{"id":44225,"date":"2023-10-15T20:28:37","date_gmt":"2023-10-15T14:58:37","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=44225"},"modified":"2025-10-13T16:16:52","modified_gmt":"2025-10-13T10:46:52","slug":"unit-v-probability-new-scheme-2023","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=44225","title":{"rendered":"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023)"},"content":{"rendered":"\n<p>SYLLABUS:&nbsp; &nbsp; Random experiment \u2013 Outcomes \u2013 Sample space \u2013 Events \u2013 Occurrence of events \u2013 \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Exhaustive events \u2013 Mutually exclusive events \u2013 Classical definition of probability \u2013 Axioms of probability \u2013 Probability of an event \u2013 Probability of \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Conditional probability \u2013 Multiplication rule \u2013 Independent events \u2013 Simple problems (Combinatorial problems excluded) \u2013 Engineering applications (not for examinations).<\/p>\n\n\n\n<p>The probability theory is an important branch of Mathematics. The word probability is used to denote the happening of certain event and likelihood of the occurrence of that event based on the past experiences.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-b031529728110d1d8dd84e2ef7e399ca\"><strong>5.1 PROBABILITY OF AN EVENT<\/strong><\/h3>\n\n\n\n<p>Experiment:&nbsp; An experiment is defines as a process for which its result is well defined.<\/p>\n\n\n\n<p>These are two types<\/p>\n\n\n\n<p>Deterministic experiment:&nbsp; An experiment whose outcomes can be predicted with certain, under identical conditions.<\/p>\n\n\n\n<p>Random experiment:&nbsp; An experiment whose all possible outcomes are know, but it is not possible to predict the outcome.<\/p>\n\n\n\n<p>Examples:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>A fair coin is \u201ctossed\u201d<\/li>\n\n\n\n<li>A die is \u201crolled\u201d.<\/li>\n\n\n\n<li>Selecting a card from a pack of cards.<\/li>\n<\/ol>\n\n\n\n<p>Outcome<\/p>\n\n\n\n<p>The result of a random experiment is called an outcome.<\/p>\n\n\n\n<p>Trial<\/p>\n\n\n\n<p>Each performance of a random experiment is called a trial.<\/p>\n\n\n\n<p>Sample space<\/p>\n\n\n\n<p>The set of all possible outcomes of a random experiment is called a sample space.&nbsp; It is denoted by S.<\/p>\n\n\n\n<p>Examples<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>If a die is rolled, then the sample space S = {1,2,3,4,5,6}<\/li>\n\n\n\n<li>A coin is tossed, then the sample space S = {H,T}<\/li>\n\n\n\n<li>Two coins are tossed once.&nbsp; Then the sample space S = {HH,HT,TH,TT}<\/li>\n<\/ol>\n\n\n\n<p>Event:&nbsp; Every non-empty subset of the sample space is an event.&nbsp; An event is the outcomes &nbsp; &nbsp;<\/p>\n\n\n\n<p>&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; of the random experiment<\/p>\n\n\n\n<p>Notations<\/p>\n\n\n\n<p>Let A and B be two events.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ A\\ \\cup\\ B\\ stands\\ for\\ the\\ occurence\\ of\\ A\\ or\\ B\\ or\\ both\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/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<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ A\\ \\cap\\ B\\ stands\\ for\\ the\\ simultaneous\\ occurence\\ of\\ A\\ and\\ B\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(iii)\\ \\bar{A}\\ or\\ A{\\prime}\\ or\\ A^c\\ stands\\ for\\ \\ non-occurence\\ of\\ A\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ (A\\ \\cap\\ \\bar{B})\\ stands\\ for\\ the\\ occurence\\ of\\ only\\ A\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1\\ .}\\ A\\ die\\ is\\ rolled\\ once,\\ \\hspace{15cm}\\]\\[\\color {red} {Find\\ the\\ probability\\ of\\ getting\\ an\\ odd\\ number.}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{A standard die has 6 faces numbered from 1 to 6. No. of possible outcomes = 6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The odd numbers on a die are 1, 3, and 5. No. of favourable outcomes = 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability P of rolling an odd number can be calculated as follows:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(odd\\ number)\\ =\\ \\frac{No.\\ of\\ favourable\\ outcomes}{No.\\ of\\ possible\\ outcomes}\\ =\\ \\frac{3}{6}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting an odd number on a rolling die is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2\\ .}\\ While\\ a\\ die\\ is\\ rolled\\ once,\\ \\hspace{15cm}\\]\\[\\color {red} {Find\\ the\\ probability\\ of\\ getting\\ an\\ even\\ number}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{A standard die has 6 faces numbered from 1 to 6. No. of possible outcomes = 6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The even numbers on a die are 2, 4, and 6. No. of favourable outcomes = 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability P of rolling an odd number can be calculated as follows:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(even\\ number)\\ =\\ \\frac{No.\\ of\\ favourable\\ outcomes}{No.