{"id":201,"date":"2023-12-06T16:14:23","date_gmt":"2023-12-06T21:14:23","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&p=201"},"modified":"2024-06-12T14:39:53","modified_gmt":"2024-06-12T18:39:53","slug":"expected-values","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/expected-values\/","title":{"raw":"Expected Values","rendered":"Expected Values"},"content":{"raw":"
\r\n

Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nCalculate and understand expected values.\r\n\r\n<\/div>\r\n<\/div>\r\n\"\"\r\n
    \r\n \t
  • The value we would expect to occur,\u00a0also known as the mean (\u03bc)<\/li>\r\n \t
  • Previous values (x'<\/em>s) and their probabilities of occurring (p<\/em>'s) are used to calculate \u03bc.<\/li>\r\n \t
  • Expected Value = [latex]\u00a0 E(X) =\\mu =\\Sigma [x_i \\cdot p(x_i)] [\/latex]<\/li>\r\n<\/ul>\r\n

    Expected Values with Coin flips (Example)<\/h1>\r\nThe concept of expected values is best illustrated by games. Let's 'play' some in the next two examples.\r\n

    Example 21.1 - Coin Flip<\/h2>\r\nSetup<\/span>:<\/span> <\/strong>This first game is rather simple, and not very exciting, but it lays the foundation for the second, more complicated game.\r\n
      \r\n \t
    • In this game, I flip a coin.<\/li>\r\n \t
    • If I flip a Head, I'll pay you a dollar.<\/li>\r\n \t
    • If I flip a Tail, you'll pay me a dollar.<\/li>\r\n<\/ul>\r\nQuestion:<\/strong> <\/span>Can either one of us EXPECT to earn anything in the long run?\r\n\r\nSolution:<\/strong> <\/span>Let us first of all define the term, \"in the long run\":\r\n
        \r\n \t
      • It simply means if we play the game many, many times.<\/li>\r\n \t
      • I guess most people will agree that the answer is NO.<\/li>\r\n \t
      • Half the time, I win a dollar.<\/li>\r\n \t
      • Half the time, I lose a dollar.<\/li>\r\n<\/ul>\r\nIn terms of mathematics, we can state the above sentence this way:\r\n\r\n\\[\\frac{1}{2} (+\\$1)+ \\frac{1}{2} (-\\$1)=\\$0\\]\r\n

        Expected Values with Dice Rolls (Exercise)<\/h1>\r\nLet us denote a more complex game in the next example. This time, we will use a six-sided die and win certain amounts of money depending on the number we roll.\r\n

        Example 21.2.1 - Die Roll<\/h2>\r\nSetup:<\/strong><\/span>\u00a0<\/strong>In this game, I will roll a fair regular 6 sided die.\r\n
          \r\n \t
        • You pay me $2 to play each round.<\/li>\r\n \t
        • Then I roll the die.<\/li>\r\n \t
        • If I roll a 6, I will pay you $5.50.<\/li>\r\n \t
        • If I roll a 5, I will pay you $3.40.<\/li>\r\n \t
        • If I roll a 4, I will pay you $2.50.<\/li>\r\n \t
        • If I roll a 3, 2, or 1, I will pay you nothing.<\/li>\r\n<\/ul>\r\nQuestion:<\/strong> <\/span>Would you like to play this game with me? Think about it for a moment.\r\n\r\nYou Try<\/strong><\/span>: Let us now analyze this game from your point of view.\r\n\r\n[h5p id=\"49\"]\r\n\r\n[h5p id=\"50\"]\r\n

          Standard Deviations of Probability Distributions<\/h1>\r\n
            \r\n \t
          • If needed, we can calculate the standard deviation of a discrete general probability distribution.<\/li>\r\n \t
          • This is not needed if we have the original data that we used to create the probability distribution<\/li>\r\n \t
          • We could just use a standard deviation calculation on the original data instead if it is available<\/li>\r\n \t
          • The standard deviation for discrete general probability distributions is:\r\n\\[ \\sigma(X) =\\sqrt{\\Sigma [(x_i-\\mu)^2 \\cdot p(x_i)]} \\]<\/li>\r\n<\/ul>\r\n

            Example 21.2.2<\/h2>\r\nSetup<\/span><\/strong>: Let us re-visit the previous example of the die rolling game.\r\n\r\nQuestion<\/strong><\/span>: What is the standard deviation of the winnings for this game?\r\n\r\nSolution<\/strong><\/span>: Let us use the formula to solve for the standard deviation:\r\n\r\n[latex]\r\n\r\n\\begin{align*}\r\n\r\n\\sigma(X) &=\\sqrt{(3.50-(-0.10))^2 \\cdot \\frac{1}{6} + (1.40-(-0.10))^2 \\cdot \\frac{1}{6}+(0.50-(-0.10))^2 \\cdot \\frac{1}{6}+(-2.00-(-0.10))^2 \\cdot \\frac{3}{6}} \\\\ \\\\\r\n\r\n&= \\sqrt{(3.60)^2 \\cdot \\frac{1}{6} + (1.50)^2 \\cdot \\frac{1}{6}+(0.60)^2 \\cdot \\frac{1}{6}+(-1.90)^2 \\cdot \\frac{3}{6}} \\\\ \\\\\r\n\r\n&= \\sqrt{12.96 \\cdot \\frac{1}{6} + 2.25 \\cdot \\frac{1}{6}+0.36 \\cdot \\frac{1}{6}+3.61 \\cdot \\frac{3}{6}} \\\\ \\\\\r\n\r\n&= \\sqrt{2.16 + 0.375 + 0.06+ 1.805} \\\\ \\\\\r\n\r\n&= \\sqrt{4.4} \\\\ \\\\\r\n\r\n&= 2.09762\r\n\r\n\\end{align*}\r\n\r\n[\/latex]\r\n

