{"id":1220,"date":"2024-05-03T23:11:22","date_gmt":"2024-05-04T03:11:22","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&p=1220"},"modified":"2024-06-12T14:39:13","modified_gmt":"2024-06-12T18:39:13","slug":"mutually-exclusive-and-independent-events","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/mutually-exclusive-and-independent-events\/","title":{"raw":"Mutually Exclusive and Independent Events","rendered":"Mutually Exclusive and Independent Events"},"content":{"raw":"
\r\n

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

\r\n\r\nUnderstand what it means for events to be mutually exclusive or independent.\r\n\r\n<\/div>\r\n<\/div>\r\n

Mutually Exclusive Events<\/h2>\r\n
    \r\n \t
  • Two events that cannot occur at the same time are considered to be mutually exclusive<\/li>\r\n \t
  • The probability of both events occurring at once will equal zero<\/li>\r\n<\/ul>\r\n

    Independent Events<\/h2>\r\n
      \r\n \t
    • Spotting independent events is a little bit more involved<\/li>\r\n \t
    • Remember, P(A|B) = the probability of A occurring given that B has already occurred.<\/li>\r\n \t
    • What happens if B has no effect on A? Then P(A|B) = P(A)<\/li>\r\n \t
    • That also means that P(A and B) = P(A|B)\u00d7P(B) =P(A)\u00d7P(B)<\/li>\r\n<\/ul>\r\n

      Testing for Mutually Exclusive Events (Example)<\/h1>\r\n

      Example 19.1<\/h2>\r\nProblem Setup<\/strong><\/span>: Let us have two possible events, 1 and 2, that can either occur or not occur. See below:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\nEvent 1<\/th>\r\nNot Event 1<\/th>\r\nTotals<\/th>\r\n<\/tr>\r\n
      Event 2<\/th>\r\n0<\/td>\r\nP(not<\/em> E1<\/sub> and E2<\/sub>)<\/td>\r\nP(E2<\/sub>)<\/td>\r\n<\/tr>\r\n
      Not Event 2<\/th>\r\nP(E1<\/sub> and not<\/em> E2<\/sub>)<\/td>\r\nP(not<\/em> E1<\/sub> and not<\/em> E2<\/sub>)<\/td>\r\nP(not<\/em> E2<\/sub>)<\/td>\r\n<\/tr>\r\n
      Totals<\/th>\r\nP(E1<\/sub>)<\/td>\r\nP(not<\/em> E1<\/sub>)<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nQuestion<\/strong><\/span>: Which events are mutually exclusive based on the above contingency table?\r\n\r\nSolutions<\/strong><\/span>:\r\n
        \r\n \t
      • Look for any 'zeros' in the table.<\/li>\r\n \t
      • In this case, P(E1<\/sub> and E2<\/sub>) = 0<\/li>\r\n \t
      • So, events 1 and 2 are mutually exclusive<\/span><\/li>\r\n<\/ul>\r\n

        Testing for Independence (Example)<\/h1>\r\n
          \r\n \t
        • \r\n
          We can test for any events where P(A|B) = P(A)<\/div><\/li>\r\n \t
        • \r\n
          Or we can test if P(A and B) =P(A)\u00d7P(B)<\/div><\/li>\r\n \t
        • \r\n
          <\/div>\r\n
          If this is true, then the events (we call them 'A' and 'B' for simplicity here) are independent<\/div><\/li>\r\n \t
        • \r\n
          See the next examples of this testing.<\/span><\/div><\/li>\r\n<\/ul>\r\n

          Example 19.2<\/h2>\r\nProblem Setup<\/strong><\/span>: Let us take a simple example of coffee and tea drinkers. Let's use the following notation:\r\n
            \r\n \t
          • P(T) = probability of drinking tea<\/li>\r\n \t
          • P(C) = probability of drinking coffee<\/li>\r\n<\/ul>\r\nSee the contingency table for coffee and tea drinkers below:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            <\/td>\r\nDrinks Coffee<\/th>\r\nDoesn't Drink Coffee<\/th>\r\nTotals<\/th>\r\n<\/tr>\r\n
            Drinks Tea<\/th>\r\n0.6<\/td>\r\n0.075<\/td>\r\n0.675<\/td>\r\n<\/tr>\r\n
            Doesn't Drink Tea<\/th>\r\n0.2<\/td>\r\n0.125<\/td>\r\n0.325<\/td>\r\n<\/tr>\r\n
            Totals<\/th>\r\n0.8<\/td>\r\n0.2<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nQuestion<\/strong><\/span>: Are drinking coffee and tea independent events?\r\n\r\nSolutions<\/strong><\/span>: Let's start by writing out the values we will use from the above table:\r\n
              \r\n \t
            • P(Drinks Coffee) = P(C) = 0.8<\/li>\r\n \t
            • P(Drinks Tea) = P(T) = 0.675<\/li>\r\n \t
            • P(Drinks Coffee and Drinks Tea) = P(C and T) = 0.6<\/li>\r\n<\/ul>\r\nLet's test, is P(C and T) = P(C)\u00d7P(T)?\r\n
                \r\n \t
              • P(C)\u00d7P(T) = 0.8\u00d70.675 = 0.54<\/li>\r\n \t
              • P(C and T) = 0.6 \u2260 0.54<\/li>\r\n<\/ul>\r\nConclusion<\/strong><\/span>: Because P(C and T) \u2260 P(C)\u00d7P(T), coffee and tea drinking are not independent events. Ie: whether someone drinks coffee has an effect on whether or not they choose to drink tea.\r\n