\\ of\\ possible\\ outcomes}\\ =\\ \\frac{3}{6}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting an odd number on a rolling die is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/SSgiSQEg1Cc\" title=\"Probability-3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ A\\ die\\ is\\ rolled\\ once,\\ \\hspace{15cm}\\]\\[\\color {red} {Find\\ the\\ probability\\ of\\ getting\\ a\\ prime\\ number.}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{A standard die has 6 faces numbered from 1 to 6. No. of possible outcomes = 6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The prime numbers on a die are 2, 3, and 5. No. of favourable outcomes = 3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{A standard die has 6 faces numbered from 1 to 6. The prime numbers on a die are 2, 3, and 5.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability P of rolling a prime number can be calculated as follows:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(prime\\ number)\\ =\\ \\frac{Number\\ of\\ favourable\\ outcomes}{Number\\ of\\ possible\\ outcomes}\\ =\\ \\frac{3}{6}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a prime number on a rolling die is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ \\text{ An integer is chosen at random from the integers 1 to 10. }\\ \\hspace{10cm}\\]\\[\\color {red} {Find\\ the\\ probability\\ that\\ it\\ is\\  an\\ even\\ number.}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Given an integer is chosen at random from 1 to 10. No. of possible outcomes = 10}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The even numbers from 1 to 10 are 2, 4, 6, 8, 10.  No. of favourable outcomes = 5}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability P of rolling a prime number can be calculated as follows:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(even\\ number)\\ =\\ \\frac{Number\\ of\\ favourable\\ outcomes}{Number\\ of\\ possible\\ outcomes}\\ =\\ \\frac{5}{10}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting an even number from the integers 1 to 10 is:}\\ \\boxed{\\frac{1}{2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5\\ .}\\ Two\\ coins\\ are\\ tossed\\ simultaneously\\ \\ \\hspace{15cm}\\]\\[\\color {red} {What\\ is\\ the\\ probability\\ of\\ getting\\ (i)\\ exactly\\ one\\ head\\ (ii)\\ atleast\\ one\\ head\\ (iii)\\ atmost\\ one\\ head}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[S\\ =\\ \\text{{HH,  HT, TH, TT}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(S)\\ =\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let A be the event of getting exactly one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ =\\ \\text{{HT, TH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(A)\\ =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let B be the event of getting atleast one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ =\\ \\text{{HH,HT, TH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(B)\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let C be the event of getting atmost one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C\\ =\\ \\text{{TT,HT, TH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(C)\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ P(A)\\ =\\ \\frac{n(A)}{n(S)}\\ =\\ \\frac{2}{4}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ P(B)\\ =\\ \\frac{n(B)}{n(S)}\\ =\\ \\frac{3}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(iii)\\ P(C)\\ =\\ \\frac{n(C)}{n(S)}\\ =\\ \\frac{3}{4}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/vEnde_vCxy0\" title=\"Probability - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6\\ .}\\ Three\\ coins\\ are\\ tossed\\ simultaneously\\  \\color {red} {Find\\ the\\ probability\\ of\\ getting}\\ \\hspace{5cm}\\]\\[(i)\\ exactly\\ one\\ head\\ \\hspace{13cm}\\]\\[(ii)\\ exactly\\ two\\ heads\\ \\hspace{13cm}\\]\\[(iii)\\ at least\\ two\\ heads\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[S\\ =\\ \\text{{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(S)\\ =\\ 8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let A be the event of getting exactly one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ =\\ \\text{{HTT, THT, TTH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(A)\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let B be the event of getting exactly two heads}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ =\\ \\text{{HHT, HTH, THH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(B)\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let C be the event of getting atleast two heads}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C\\ =\\ \\text{{HHT, HTH, THH, HHH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(C)\\ =\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ P(A)\\ =\\ \\frac{n(A)}{n(S)}\\ =\\ \\frac{3}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ P(B)\\ =\\ \\frac{n(B)}{n(S)}\\ =\\ \\frac{3}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(iii)\\ P(C)\\ =\\ \\frac{n(C)}{n(S)}\\ =\\ \\frac{4}{8}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7\\ .}\\ Three\\ coins\\ are\\ tossed\\ simultaneously\\  \\color {red} {Find\\ the\\ probability\\ of\\ getting}\\ \\hspace{5cm}\\]\\[(i)\\ at\\ least\\ one\\ head\\ \\hspace{13cm}\\]\\[(ii)\\ at\\ most\\ one\\ head\\ \\hspace{13cm}\\]\\[(iii)\\ exactly\\ one\\ head.