{"id":933,"date":"2020-02-18T06:58:50","date_gmt":"2020-02-18T06:58:50","guid":{"rendered":"https:\/\/bcisnotes.com\/thirdsemester\/?p=933"},"modified":"2020-02-18T06:58:50","modified_gmt":"2020-02-18T06:58:50","slug":"approaches-and-laws-of-probability-probability-bcis-notes","status":"publish","type":"post","link":"https:\/\/bcisnotes.com\/thirdsemester\/business-statistics\/approaches-and-laws-of-probability-probability-bcis-notes\/","title":{"rendered":"Approaches and Laws of Probability || Probability || Bcis Notes"},"content":{"rendered":"<h2>Approaches to Probability<\/h2>\n<p>The following are the approaches to probability:<\/p>\n<ol>\n<li>Classical Approach<\/li>\n<li>Empirical Approach<\/li>\n<li>Subjective Approach<\/li>\n<li>Axiomatic Approach<\/li>\n<\/ol>\n<h3>\u00a0Classical Approach<\/h3>\n<p>If there are &#8216;n&#8217; mutually exclusive and mutually likely cases and &#8216;m&#8217; of them are favorable to an event &#8216;E&#8217;. Then, the probability of the happening of an event &#8216;E&#8217; denoted by probability is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-934\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_144128.png\" alt=\"Classical Approach\" width=\"143\" height=\"50\" \/><\/p>\n<h4>Remarks:<\/h4>\n<ol>\n<li>The probability takes a value between 0 and 1.<\/li>\n<li>If &#8216;E&#8217; is an impossible event then, the probability is 0.<\/li>\n<li>The &#8216;E&#8217; is sure then P(E)=1.<\/li>\n<li>The sum of the probabilities of occurrence and non-occurrence is 0 and 1<\/li>\n<\/ol>\n<p>ie. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-935\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_144759.png\" alt=\"Sum of Probabilities\" width=\"191\" height=\"27\" \/><\/p>\n<p>p + q = 1<\/p>\n<h2>Laws (Rules) of Probability<\/h2>\n<h3>1. Additive Law<\/h3>\n<p>If A and B are two events with their respective probabilities P(A) and P(B) then the probability of occurrence <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-937\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_145305.png\" alt=\"Occurance\" width=\"23\" height=\"16\" \/> at least one \u00a0of these two events given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-938\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_145457-300x14.png\" alt=\"Additive Law\" width=\"450\" height=\"21\" srcset=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_145457-300x14.png 300w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_145457.png 514w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>where P(A U B) is the probability of the simultaneous occurrence of events A and B.<\/p>\n<h4>Remarks:<\/h4>\n<p>1. If A &amp; B are mutually exclusive event then, <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-941\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_150148.png\" alt=\"Mutually Exclusive \" width=\"141\" height=\"19\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-942\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_150337-300x21.png\" alt=\"Exclusive\" width=\"300\" height=\"21\" srcset=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_150337-300x21.png 300w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_150337-326x24.png 326w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_150337.png 345w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>2. If A &amp; B &amp; C are 3 events then the &#8216;P&#8217; of occurrence of at least one of these events is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-944\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151000-300x7.png\" alt=\"Events\" width=\"557\" height=\"13\" srcset=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151000-300x7.png 300w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151000-1024x25.png 1024w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151000-768x19.png 768w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151000-1536x37.png 1536w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151000-2048x50.png 2048w\" sizes=\"(max-width: 557px) 100vw, 557px\" \/><\/p>\n<p>If A &amp; B &amp; C are mutually exclusive then, the probability is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-947\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151205-300x14.png\" alt=\"Exclusive\" width=\"300\" height=\"14\" srcset=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151205-300x14.png 300w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_16022020_151205.png 515w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3>2. Multiplicative Law<\/h3>\n<p>If two events A &amp; B are independent then the probability of their simultaneous occurrence is equal to their individual probabilities. If P(A) and P(B) be the probabilities of occurrence