            Applied Expected Value Example (Video & Exercise)<\/h1>\r\nLet us finish this section with an applied example. This is from a forestry course. We used expected values to calculate expected numbers of defective boards.\r\n\r\n[video width=\"1716\" height=\"1080\" mp4=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/Probs-9.mp4\"][\/video]\r\n

            Example 21.3.2<\/h2>\r\nSetup<\/strong><\/span>: What if, instead, you had the following:\r\n
              \r\n \t
            • 1<\/sup>\u20445<\/sub> of the boards have a defect<\/li>\r\n \t
            • 1<\/sup>\u20442<\/sub>\u00a0of the boards with a defect have rot<\/li>\r\n \t
            • 1<\/sup>\u20444<\/sub>\u00a0of the boards with rot are also bent<\/li>\r\n<\/ul>\r\nQuestion<\/strong><\/span>: Calculate the expected number of boards for each scenario below if there are 3,000 boars in the yards.\r\n\r\nYou Try<\/strong><\/span>:\r\n\r\n[h5p id=\"51\"]\r\n\r\n[h5p id=\"52\"]\r\n\r\n[h5p id=\"53\"]\r\n

              Key Takeaways (EXERCISE)<\/h1>\r\n
              \r\n

              Key Takeaways: Expected Values<\/p>\r\n\r\n<\/header>\r\n

              \r\n\r\n[h5p id=\"54\"]\r\n\r\n[h5p id=\"55\"]\r\n\r\n<\/div>\r\n<\/div>\r\n

              Your Own Notes (EXERCISE)<\/h1>\r\n
                \r\n \t
              • Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t
              • These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t
              • Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"
                \n
                \n

                Learning Objectives<\/p>\n<\/header>\n

                \n

                Calculate and understand expected values.<\/p>\n<\/div>\n<\/div>\n

                \"\"<\/p>\n

                  \n
                • The value we would expect to occur,\u00a0also known as the mean (\u03bc)<\/li>\n
                • Previous values (x’<\/em>s) and their probabilities of occurring (p<\/em>‘s) are used to calculate \u03bc.<\/li>\n
                • Expected Value = [latex]\u00a0 E(X) =\\mu =\\Sigma [x_i \\cdot p(x_i)][\/latex]<\/li>\n<\/ul>\n

                  Expected Values with Coin flips (Example)<\/h1>\n

                  The concept of expected values is best illustrated by games. Let’s ‘play’ some in the next two examples.<\/p>\n

                  Example 21.1 – Coin Flip<\/h2>\n

                  Setup<\/span>:<\/span> <\/strong>This first game is rather simple, and not very exciting, but it lays the foundation for the second, more complicated game.<\/p>\n

                    \n
                  • In this game, I flip a coin.<\/li>\n
                  • If I flip a Head, I’ll pay you a dollar.<\/li>\n
                  • If I flip a Tail, you’ll pay me a dollar.<\/li>\n<\/ul>\n

                    Question:<\/strong> <\/span>Can either one of us EXPECT to earn anything in the long run?<\/p>\n

                    Solution:<\/strong> <\/span>Let us first of all define the term, “in the long run”:<\/p>\n

                      \n
                    • It simply means if we play the game many, many times.<\/li>\n
                    • I guess most people will agree that the answer is NO.<\/li>\n
                    • Half the time, I win a dollar.<\/li>\n
                    • Half the time, I lose a dollar.<\/li>\n<\/ul>\n

                      In terms of mathematics, we can state the above sentence this way:<\/p>\n

                      \\[\\frac{1}{2} (+\\$1)+ \\frac{1}{2} (-\\$1)=\\$0\\]<\/p>\n

                      Expected Values with Dice Rolls (Exercise)<\/h1>\n

                      Let us denote a more complex game in the next example. This time, we will use a six-sided die and win certain amounts of money depending on the number we roll.<\/p>\n

                      Example 21.2.1 – Die Roll<\/h2>\n

                      Setup:<\/strong><\/span>\u00a0<\/strong>In this game, I will roll a fair regular 6 sided die.<\/p>\n

                        \n
                      • You pay me $2 to play each round.<\/li>\n
                      • Then I roll the die.<\/li>\n
                      • If I roll a 6, I will pay you $5.50.<\/li>\n
                      • If I roll a 5, I will pay you $3.40.<\/li>\n
                      • If I roll a 4, I will pay you $2.50.<\/li>\n
                      • If I roll a 3, 2, or 1, I will pay you nothing.<\/li>\n<\/ul>\n

                        Question:<\/strong> <\/span>Would you like to play this game with me? Think about it for a moment.<\/p>\n

                        You Try<\/strong><\/span>: Let us now analyze this game from your point of view.<\/p>\n

                        \n