\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[S\\ =\\ \\text{{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(S)\\ =\\ 8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let A be the event of getting at least one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ =\\ \\text{{HTT, TTH, HHT, HTH, THT, THH, HHH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(A)\\ =\\ 7\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let B be the event of getting at most one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ =\\ \\text{{TTT, HTT, THT, TTH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(B)\\ =\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Let C be the event of getting exactly one head}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[C\\ =\\ \\text{{HTT, THT, TTH}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[n(C)\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(i)\\ P(A)\\ =\\ \\frac{n(A)}{n(S)}\\ =\\ \\frac{7}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ P(B)\\ =\\ \\frac{n(B)}{n(S)}\\ =\\ \\frac{4}{8}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(iii)\\ P(C)\\ =\\ \\frac{n(C)}{n(S)}\\ =\\ \\frac{3}{8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8\\ .}\\ A\\ card\\ is\\ picked\\ randomly\\ from\\ a\\  pack\\ of\\ 52\\ cards.\\ \\hspace{8cm}\\]\\[\\color {red} {Find\\ the\\ probability\\ of\\ getting\\ a\\ King}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>In a standard deck of 52 playing cards, there are 4 Kings (one from each suit: hearts, diamonds, clubs, and spades.<\/p>\n\n\n\n<p>The probability&nbsp;PP&nbsp;of drawing a King from a deck of 52 cards can be calculated as follows:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(King)\\ =\\ \\frac{Number\\ of\\ Kings}{Total\\ number\\ of\\ cards}\\ =\\ \\frac{4}{52}\\ =\\ \\frac{1}{13}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of picking a King from a standard deck of 52 cards is:}\\ \\boxed{\\frac{1}{13}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 9\\ .}\\ A\\ card\\ is\\ picked\\ randomly\\ from\\ a\\  pack\\ of\\ 52\\ cards\\ randomly.\\ \\color {red} {Find\\ the\\ probability}\\ \\hspace{8cm}\\]\\[of\\ getting\\ a\\ queen\\ card\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>In a standard deck of 52 playing cards, there are 4 queens(one from each suit: hearts, diamonds, clubs, and spades).<\/p>\n\n\n\n<p>The probability&nbsp;PP&nbsp;of drawing a queen from a deck of 52 cards can be calculated as follows:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(King)\\ =\\ \\frac{Number\\ of\\ queens}{Total\\ number\\ of\\ cards}\\ =\\ \\frac{4}{52}\\ =\\ \\frac{1}{13}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of picking a queenfrom a standard deck of 52 cards is:}\\ \\boxed{\\frac{1}{13}}\\]<\/div>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-324f0cea881dbefc97fb0029120897c7\">5.2 <strong>PROBABILITY OF \u2018not\u2019, \u2018and\u2019, \u2018or\u2019 EVENTS<\/strong><\/h3>\n\n\n\n<h5 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-3d8f1fe0e4bc338a6c76ede3ced59fad\">Basic Results<\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ P(\\Phi)\\ =\\ 0\\ \\hspace{19cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ If\\ \\bar{A}\\ is\\ the\\ complimentary\\ event\\ of\\ A,\\ then\\ P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ &#8211; P(A\\ \\cap\\ B)\\ \\hspace{12cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10\\ .}\\ If\\ A\\ and\\ B\\ are\\ two\\ events\\ such\\ that\\ P(A)\\ =\\ 0.42,\\ and\\ P(B)\\ =\\ 0.48,\\ \\hspace{10cm}\\]\\[\\color{red}{find\\ P(\\bar{A})\\ and\\ P(\\bar{B})}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024,\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ P(\\bar{B})\\ =\\ 1\\ -\\ P(B)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A)\\ =\\ 0.42\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(B)\\ =\\ 0.48\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\ =\\ 1\\ -\\ 0.42\\ =\\ 0.58\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(\\bar{B})\\ =\\ 1\\ -\\ P(B)\\ =\\ 1\\ -\\ 0.48\\ =\\ 0.52\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11\\ .}\\ If\\ A\\ and\\ B\\ are\\ two\\ events\\ such\\ that\\ P(A)\\ =\\ 0.35,\\ and\\ P(B)\\ =\\ 0.62,\\ \\hspace{10cm}\\]\\[\\color{red}{find\\ P(\\bar{A})\\ and\\ P(\\bar{B})}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ P(\\bar{B})\\ =\\ 1\\ -\\ P(B)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A)\\ =\\ 0.35\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(B)\\ =\\ 0.62\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(\\bar{A})\\ =\\ 1\\ -\\ P(A)\\ =\\ 1\\ -\\ 0.35\\ =\\ 0.65\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(\\bar{B})\\ =\\ 1\\ -\\ P(B)\\ =\\ 1\\ -\\ 0.62\\ =\\ 0.38\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12\\ .