{"id":154094,"date":"2022-03-25T23:09:12","date_gmt":"2022-03-25T17:39:12","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/total-probability-theorem\/"},"modified":"2022-12-31T17:02:49","modified_gmt":"2022-12-31T11:32:49","slug":"total-probability-theorem","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/","title":{"rendered":"Total Probability Theorem"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 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\">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\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><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><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#Total_Law_of_Probability_and_Decision_Trees\" title=\"Total Law of Probability and Decision Trees\">Total Law of Probability and Decision Trees<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#Total_Probability_Theorem\" title=\"Total Probability Theorem:\">Total Probability Theorem:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#How_to_State_and_Prove_Total_Probability_Theorem\" title=\"How to State and Prove Total Probability Theorem:\">How to State and Prove Total Probability Theorem:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#Total_Probability_Theorem_Proof\" title=\"Total Probability Theorem Proof:\">Total Probability Theorem Proof:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#Total_Probability_Theorem_Examples\" title=\"Total Probability Theorem Examples:\">Total Probability Theorem Examples:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#about_Total_Probability_Theorem-proof\" title=\"about Total Probability Theorem-proof:\">about Total Probability Theorem-proof:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#To_understand_how_you_can_use_the_decision_tree_in_calculating_full_potential_consider_the_following_example\" title=\"To understand how you can use the decision tree in calculating full potential, consider the following example:\">To understand how you can use the decision tree in calculating full potential, consider the following example:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#Then_the_chances_of_the_second_card_becoming_king_or_not_will_be_represented_by_the_law_of_full_probabilities_such_as\" title=\"Then the chances of the second card becoming king or not will be represented by the law of full probabilities such as:\">Then the chances of the second card becoming king or not will be represented by the law of full probabilities such as:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/total-probability-theorem\/#Explanation_of_Total_Probability_Theorem_with_the_Following_Examples\" title=\"Explanation of Total Probability Theorem with the Following Examples:\">Explanation of Total Probability Theorem with the Following Examples:<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Total_Law_of_Probability_and_Decision_Trees\"><\/span>Total Law of Probability and Decision Trees<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The total law of probability states that the probability of two or more independent events occurring is the product of their individual probabilities. Decision trees are a graphical representation of the total law of probability. They allow you to visualize the possible outcomes of a decision and the probability of each outcome.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154093 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/total-probability-theorem.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/total-probability-theorem.jpg?v=1648229947 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/total-probability-theorem-300x212.jpg?v=1648229947 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Total_Probability_Theorem\"><\/span>Total Probability Theorem:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The total probability theorem states that the probability of an event occurring is the sum of the probabilities of the individual events that make up the event.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_to_State_and_Prove_Total_Probability_Theorem\"><\/span>How to State and Prove Total Probability Theorem:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The total probability theorem states that the probability of an event is the sum of the probabilities of the individual outcomes that make up the event. To prove the theorem, we use the principle of addition.<\/p>\n<p>The principle of addition states that the probability of two events occurring is the sum of the probabilities of the individual events occurring.<\/p>\n<p>For example, the probability of flipping a coin and getting heads is 1\/2, the probability of flipping a coin and getting tails is 1\/2, and the probability of flipping two coins and getting two heads is 1\/4.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Total_Probability_Theorem_Proof\"><\/span>Total Probability Theorem Proof:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The proof of the theorem is simple. We will use the fact that the probability of any event is the sum of the probabilities of the individual outcomes that make up the event.