of events A &amp; B respectively then the probability of their simultaneous occurrence is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1000\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_121444-300x13.png\" alt=\"Mutiplicative Law\" width=\"406\" height=\"21\" \/><\/p>\n<h4>Remarks:<\/h4>\n<ol>\n<li>If A, B, &amp; C are three independent events then the probabilities of their simultaneous occurrence is given by:<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1001\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_121840-300x15.png\" alt=\"Three independent Events\" width=\"369\" height=\"19\" srcset=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_121840-300x15.png 300w, https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_121840.png 466w\" sizes=\"(max-width: 369px) 100vw, 369px\" \/><\/p>\n<ol>\n<li>If A &amp; B are two dependent events then:<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1002\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_122026-300x12.png\" alt=\"Two dependent Events\" width=\"373\" height=\"26\" \/><\/p>\n<p>where,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1003\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_122126.png\" alt=\"B Given A\" width=\"79\" height=\"19\" \/>is the probability of occurrence of event B given that A has already occurred.<\/p>\n<h2>Marginal and Joint Probability<\/h2>\n<table>\n<tbody>\n<tr>\n<td width=\"156\">&nbsp;<\/td>\n<td width=\"156\">A<\/td>\n<td width=\"156\">B<\/td>\n<td width=\"156\">Total<\/td>\n<\/tr>\n<tr>\n<td width=\"156\">C<\/td>\n<td width=\"156\">N1<\/td>\n<td width=\"156\">N2<\/td>\n<td width=\"156\">N1 + N2<\/td>\n<\/tr>\n<tr>\n<td width=\"156\">D<\/td>\n<td width=\"156\">N3<\/td>\n<td width=\"156\">N4<\/td>\n<td width=\"156\">N3 + N4<\/td>\n<\/tr>\n<tr>\n<td width=\"156\">Total<\/td>\n<td width=\"156\">N1 + N2<\/td>\n<td width=\"156\">N2 + N4<\/td>\n<td width=\"156\">N<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The probability of each event given by a row or column in the contingency table is called marginal probability.<\/p>\n<p><span style=\"text-decoration: underline;\">The marginal probabilities are:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1004\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123008.png\" alt=\"P(A)\" width=\"233\" height=\"57\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1005\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123050.png\" alt=\"P(B)\" width=\"233\" height=\"56\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1006\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123207.png\" alt=\"P(C)\" width=\"233\" height=\"56\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1007\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123254.png\" alt=\"P(D)\" width=\"235\" height=\"56\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The probability of the joint event whose frequency is given in the cells in the table is called joint probability.<\/p>\n<p><span style=\"text-decoration: underline;\">The joint probabilities are:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1008\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123520.png\" alt=\"P(A)\" width=\"231\" height=\"56\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1009\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123555.png\" alt=\"P(B)\" width=\"231\" height=\"56\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1010\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123640.png\" alt=\"P(C)\" width=\"233\" height=\"56\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1011\" src=\"https:\/\/bcisnotes.com\/thirdsemester\/wp-content\/uploads\/2020\/02\/vme_img_18022020_123745.png\" alt=\"P(D)\" width=\"233\" height=\"56\" \/><\/p>\n<p>You may also like: <a href=\"https:\/\/bcisnotes.com\/thirdsemester\/business-statistics\/probability-and-permutation-probability-bcis-notes\/\" target=\"_blank\" rel=\"noopener noreferrer\">Probability and Permutation || Probability\u00a0<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<div class=\"mh-excerpt\"><p>Approaches to Probability The following are the approaches to probability: Classical Approach Empirical Approach Subjective Approach Axiomatic Approach \u00a0Classical Approach If there are &#8216;n&#8217; mutually <a class=\"mh-excerpt-more\" href=\"https:\/\/bcisnotes.com\/thirdsemester\/business-statistics\/approaches-and-laws-of-probability-probability-bcis-notes\/\" title=\"Approaches and Laws of Probability || Probability || Bcis Notes\">[&#8230;]<\/a><\/p>\n<\/div>","protected":false},"author":4,"featured_media":1014,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"yoast_head":"<!-- This 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