}\\ If\\ P(A)\\ =\\ 0.15,\\ P(B)\\ =\\ 0.25,\\ and\\  P(A\\ \\cap\\ B)\\ =\\ 0.10,\\ \\hspace{10cm}\\]\\[\\color {red} {find\\ the\\ value\\ of\\  P(A\\ \\cup\\ B)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ Supp\\ June\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Given P(A) = 0.15,  P(B) = 0.25 and}\\ P(A\\ \\cap\\ B)\\ =\\ 0.10\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ know\\ that\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B\\ =\\ P(A)\\ +\\ P(B)\\ &#8211; P(A\\ \\cap\\ B)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 0.15\\ +\\ 0.25\\ -\\ 0.10\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 0.40\\ -\\ 0.10\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{P(A\\ \\cup\\ B)\\ =\\ 0.3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13\\ .}\\ If\\ P(A)\\ =\\ 0.5,\\ P(B)\\ =\\ 0.3,\\ and\\ A\\ \\cap\\ B\\ is\\ empty,\\ \\hspace{10cm}\\]\\[\\color {red} {find\\ P(A\\ \\cup\\ B)}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Given P(A) = 0.5,  P(B) = 0.3 and} A \\cap\\ B\\ is\\ empty\\  (\\text{ which means events A and B are mutually exclusive),}\\] \n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ can\\ find\\ P(A\\ \\cup\\ B)\\ \\text{using the formula for the union of two mutually exclusive events:}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B\\ =\\ P(A)\\ +\\ P(B)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{2cm}\\ =\\ 0.5\\ +\\ 0.3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{P(A\\ \\cup\\ B)\\ =\\ 0.8}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14\\ .}\\ Two\\ dice\\ are\\ thrown\\ simultaneously\\  \\color {red} {Find\\ the\\ probability}\\ \\hspace{10cm}\\\\\\ of\\ getting\\ a\\ sum\\ 5\\ or\\ same\\ number\\ on\\ both\\ dice\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ two\\ dice\\ are\\ thrown\\ simultaneously,\\ each\\ die\\ has\\ 6\\ faces,\\ so\\ there\\ are\\ a\\ \\hspace{2cm}\\\\\\ total\\ of\\ 6\\ \\times\\ 6\\ =\\ 36\\ possible\\ outcomes\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ a\\ sum\\ of\\ 6}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the sum of the two dice is 5 are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 4 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ the\\ same\\ number\\ on\\ both\\ dice}}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the numbers on both dice are the same are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (6,\\ 6)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 6 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\bf{\\color{green}{probability\\ of\\ either\\ getting\\ a\\ \\ sum\\ of\\ 5\\ or\\ the\\ same\\ number\\ on\\ both\\ dice}}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 5\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ denotes\\ the\\ event\\ of\\ getting\\ the\\ same\\ number\\ on\\ both\\ dice\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ needed\\ to\\ find\\ P(A\\ \\cup\\ B) =\\ ?\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ -\\ P(A\\ \\cap\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{4}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{6}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(P(A\\ \\cap\\ B))\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{4}{36}\\ +\\ \\frac{6}{36}\\ -\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{10}{36}\\ =\\ \\frac{5}{18}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a sum of 5 or the same number on both dice is:}\\ =\\ \\frac{5}{18}\\ \\hspace{1cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 15\\ .}\\ Two\\ dice\\ are\\ thrown\\ simultaneously\\  \\color {red} {Find\\ the\\ probability}\\ \\hspace{10cm}\\\\\\ of\\ getting\\ a\\ sum\\ 6\\ or\\ same\\ number\\ on\\ both\\ dice\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\ Supp(June)\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ two\\ dice\\ are\\ thrown\\ simultaneously,\\ each\\ die\\ has\\ 6\\ faces,\\ so\\ there\\ are\\ a\\ \\hspace{2cm}\\\\\\ total\\ of\\ 6\\ \\times\\ 6\\ =\\ 36\\ possible\\ outcomes\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ a\\ sum\\ of\\ 6}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the sum of the two dice is 6 are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 5 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ the\\ same\\ number\\ on\\ both\\ dice}}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the numbers on both dice are the same are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (6,\\ 6)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 6 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\bf{\\color{green}{probability\\ of\\ either\\ getting\\ a\\ \\ sum\\ of\\ 6\\ or\\ the\\ same\\ number\\ on\\ both\\ dice}}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 6\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ denotes\\ the\\ event\\ of\\ getting\\ the\\ same\\ number\\ on\\ both\\ dice\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ needed\\ to\\ find\\ P(A\\ \\cup\\ B) =\\ ?\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ -\\ P(A\\ \\cap\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{5}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{6}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(P(A\\ \\cap\\ B))\\ =\\ \\frac{1}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{5}{36}\\ +\\ \\frac{6}{36}\\ -\\ \\frac{1}{36}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{10}{36}\\ =\\ \\frac{5}{18}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a sum of 6 or the same number on both dice is:}\\ =\\ \\frac{5}{18}\\ \\hspace{1cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16\\ .