><mo>\u2212<\/mo><mi>\ud835\udd3c<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">Var(X) = \\mathbb{E}\\left[\\left(X -\\mathbb{E}(X)\\right)^2\\right]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Or the much easier to compute version<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\ud835\udd3c<\/mi><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><msup><mi>X<\/mi><mn>2<\/mn><\/msup><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo>\u2212<\/mo><mi>\ud835\udd3c<\/mi><msup><mrow><mo fence=\"true\" form=\"prefix\">[<\/mo><mi>X<\/mi><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">Var(X) = \\mathbb{E}\\left[X^2\\right] &#8211; \\mathbb{E}\\left[X\\right]^2<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">One could then choose to square root this and get the standard deviation, denoted <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.samuelgill.net\/wp-content\/ql-cache\/quicklatex.com-8c2bfc5932b7c407bf6c718186ea1ed5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#103;&#109;&#97;&#95;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\"\/>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The concept of probability is ubiquitous in our modern world, but how does it work, and how do we construct these tools from the ground up? The Probability Space The underlying tool we will use for probability is our probability space, this consists of 3 main items Example Let&#8217;s take the example of a $6$ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-362","post","type-post","status-publish","format-standard","hentry","category-uncategorised"],"_links":{"self":[{"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/posts\/362","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/comments?post=362"}],"version-history":[{"count":56,"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/posts\/362\/revisions"}],"predecessor-version":[{"id":425,"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/posts\/362\/revisions\/425"}],"wp:attachment":[{"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/media?parent=362"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/categories?post=362"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.samuelgill.net\/index.php\/wp-json\/wp\/v2\/tags?post=362"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}