{"id":9370,"date":"2018-02-09T16:45:14","date_gmt":"2018-02-09T11:15:14","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/?p=9370"},"modified":"2019-03-02T18:44:45","modified_gmt":"2019-03-02T13:14:45","slug":"probability-class-12-notes-mathematics","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/","title":{"rendered":"Probability Class 12 Notes Mathematics"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/#CBSE_Guide_Probability_class_12_Notes_Mathematics\" >CBSE Guide Probability class 12 Notes Mathematics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/#Class_12_Mathematics_notes_Chapter_13_Probability\" >Class 12 Mathematics notes Chapter 13 Probability<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/#CBSE_Class_12_Mathematics_Chapter_13_Probability\" >CBSE Class 12 Mathematics Chapter 13 Probability<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/#Probability_class_12_Notes\" >Probability class 12 Notes<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/#CBSE_Class-12_Revision_Notes_and_Key_Points\" >CBSE Class-12 Revision Notes and Key Points<\/a><\/li><\/ul><\/nav><\/div>\n<p><strong>Probability Class 12 Notes Mathematics<\/strong> in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Probability class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 13 Probability are also available for download in CBSE Guide website.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Guide_Probability_class_12_Notes_Mathematics\"><\/span><strong>CBSE Guide Probability class 12 Notes Mathematics<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>CBSE guide notes are the comprehensive notes which covers the latest syllabus of CBSE and NCERT. It includes all the topics given in NCERT class 12 Mathematics textbook. Users can download CBSE guide quick revision notes from myCBSEguide mobile app and my CBSE guide website.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Class_12_Mathematics_notes_Chapter_13_Probability\"><\/span><strong>Class 12 Mathematics notes Chapter 13 Probability<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Download CBSE class 12th revision notes for chapter 13 Probability in PDF format for free. Download revision notes for Probability class 12 Notes and score high in exams. These are the Probability class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 13 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.<\/p>\n<p style=\"text-align: center;\"><strong><a class=\"button\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Download Revision Notes as PDF<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Class_12_Mathematics_Chapter_13_Probability\"><\/span><strong>CBSE Class 12 Mathematics Chapter 13 Probability<br \/>\n<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li><strong>Sample Space<\/strong>: The set of all possible outcomes of a random experiment. It is denoted by the symbol S.<\/li>\n<li><strong>Sample points<\/strong>: Elements of the sample space.<\/li>\n<li><strong>Event<\/strong>: A subset of the sample space.<\/li>\n<li><strong>Impossible Event<\/strong>: The empty set.<\/li>\n<li><strong>Sure Event<\/strong>: The whole sample space.<\/li>\n<li><strong>Complementary event or &#8220;not event&#8221;: <\/strong>The set &#8220;S&#8221; or S &#8211; A.<\/li>\n<li><strong>The event A or B<\/strong>: The set A <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cup\" \/><\/span><\/span>\u00a0B.<\/li>\n<li><strong>The event A and B<\/strong>: The set A <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cap\" \/><\/span><\/span>\u00a0B.<\/li>\n<li><strong>The event A but not B<\/strong>: A &#8211; B.<\/li>\n<li><strong>Mutually exclusive events<\/strong>: A and B are mutually exclusive if A <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cap\" \/><\/span><\/span>\u00a0B = <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\phi\" \/><\/span><\/span>.<\/li>\n<li><strong>Exhaustive and Mutually exclusive events<\/strong>: Events E<sub>1<\/sub>, E<sub><span style=\"font-size: small;\">2<\/span><\/sub>,&#8230;&#8230;.., E<sub><span style=\"font-size: small;\">n<\/span><\/sub> are mutually exclusive and exhaustive if E<sub><span style=\"font-size: small;\">1<\/span><\/sub><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cup\" \/><\/span><\/span>\u00a0E<sub><span style=\"font-size: small;\">2<\/span><\/sub> <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cup\" \/><\/span><\/span>&#8230;&#8230;.<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cup\" \/><\/span><\/span>\u00a0E<sub><span style=\"font-size: small;\">n<\/span><\/sub> = S and E<sub><span style=\"font-size: small;\">i<\/span><\/sub>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cap\" \/><\/span><\/span>\u00a0E<sub><span style=\"font-size: small;\">j<\/span><\/sub>\u00a0= <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\phi\" \/><\/span><\/span>\u00a0for all <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?i \\ne j\" \/><\/span><\/span>.<\/li>\n<li><strong>Exiomatic approach to probability<\/strong>: To assign probabilities to various events, some axioms or rules have been described.<\/li>\n<\/ul>\n<p>Let S be the sample space of a random experiment. The probability P is a real values function whose domain is the power set of S and range is the interval [0, 1] satisfying the following axioms:<\/p>\n<p>(a) For any event E, P(E) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\ge\" \/><\/span><\/span>\u00a00<\/p>\n<p>(b) P(S) = 1<\/p>\n<p>(c) If E and F are mutually exclusive event, then P(E <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\cup\" \/><\/span><\/span>\u00a0F)\u00a0= P(E) + P(F)<\/p>\n<p>If E<sub><span style=\"font-size: small;\">1<\/span><\/sub>, E<sub><span style=\"font-size: small;\">2<\/span><\/sub>,\u00a0E<sub><span style=\"font-size: small;\">3<\/span><\/sub>&#8230;&#8230;&#8230;&#8230; are n mutually exclusive events, then <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{P}}\\left( {\\mathop \\cup \\limits_{i = 1}^n {{\\rm{E}}_i}} \\right) = \\sum\\limits_{i = 1}^n {{\\rm{P}}\\left( {{{\\rm{E}}_i}} \\right)}\" \/><\/span><\/span><\/p>\n<ul>\n<li><strong>Probability of an event in terms of the probabilities of the same points (outcomes): <\/strong>Let S be the sample space containing n exhaustive outcomes <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{\\rm{W}}_1},{{\\rm{W}}_2},{{\\rm{W}}_3},......