}\\ Two\\ dice\\ are\\ rolled\\ together.\\  \\color {red} {Find\\ the\\ probability}\\ \\hspace{10cm}\\\\\\ for\\ getting\\ the\\ sum\\ of\\ the\\ numbers\\ on\\ the\\ faces\\ are\\ 10\\ and\\ 7\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[When\\ two\\ dice\\ are\\ thrown\\ simultaneously,\\ each\\ die\\ has\\ 6\\ faces,\\ so\\ there\\ are\\ a\\ \\hspace{2cm}\\\\\\ total\\ of\\ 6\\ \\times\\ 6\\ =\\ 36\\ possible\\ outcomes\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ a\\ sum\\ of\\ 10}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the sum of the two dice is 6 are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 6)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (6,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 3 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 10\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{3}{36}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\bf{\\color{green}{probability\\ of\\ getting\\ a\\ sum\\ of\\ 7}\\ \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The possible outcomes where the numbers on both dice are the same are:}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (1,\\ 6)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (2,\\ 5)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (3,\\ 4)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (4,\\ 3)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (5,\\ 2)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ (6,\\ 1)\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{There are 6 such outcomes.}\\ \\hspace{10cm}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ denotes\\ the\\ event\\ of\\ getting\\ a\\ sum\\ of\\ 10\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{6}{36}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\bf{\\color{green}{probability\\ of\\ sum\\ of\\ the\\ numbers\\ on\\ on\\ the\\ faces\\ are\\ 10\\ and\\ 7}}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ needed\\ to\\ find\\ P(A\\ \\cup\\ B) =\\ ?\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ -\\ P(A\\ \\cap\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{3}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{6}{36}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(P(A\\ \\cap\\ B))\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{3}{36}\\ +\\ \\frac{6}{36}\\ -\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{9}{36}\\ =\\ \\frac{1}{4}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of getting a sum of 10 and 7 on both dice is:}\\ =\\ \\frac{1}{4}\\ \\hspace{1cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 17\\ .}\\ A\\ card\\ is\\ selected\\ at\\ random\\ from\\ a\\ pack\\ of\\ 52\\ cards.\\ \\color{red}{Find\\ the}\\ \\hspace{8cm}\\]\\[\\color{red}{probability\\ that\\ the\\ card\\ is\\ either\\ a\\  black\\ card\\ or\\ a\\ card\\ with\\ number\\ 6.}\\ \\hspace{2cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024,\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{To find the probability of drawing either a black card or a card with the number 6 from a standard deck of 52 cards,}\\\\ \\text{we need to consider both events and their possible overlap.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\text{Total number of cards in a deck: 52}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ 2.\\ \\text{Number of black cards: There are 26 black cards in the deck (13 spades +}\\\\ \\text{13 clubs}).\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ 3.\\ \\text{Number of cards with the number 6: There are 4 cards with the number 6}\\\\ \\text{(one in each suit: hearts, diamonds, clubs, and spades}).\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ denotes\\ the\\ event\\ of\\ drawing\\ a\\ black\\ card\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B\\ denotes\\ the\\ event\\ of\\ drawing\\ a\\ card\\ with\\ on\\ number\\ 6\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[we\\ needed\\ to\\ find\\ P(A\\ \\cup\\ B) =\\ ?\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ -\\ P(A\\ \\cap\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A)\\ =\\ \\frac{26}{52}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B)\\ =\\ \\frac{4}{52}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(P(A\\ \\cap\\ B))\\ =\\ \\frac{2}{52}\\ (2\\ such\\ cards\\ with \\ black\\ card\\ and\\ number\\ 6\\ from\\ spades\\ and\\ clubs)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ \\frac{26}{52}\\ +\\ \\frac{4}{52}\\ -\\ \\frac{2}{52}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{28}{52}\\ =\\ \\frac{7}{13}\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{So, the probability of drawing either a black card or a card with the}\\\\ \\text{number 6 from a standard deck of 52 cards is:}  \\frac{7}{13}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-b7fbed1c36957bd6a4923b1942dbed13\">5.3 <strong>CONDITIONAL PROBABILITY<\/strong><\/h3>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{The probability of occurrence of A when B has occurred is called the conditional probability}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ given\\ B,\\ denoted\\ as\\ P(A\/B),\\ is\\ defied\\ by\\ P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B}{P(B)},\\ provided\\ P(B)\\ \\neq\\ 0.