<\/p>\n<p>We will start with a simple example. Suppose we have a die and we want to find the probability of getting a three. There are six possible outcomes, and only one of them is a three. So, the probability of getting a three is 1\/6.<\/p>\n<p>Now, let&#8217;s suppose we have a deck of cards and we want to find the probability of getting a seven. There are 52 possible outcomes, and only one of them is a seven. So, the probability of getting a seven is 1\/52.<\/p>\n<p>Now, let&#8217;s suppose we have a bag of marbles and we want to find the probability of getting a blue marble. There are six possible outcomes, and only one of them is blue. So, the probability of getting a blue marble is 1\/6.<\/p>\n<p>Now, let&#8217;s suppose we have a bag of marbles and we want to find the probability of getting a red marble. There are six possible outcomes, and two of them are red. So, the probability of getting a red marble is 2\/6.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Total_Probability_Theorem_Examples\"><\/span>Total Probability Theorem Examples:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>(1) A fair coin is flipped three times. What is the probability of getting at least two heads?<\/p>\n<p>There are three possible outcomes for the first flip: heads, tails, or a tie. There are two possible outcomes for the second flip: heads or tails. Finally, there is only one possible outcome for the third flip: heads or tails. Therefore, the probability of getting at least two heads is 3\/8.<\/p>\n<p>(2) A jar contains ten green and ten red balls. If a ball is selected at random from the jar, what is the probability that it is red?<\/p>\n<p>There are ten possible outcomes for the ball selected: nine red balls and one green ball. Therefore, the probability that the ball is red is 9\/10.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"about_Total_Probability_Theorem-proof\"><\/span>about Total Probability Theorem-proof:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let \\(A\\) and \\(B\\) be any two events in a sample space \\(S\\). The total probability theorem states that<\/p>\n<p>\\[P(A\\cup B) = P(A) + P(B) &#8211; P(A\\cap B)\\]\n<p>Proof:<\/p>\n<p>We will use mathematical induction to prove the theorem.<\/p>\n<p>The base case is when \\(A\\) and \\(B\\) are mutually exclusive, meaning that \\(A\\cap B = \\emptyset\\). In this case,<\/p>\n<p>\\[P(A\\cup B) = P(A) + P(B) &#8211; P(A\\cap B) = P(A) + P(B) = 1\\]\n<p>Now we will assume the theorem is true for all pairs of events \\(A\\) and \\(B\\) that are subsets of a given sample space \\(S\\), and show that it is also true for the pair \\(A\\) and \\(B\\) itself.<\/p>\n<p>Let \\(A\\) and \\(B\\) be any two events in a sample space \\(S\\). We will show that<\/p>\n<p>\\[P(A\\cup B) = P(A) + P(B) &#8211; P(A\\cap B)\\]\n<p>By definition,<\/p>\n<p>\\[P(A\\cup B) = P(A) + P(B) &#8211; P(A\\cap B) = P(<\/p>\n<h2><span class=\"ez-toc-section\" id=\"To_understand_how_you_can_use_the_decision_tree_in_calculating_full_potential_consider_the_following_example\"><\/span>To understand how you can use the decision tree in calculating full potential, consider the following example:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>You are considering investing in a new software system that will help your business to automate some of its operations.<\/p>\n<p>The system has two options:<\/p>\n<p>1) Buy the software outright for a one-time cost of $10,000<\/p>\n<p>2) Rent the software for a monthly cost of $100<\/p>\n<p>Which option should you choose?<\/p>\n<p>The decision tree can help you to calculate the full potential of each option.<\/p>\n<p>In option 1, you would pay a one-time cost of $10,000 for the software. In option 2, you would pay a monthly cost of $100 for the software.<\/p>\n<p>If you use the software for 10 months, you would pay a total of $1,000 for the software in option 1, and you would pay a total of $1,200 for the software in option 2.<\/p>\n<p>So, in option 2, you would pay an extra $200 for the software.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Then_the_chances_of_the_second_card_becoming_king_or_not_will_be_represented_by_the_law_of_full_probabilities_such_as\"><\/span>Then the chances of the second card becoming king or not will be represented by the law of full probabilities such as:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>P(K|H) = 1\/2<\/p>\n<p>P(K|H) = 1\/2<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Explanation_of_Total_Probability_Theorem_with_the_Following_Examples\"><\/span>Explanation of Total Probability Theorem with the Following Examples:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Example 1:<\/p>\n<p>In a jar there are 5 red balls and 10 blue balls. What is the probability that a randomly chosen ball is blue?<\/p>\n<p>There are 10 blue balls and 15 balls in total. So, the probability of choosing a blue ball is 10\/15.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Total Law of Probability and Decision Trees The total law of probability states that the probability of two or more [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Total Probability Theorem","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn Total Probability Theorem topic of Maths in details explained by subject experts on infinitylearn.com. 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