{{\\rm{W}}_n}\" \/><\/span><\/span>\u00a0i.e., S = <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {{{\\rm{W}}_1},{{\\rm{W}}_2},{{\\rm{W}}_3},......{{\\rm{W}}_n}} \\right)\" \/><\/span><\/span><\/li>\n<\/ul>\n<p>Now from the axiomatic definition of the probability:<\/p>\n<p>(a) 0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\le\" \/><\/span><\/span>\u00a0P(W<sub>i<\/sub>) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\le\" \/><\/span><\/span>1, for each <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{\\rm{W}}_i} \\in {\\rm{S}}\" \/><\/span><\/span>.<\/p>\n<p>(b) P(W<sub><span style=\"font-size: small;\">1<\/span><\/sub>) + P(W<sub><span style=\"font-size: small;\">2<\/span><\/sub>) + &#8230;&#8230;.+ P(W<sub><span style=\"font-size: small;\">3<\/span><\/sub>) = P(S) = 1<\/p>\n<p>(c) For any event A, P(A) = <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\sum {{\\rm{P}}\\left( {{{\\rm{W}}_i}} \\right),}\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{\\rm{W}}_i} \\in {\\rm{A}}\" \/><\/span><\/span><\/p>\n<ul>\n<li><strong>Equally likely outcomes<\/strong>: All outcomes with equal probability.<\/li>\n<li><strong>Classical definition of the probability of an event<\/strong>: For a finite sample space with equally likely outcome, probability of an event A.<\/li>\n<\/ul>\n<p>P(A) = <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{n\\left( {\\rm{A}} \\right)} \\over {n\\left( {\\rm{S}} \\right)}}\" \/><\/span><\/span><\/p>\n<p>where <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?n\\left( {\\rm{A}} \\right)\" \/><\/span><\/span>\u00a0= Number of elements in the set A. and <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?n\\left( {\\rm{S}} \\right)\" \/><\/span><\/span>\u00a0= Number of elements in set S.<\/p>\n<ul>\n<li>If A is any event, then P(not A) = 1 &#8211; P(A) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\Rightarrow\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{P}}\\left( {\\overline {\\rm{A}} } \\right) = 1 - {\\rm{P}}\\left( {\\rm{A}} \\right)\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\Rightarrow\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{P}}\\left( {{\\rm{A'}}} \\right) = 1 - {\\rm{P}}\\left( {\\rm{A}} \\right)\" \/><\/span><\/span><\/li>\n<li>The conditional probability of an event E, given the occurrence of the event F is given by <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image001.png\" \/><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image002.png\" \/><\/li>\n<li>\u00b7\u00a0\u00a0 <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image003.png\" \/><\/li>\n<li>\u00b7\u00a0\u00a0 <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image004.png\" \/><\/li>\n<li><strong>Theorem of total probability:<\/strong><\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image005.png\" \/>\u00a0be a partition of a sample space and suppose that each of <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image006.png\" \/>has non zero probability.\u00a0\u00a0 \u00a0Let A be any event associated with S, then<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image007.png\" \/><\/p>\n<ul>\n<li><strong>Bayes&#8217; theorem:<\/strong> If <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image006.png\" \/>are events which constitute a partition of sample space S, i.e. <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image006.png\" \/>are pairwise disjoint and <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image008.png\" \/>be any event with non-zero probability, then, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image009.png\" \/><\/li>\n<li><strong>Random variable<\/strong>: A random variable is a real valued function whose domain is the sample space of a random experiment.<\/li>\n<li><strong>Probability distribution<\/strong>: The probability distribution of a random variable X is the system of numbers<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image010.png\" \/><\/p>\n<p>Where,\u00a0<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image011.png\" \/><\/p>\n<ul>\n<li><strong>Mean of a probability distribution<\/strong>: Let X be a random variable whose possible values <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image012.png\" \/>occur with probabilities\u00a0 \u00a0<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image013.png\" \/>respectively.\u00a0 The mean of X, denoted by <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image014.png\" \/>\u00a0is the number <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image015.png\" \/>. The mean of a random variable X is also called the expectation of X, denoted by E (X).<\/li>\n<li><strong>Variance<\/strong>: Let X be a random variable whose possible values <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image012.png\" \/>occur with probabilities\u00a0\u00a0 <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image016.png\" \/>\u00a0respectively. Let <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image017.png\" \/>\u00a0be the mean of X. The variance of X, denoted by Var (X) or<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image018.png\" \/>\u00a0\u00a0is defined as \u00a0 <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image019.png\" \/><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image020.png\" \/>or equivalently\u00a0 <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image021.png\" \/>. The non-negative number, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image022.png\" \/>\u00a0is called the <strong>standard deviation <\/strong>of the random variable X.