\\ Similarly,\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B\/A)\\ =\\ \\frac{P(A\\ \\cap\\ B}{P(A)},\\ provided\\ P(A)\\ \\neq\\ 0.\\]<\/div>\n\n\n\n<h4 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-21bcb523871b61a35c9ec4fd460fa871\"><strong>Independent events<\/strong><\/h4>\n\n\n\n<p>Two events are said to be independent events if the occurrence of any one event does not affect&nbsp;the probability of occurrence of another event. Two events A and B are said to be independent if and only if P(A&nbsp;\u2229&nbsp;B) = P(A)P(B).<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 18\\ .}\\ \\color{red}{Find\\ P(A\/B)},\\ If\\ P(B)\\ =\\ 0.5,\\ and\\ P(A\\ \\cap\\ B)\\ =\\ 0.2,\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ the\\ conditional\\ probability\\ P(A\/B),\\ \\text{We use the definitions of conditional probability.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(B)\\ =\\ 0.5\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cap\\ B)\\ =\\ \\ 0.2\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\ =\\ \\frac{0.2} {0.5}\\ =\\  \\frac{2}{5}\\ =\\ 0.4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 19\\ .}\\ If\\ P(A)\\ =\\ \\frac{1}{3},\\ P(B)\\ =\\ \\frac{3}{4},\\ P(A\\ \\cap\\ B)\\ =\\ \\frac{1}{6},\\ \\hspace{10cm}\\]\\[\\color{red}{find\\ P(A\/B)\\ and\\ P(B\/A)}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\ April\\ 25\\ Supp\\ June\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ the\\ conditional\\ probabilities\\ P(A\/B)\\ and\\ P(B\/A),\\ \\text{We use the definitions of conditional probability.}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ P(B\/A)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(A)}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A)\\ =\\ \\frac{1}{3}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(B)\\ =\\ \\frac{3}{4}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cap\\ B)\\ =\\ \\frac{1}{6}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\/B)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(B)}\\ =\\ \\frac{\\frac{1}{6}} {\\frac{3}{4}}\\ =\\ \\frac{1}{6}\\ \\times\\ \\frac{4}{3}\\ =\\ \\frac{2}{9}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(B\/A)\\ =\\ \\frac{P(A\\ \\cap\\ B)}{P(A)}\\ =\\ \\frac{\\frac{1}{6}} {\\frac{1}{3}}\\ =\\ \\frac{1}{6}\\ \\times\\ \\frac{3}{1}\\ =\\ \\frac{1}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 20\\ .}\\ If\\ A\\ and\\ B\\ are\\ two\\ independent\\ events\\ such\\ that\\ P(A)\\ =\\ 0.4\\ and\\ P(A\\ \\cup\\ B)\\ =\\ 0.9.\\ \\hspace{10cm}\\]\\[\\color {red} {Find\\ P(B)}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A)\\ =\\ 0.4\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cup\\ B)\\ =\\ 0.9\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\bullet\\ P(A\\ \\cap\\ B)\\ =\\ P(A)\\ P(B)\\ (A\\ and\\ B\\ are\\ independent\\ events)\\ \\hspace{5cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[P(A\\ \\cup\\ B)\\ =\\ P(A)\\ +\\ P(B)\\ -\\ P(A\\ \\cap\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[0.9\\ =\\ 0.4\\ +\\ P(B)\\ -\\ P(A)\\ P(B)\\ \\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[0.9\\ -\\ 0.4\\ =\\ P(B) (1\\ -\\ P(A))\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[0.5\\ =\\ P(B) (1\\ -\\ 0.4)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{P(B)\\ =\\ 5\/6}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/_exTvmM8uaM\" title=\"Probability - 11\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- 350 x 250 ad -->\n<ins class=\"adsbygoogle\"\n     style=\"display:block\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"3583972194\"\n     data-ad-format=\"auto\"\n     data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 21\\ .}\\ A\\ problem\\ in\\ statistics\\ is\\ given\\ to\\ two\\ students\\ A\\ and\\ B.\\  The\\ \\hspace{10cm}\\\\\\ probability\\ of\\ A\\ solves\\ the\\ problem\\ is\\ \\frac{1}{2}\\ and\\ that\\ of\\ B\\ solves\\ the\\ \\hspace{8cm}\\\\\\ problem\\ is\\ \\frac{2}{3}.\\ If\\ the\\ students\\ solve\\ the\\ problems\\ independently.\\ \\hspace{8cm}\\\\\\ \\color{red}{find\\ the\\ probability\\ that\\ the\\ problem\\ is\\ solved.}\\ \\hspace{12cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\ April\\ 25\\ Supp\\ June\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given,\\ P(A)\\ =\\ \\frac{1}{2}\\  and\\  P(B)\\ =\\ \\frac{2}{3}\\ \\hspace{10cm}\\\\\\ \\text{Probability that the problem is solved = Probability that A solves the problem or B solves the problem}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A\\ \\cup\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A)\\ +\\  P(B)\\ -\\ (P(A\\ \\cap\\ B)\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A)\\ +\\ P(B)\\ -\\ P(A)\\ \\cdot \\ P(B)\\ Since A\\ and\\ B\\ are\\ independent\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ +\\ \\frac{2}{3}\\ -\\ \\frac{1}{2}\\ \\cdot \\ \\frac{2}{3}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{2}\\ +\\ \\frac{2}{3}\\ -\\ \\frac{1}{3}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{3\\ +4\\ -\\ 2}{6}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5}{6}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Thus, the probability that the problem is solved by either student A, student B, or both is:}\\ =\\ \\frac{5}{6}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 22\\ .