{"id":1175,"date":"2024-05-03T18:32:05","date_gmt":"2024-05-03T22:32:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&p=1175"},"modified":"2024-06-12T14:38:48","modified_gmt":"2024-06-12T18:38:48","slug":"calculating-probabilities-using-contingency-tables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/calculating-probabilities-using-contingency-tables\/","title":{"raw":"Calculating Probabilities Using Contingency Tables","rendered":"Calculating Probabilities Using Contingency Tables"},"content":{"raw":"
\r\n

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

\r\n\r\nUse contingency tables to calculate probabilities.\r\n\r\n<\/div>\r\n<\/div>\r\n

Possible Calculations Using Contingency Tables:<\/h2>\r\nContingency tables give us the following values:\r\n
    \r\n \t
  • 'Singular' probabilities: P(A), P(B), P(\u0100), P(B\u0305), ...<\/li>\r\n \t
  • 'Joint' probabilities: P(A and B), P(A and B\u0305), P(\u0100 and B), P(\u0100 and B\u0305), ...<\/li>\r\n<\/ul>\r\nWe can use these probabilities and our probability rule to calculate many probabilities. Ex:\r\n
      \r\n \t
    • P(At least one event occurring):[latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B) [\/latex]<\/li>\r\n \t
    • P(Neither event occurring):[latex]P(\\text{neither }A \\text{ nor } B) = P(\\overline{A} \\text{ and } \\overline{B}) [\/latex]<\/li>\r\n \t
    • P(One event occurring given another event occurred):[latex]P(A|B) = \\frac{P(A \\text{ and } B)}{P(B)} [\/latex]<\/li>\r\n<\/ul>\r\n

      Calculting Either\/Or, Neither\/Nor (Exercise)<\/h1>\r\nLet us explore understanding multiple ways to use the table to calculate:\r\n
        \r\n \t
      • P(A or B) = P(at least one of event A or B occurs)<\/li>\r\n \t
      • P(neither A nor B) = P(neither event A nor event B occurs)<\/li>\r\n<\/ul>\r\n

        Example 18.1.1<\/h2>\r\nProblem Setup<\/span><\/strong>: Let us continue with the two marketing campaigns example (Example 18.1):\r\n
          \r\n \t
        • P(A) = the click-through rate for ad campaign 'A'<\/li>\r\n \t
        • P(B) = the click-through rate for ad campaign 'B'<\/li>\r\n \t
        • P(\u0100) = probability of people not clicking through after viewing ad 'A'<\/li>\r\n \t
        • P(B\u0305) = probability of people not clicking through after viewing ad 'B'<\/li>\r\n \t
        • The 2x2 (contingency) table for this example is below:<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          <\/td>\r\nA<\/strong><\/td>\r\nnot A<\/strong><\/td>\r\nTotals<\/strong><\/td>\r\n<\/tr>\r\n
          B<\/strong><\/td>\r\n0.003<\/td>\r\n0.047<\/td>\r\n0.05<\/td>\r\n<\/tr>\r\n
          not B<\/strong><\/td>\r\n0.017<\/td>\r\n0.933<\/td>\r\n0.95<\/td>\r\n<\/tr>\r\n
          Totals<\/strong><\/td>\r\n0.02<\/td>\r\n0.98<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nQuestions<\/strong><\/span>: Use the above table to answer the following:\r\n
            \r\n \t
          1. What percent of people click through on at least one of the ads?<\/li>\r\n \t
          2. What percent of people click through on neither 'A' nor 'B'?<\/li>\r\n<\/ol>\r\nSolution<\/strong><\/span>: Click below to reveal the answers (after trying it yourself):\r\n\r\n[h5p id=\"44\"]\r\n

            Calculating COnditional Probabilities (Exercise)<\/h1>\r\nTo get the conditional\/given probabilities:\r\n
              \r\n \t
            • Take any of the AND<\/em>'s in our table and divide by an overall probabilities.<\/li>\r\n \t
            • Ex: [latex]P(A|B) = \\frac{P(A \\text{ and } B)}{P(B)} [\/latex]<\/li>\r\n<\/ul>\r\n

              Example 18.1.2<\/h2>\r\nProblem Setup<\/span><\/strong>: Let us continue using the table given in Example 18.1.1.\r\n\r\nQuestions<\/strong><\/span>: Calculate the conditional probabilities listed below:\r\n
                \r\n \t
              1. What percent of people who clicked through on ad 'A' then click through on 'B'?<\/li>\r\n \t
              2. Given someone doesn't click through 'A', what is the probability they click through 'B'?<\/li>\r\n \t
              3. Out of those who do not click through on 'B', what percent click through on 'A'?<\/li>\r\n \t
              4. Who should the company running these ads target for ad 'B'? Those who clicked through on 'A' or those who did not?<\/li>\r\n<\/ol>\r\nInstructions: <\/strong><\/span>Try the questions yourself. Once done, click below to reveal the answers for each part.\r\n
                \r\n

                Solution # 18.1.2-1 (Click To ReveAL)<\/h1>\r\nThink of what comes 'first' in this conditional probability. This event goes behind the line<\/span>\r\n
                  \r\n \t
                • In this case, clicking through on 'A' comes first.<\/li>\r\n \t
                • So, we are looking for P(B|A)<\/li>\r\n \t
                • Ie: [latex]P(B|A) = \\frac{P(A \\text{ and } B)}{P(A)}=\\frac{0.003}{0.02}= 0.15=15\\%[\/latex]<\/li>\r\n \t
                • This was actually the probability we were given at the start of this example in the last section!<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                  \r\n