<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image023.png\" \/><\/p>\n<ul>\n<li><strong>Bernoulli Trials<\/strong>: Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions:<\/li>\n<\/ul>\n<p>(i)\u00a0\u00a0 There should be a finite number of trials.<\/p>\n<p>(ii)\u00a0\u00a0 The trials should be independent.<\/p>\n<p>(iii)\u00a0\u00a0 Each trial has exactly two outcomes: success or failure.<\/p>\n<p>(iv)\u00a0\u00a0 The probability of success remains the same in each trial.<\/p>\n<p>For Binomial distribution <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch13\/image024.png\" \/><strong>\u00a0 \u00a0 \u00a0\u00a0<\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Probability_class_12_Notes\"><\/span><strong>Probability class 12 Notes<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>CBSE Revision notes (PDF Download) Free<\/li>\n<li>CBSE Revision notes for Class 12 Mathematics PDF<\/li>\n<li>CBSE Revision notes Class 12 Mathematics \u2013 CBSE<\/li>\n<li>CBSE Revisions notes and Key Points Class 12 Mathematics<\/li>\n<li>Summary of the NCERT books all chapters in Mathematics class 12<\/li>\n<li>Short notes for CBSE class 12th Mathematics<\/li>\n<li>Key notes and chapter summary of Mathematics class 12<\/li>\n<li>Quick revision notes for CBSE board exams<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Class-12_Revision_Notes_and_Key_Points\"><\/span><strong>CBSE Class-12 Revision Notes and Key Points<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Probability class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Math\u2019s, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.<\/p>\n<ul>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physics\/1251\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Physics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-chemistry\/1267\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Chemistry<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Mathematics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-biology\/1298\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Biology<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-accountancy\/1315\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Accountancy<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-economics\/1327\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Economics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-business-studies\/1727\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Business Studies<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-computer-science\/1851\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Computer Science<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-informatics-practices\/1873\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Informatics Practices<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-english-core\/1855\/cbse-revision-notes\/7\/\">Revision Notes for class-12 English Core<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-history\/1869\/cbse-revision-notes\/7\/\">Revision Notes for class-12 History<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physical-education\/1877\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Physical Education<\/a><\/li>\n<\/ul>\n<p>To download Probability class 12 Notes Mathematics, sample paper for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT Probability, NCERT Exemplar Probability, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n<ul>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/\">Relations and Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/inverse-trigonometric-functions-class-12-notes-mathematics\/\">Inverse Trigonometric Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/matrices-class-12-notes-mathematics\/\">Matrices class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/determinants-class-12-notes-mathematics\/\">Determinants class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/continuity-differentiability-class-12-notes-mathematics\/\">Continuity and Differentiability class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-derivatives-class-12-notes-mathematics\/\">Application of Derivatives class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/integrals-class-12-notes-mathematics\/\">Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-integrals-class-12-notes-mathematics\/\">Application of Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/differential-equations-class-12-notes-mathematics\/\">Differential Equations class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/vector-algebra-class-12-notes-mathematics\/\">Vector Algebra class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/three-dimensional-geometry-class-12-notes-mathematics\/\">Three Dimensional Geometry class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/linear-programming-class-12-notes-mathematics\/\">Linear Programming class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/\">Probability class 12 Notes Mathematics<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Probability Class 12 Notes Mathematics in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Probability class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 13 Probability are also available for &#8230; <a title=\"Probability Class 12 Notes Mathematics\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/\" aria-label=\"More on Probability Class 12 Notes Mathematics\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":9266,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[48,456],"tags":[457,150,461,500,426,240],"class_list":["post-9370","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cbse-class-12","category-revision-notes","tag-cbse-notes","tag-cbse-notes-and-key-points","tag-mathematics-notes","tag-probability","tag-quick-revision","tag-quick-revision-notes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Probability Class 12 Notes Mathematics | myCBSEguide<\/title>\n<meta name=\"description\" content=\"Probability class 12 Notes Mathematics chapter 13 in PDF format for free download. 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