}\\ A\\ problem\\ in\\ statistics\\ is\\ given\\ to\\ two\\ students\\ A\\ and\\ B.\\  The\\ \\hspace{10cm}\\\\\\ probability\\ of\\ A\\ solves\\ the\\ problem\\ is\\ \\frac{1}{4}\\ and\\ that\\ of\\ B\\ solves\\ the\\ \\hspace{8cm}\\\\\\ problem\\ is\\ \\frac{2}{5}.\\ If\\ they\\ solve\\ the\\ problem\\ independently,\\ \\hspace{8cm}\\\\\\ \\color{red}{find\\ the\\ probability\\ that\\ the\\ problem\\ is\\ solved.}\\ \\hspace{12cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ April\\ 2024,\\ October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue} {Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given,\\ P(A)\\ =\\ \\frac{1}{4}\\  and\\  P(B)\\ =\\ \\frac{2}{5}\\ \\hspace{10cm}\\\\\\ \\text{Probability that the problem is solved = Probability that A solves the problem or B solves the problem}\\]&nbsp;<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A\\ \\cup\\ B)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A)\\ +\\  P(B)\\ -\\ (P(A\\ \\cap\\ B)\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ P(A)\\ +\\ P(B)\\ -\\ P(A)\\ \\cdot \\ P(B)\\ Since A\\ and\\ B\\ are\\ independent\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{4}\\ +\\ \\frac{2}{5\n}\\ -\\ \\frac{1}{4}\\ \\cdot \\ \\frac{2}{5}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{1}{4}\\ +\\ \\frac{2}{5}\\ -\\ \\frac{1}{10}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{5\\ +\\ 8\\ -\\ 2}{20}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{11}{20}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\text{Thus, the probability that the problem is solved by either student A, student B, or both is:}\\ =\\ \\frac{11}{20}\\]<\/div>\n","protected":false},"excerpt":{"rendered":"<p>SYLLABUS:&nbsp; &nbsp; Random experiment \u2013 Outcomes \u2013 Sample space \u2013 Events \u2013 Occurrence of events \u2013 \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Exhaustive events \u2013 Mutually exclusive events \u2013 Classical definition of probability \u2013 Axioms of probability \u2013 Probability of an event \u2013 Probability of \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Conditional probability \u2013 Multiplication [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":52223,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788674],"tags":[],"class_list":["post-44225","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-basic-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=44225\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"SYLLABUS:&nbsp; &nbsp; Random experiment \u2013 Outcomes \u2013 Sample space \u2013 Events \u2013 Occurrence of events \u2013 \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Exhaustive events \u2013 Mutually exclusive events \u2013 Classical definition of probability \u2013 Axioms of probability \u2013 Probability of an event \u2013 Probability of \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Conditional probability \u2013 Multiplication [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=44225\" \/>\n<meta property=\"og:site_name\" content=\"YANAMTAKSHASHILA\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:published_time\" content=\"2023-10-15T14:58:37+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-13T10:46:52+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png\" \/>\n\t<meta property=\"og:image:width\" content=\"300\" \/>\n\t<meta property=\"og:image:height\" content=\"168\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"rajuviswa\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rajuviswa\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"18 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023)\",\"datePublished\":\"2023-10-15T14:58:37+00:00\",\"dateModified\":\"2025-10-13T10:46:52+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225\"},\"wordCount\":3682,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1\",\"articleSection\":[\"Basic Mathematics\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225\",\"name\":\"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1\",\"datePublished\":\"2023-10-15T14:58:37+00:00\",\"dateModified\":\"2025-10-13T10:46:52+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#primaryimage\",\"url\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1\",\"contentUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1\",\"width\":300,\"height\":168},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=44225#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/\",\"name\":\"yanamtakshashila.com\",\"description\":\"one stop