                  Solution #18.1.2-2 (Click to reveal)<\/h1>\r\nNot clicking through after viewing ad 'A' comes first for this conditional probability.\r\n
                    \r\n \t
                  • Ie: We are looking for P(B| not A) = P(B| \u0100)<\/li>\r\n \t
                  • [latex]P(B|\\overline{A}) = \\frac{P(\\overline{A} \\text{ and } B)}{P(\\overline{A})}=\\frac{0.047}{0.98}= 0.0480=4.8\\%[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                    \r\n

                    Solution #18.1.2-3 (Click To reveal)<\/h1>\r\nNot clicking through after viewing ad 'B' comes first for this conditional probability. Ie:\r\n
                      \r\n \t
                    • We are looking for P(A| not B) = P(A | B\u0305)<\/li>\r\n \t
                    • [latex]P(A|\\overline{B}) = \\frac{P(A \\text{ and } \\overline{B})}{P(\\overline{B})}=\\frac{0.017}{0.95}= 0.0179=1.79\\%[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                      \r\n

                      Solution #18.1.2-4 (Click To reveal)<\/h1>\r\nIn questions 1 and 2, we examined the following:\r\n
                        \r\n \t
                      • Percent who clicked on 'B' after they clicked on 'A'= P(B|A) =15%<\/li>\r\n \t
                      • Percent who clicked on 'B' after not clicking on 'A'= P(B|\u0100) =4.8%<\/li>\r\n \t
                      • We can tell that those who clicked on 'A' first are quite a bit more likely to click on 'B'<\/li>\r\n \t
                      • So, the company should target those who already clicked on A.<\/li>\r\n \t
                      • We will revisit this idea later in the course with 'Chi-squared' testing<\/li>\r\n<\/ul>\r\n<\/div>\r\n

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

                        Key Takeaways: Calculating Probabilities Using Contingency Tables<\/p>\r\n \r\n\r\n<\/header>\r\n

                        \r\n\r\n[h5p id=\"46\"]\r\n\r\n[h5p id=\"45\"]\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

                          Use contingency tables to calculate probabilities.<\/p>\n<\/div>\n<\/div>\n

                          Possible Calculations Using Contingency Tables:<\/h2>\n

                          Contingency tables give us the following values:<\/p>\n

                            \n
                          • ‘Singular’ probabilities: P(A), P(B), P(\u0100), P(B\u0305), …<\/li>\n
                          • ‘Joint’ probabilities: P(A and B), P(A and B\u0305), P(\u0100 and B), P(\u0100 and B\u0305), …<\/li>\n<\/ul>\n

                            We can use these probabilities and our probability rule to calculate many probabilities. Ex:<\/p>\n

                              \n
                            • P(At least one event occurring):[latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)[\/latex]<\/li>\n
                            • P(Neither event occurring):[latex]P(\\text{neither }A \\text{ nor } B) = P(\\overline{A} \\text{ and } \\overline{B})[\/latex]<\/li>\n
                            • P(One event occurring given another event occurred):[latex]P(A|B) = \\frac{P(A \\text{ and } B)}{P(B)}[\/latex]<\/li>\n<\/ul>\n

                              Calculting Either\/Or, Neither\/Nor (Exercise)<\/h1>\n

                              Let us explore understanding multiple ways to use the table to calculate:<\/p>\n

                                \n
                              • P(A or B) = P(at least one of event A or B occurs)<\/li>\n
                              • P(neither A nor B) = P(neither event A nor event B occurs)<\/li>\n<\/ul>\n

                                Example 18.1.1<\/h2>\n

                                Problem Setup<\/span><\/strong>: Let us continue with the two marketing campaigns example (Example 18.1):<\/p>\n

                                  \n
                                • P(A) = the click-through rate for ad campaign ‘A’<\/li>\n
                                • P(B) = the click-through rate for ad campaign ‘B’<\/li>\n
                                • P(\u0100) = probability of people not clicking through after viewing ad ‘A’<\/li>\n
                                • P(B\u0305) = probability of people not clicking through after viewing ad ‘B’<\/li>\n
                                • The 2×2 (contingency) table for this example is below:<\/li>\n<\/ul>\n\n\n\n\n\n\n
                                  <\/td>\nA<\/strong><\/td>\nnot A<\/strong><\/td>\nTotals<\/strong><\/td>\n<\/tr>\n
                                  B<\/strong><\/td>\n0.003<\/td>\n0.047<\/td>\n0.05<\/td>\n<\/tr>\n
                                  not B<\/strong><\/td>\n0.017<\/td>\n0.933<\/td>\n0.95<\/td>\n<\/tr>\n
                                  Totals<\/strong><\/td>\n0.02<\/td>\n0.98<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                  Questions<\/strong><\/span>: Use the above table to answer the following:<\/p>\n

                                    \n
                                  1. What percent of people click through on at least one of the ads?<\/li>\n
                                  2. What percent of people click through on neither ‘A’ nor ‘B’?<\/li>\n<\/ol>\n

                                    Solution<\/strong><\/span>: Click below to reveal the answers (after trying it yourself):<\/p>\n

                                    \n