solutions\",\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/yanamtakshashila.com\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\",\"name\":\"rajuviswa\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"url\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"contentUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"width\":3600,\"height\":3600,\"caption\":\"rajuviswa\"},\"logo\":{\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\"},\"sameAs\":[\"http:\\\/\\\/yanamtakshashila.wordpress.com\",\"https:\\\/\\\/www.facebook.com\\\/profile.php?id=100063680185552\",\"https:\\\/\\\/www.instagram.com\\\/rajuviswa\\\/?hl=en\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCjJ2KWWvsFm6F42UtMdbxzw\"],\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?author=187055548\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/yanamtakshashila.com\/?p=44225","og_locale":"en_US","og_type":"article","og_title":"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA","og_description":"SYLLABUS:&nbsp; &nbsp; Random experiment \u2013 Outcomes \u2013 Sample space \u2013 Events \u2013 Occurrence of events \u2013 \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Exhaustive events \u2013 Mutually exclusive events \u2013 Classical definition of probability \u2013 Axioms of probability \u2013 Probability of an event \u2013 Probability of \u2018not\u2019, \u2018and\u2019 and \u2018or\u2019 events \u2013 Conditional probability \u2013 Multiplication [&hellip;]","og_url":"https:\/\/yanamtakshashila.com\/?p=44225","og_site_name":"YANAMTAKSHASHILA","article_publisher":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_author":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_published_time":"2023-10-15T14:58:37+00:00","article_modified_time":"2025-10-13T10:46:52+00:00","og_image":[{"width":300,"height":168,"url":"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png","type":"image\/png"}],"author":"rajuviswa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rajuviswa","Est. reading time":"18 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=44225#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=44225"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023)","datePublished":"2023-10-15T14:58:37+00:00","dateModified":"2025-10-13T10:46:52+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=44225"},"wordCount":3682,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=44225#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1","articleSection":["Basic Mathematics"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=44225#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=44225","url":"https:\/\/yanamtakshashila.com\/?p=44225","name":"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023) - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=44225#primaryimage"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=44225#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1","datePublished":"2023-10-15T14:58:37+00:00","dateModified":"2025-10-13T10:46:52+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=44225#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=44225"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/yanamtakshashila.com\/?p=44225#primaryimage","url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1","contentUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1","width":300,"height":168},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=44225#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"UNIT \u2013 V PROBABILITY (NEW SCHEME \u2013 2023)"}]},{"@type":"WebSite","@id":"https:\/\/yanamtakshashila.com\/#website","url":"https:\/\/yanamtakshashila.com\/","name":"yanamtakshashila.com","description":"one stop solutions","publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/yanamtakshashila.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":["Person","Organization"],"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e","name":"rajuviswa","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","contentUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","width":3600,"height":3600,"caption":"rajuviswa"},"logo":{"@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1"},"sameAs":["http:\/\/yanamtakshashila.wordpress.com","https:\/\/www.facebook.com\/profile.php?id=100063680185552","https:\/\/www.instagram.com\/rajuviswa\/?hl=en","https:\/\/www.youtube.com\/channel\/UCjJ2KWWvsFm6F42UtMdbxzw"],"url":"https:\/\/yanamtakshashila.com\/?author=187055548"}]}},"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/10\/yanamtakshashila.com_.png?fit=300%2C168&ssl=1","jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/pc3kmt-bvj","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/44225","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/users\/187055548"}],"replies":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=44225"}],"version-history":[{"count":100,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/44225\/revisions"}],"predecessor-version":[{"id":61818,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/44225\/revisions\/61818"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/media\/52223"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=44225"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=44225"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=44225"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}