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

                Key Takeaways: Mutually Exclusive and Independent Events<\/p>\r\n \r\n\r\n<\/header>\r\n

                \r\n\r\n[h5p id=\"47\"]\r\n\r\n[h5p id=\"48\"]\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

                  Understand what it means for events to be mutually exclusive or independent.<\/p>\n<\/div>\n<\/div>\n

                  Mutually Exclusive Events<\/h2>\n
                    \n
                  • Two events that cannot occur at the same time are considered to be mutually exclusive<\/li>\n
                  • The probability of both events occurring at once will equal zero<\/li>\n<\/ul>\n

                    Independent Events<\/h2>\n
                      \n
                    • Spotting independent events is a little bit more involved<\/li>\n
                    • Remember, P(A|B) = the probability of A occurring given that B has already occurred.<\/li>\n
                    • What happens if B has no effect on A? Then P(A|B) = P(A)<\/li>\n
                    • That also means that P(A and B) = P(A|B)\u00d7P(B) =P(A)\u00d7P(B)<\/li>\n<\/ul>\n

                      Testing for Mutually Exclusive Events (Example)<\/h1>\n

                      Example 19.1<\/h2>\n

                      Problem Setup<\/strong><\/span>: Let us have two possible events, 1 and 2, that can either occur or not occur. See below:<\/p>\n\n\n\n\n\n\n
                      <\/td>\nEvent 1<\/th>\nNot Event 1<\/th>\nTotals<\/th>\n<\/tr>\n
                      Event 2<\/th>\n0<\/td>\nP(not<\/em> E1<\/sub> and E2<\/sub>)<\/td>\nP(E2<\/sub>)<\/td>\n<\/tr>\n
                      Not Event 2<\/th>\nP(E1<\/sub> and not<\/em> E2<\/sub>)<\/td>\nP(not<\/em> E1<\/sub> and not<\/em> E2<\/sub>)<\/td>\nP(not<\/em> E2<\/sub>)<\/td>\n<\/tr>\n
                      Totals<\/th>\nP(E1<\/sub>)<\/td>\nP(not<\/em> E1<\/sub>)<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                      Question<\/strong><\/span>: Which events are mutually exclusive based on the above contingency table?<\/p>\n

                      Solutions<\/strong><\/span>:<\/p>\n

                        \n
                      • Look for any ‘zeros’ in the table.<\/li>\n
                      • In this case, P(E1<\/sub> and E2<\/sub>) = 0<\/li>\n
                      • So, events 1 and 2 are mutually exclusive<\/span><\/li>\n<\/ul>\n

                        Testing for Independence (Example)<\/h1>\n
                          \n
                        • \n
                          We can test for any events where P(A|B) = P(A)<\/div>\n<\/li>\n
                        • \n
                          Or we can test if P(A and B) =P(A)\u00d7P(B)<\/div>\n<\/li>\n
                        • \n
                          <\/div>\n
                          If this is true, then the events (we call them ‘A’ and ‘B’ for simplicity here) are independent<\/div>\n<\/li>\n
                        • \n
                          See the next examples of this testing.<\/span><\/div>\n<\/li>\n<\/ul>\n

                          Example 19.2<\/h2>\n

                          Problem Setup<\/strong><\/span>: Let us take a simple example of coffee and tea drinkers. Let’s use the following notation:<\/p>\n

                            \n
                          • P(T) = probability of drinking tea<\/li>\n
                          • P(C) = probability of drinking coffee<\/li>\n<\/ul>\n

                            See the contingency table for coffee and tea drinkers below:<\/p>\n\n\n\n\n\n\n
                            <\/td>\nDrinks Coffee<\/th>\nDoesn’t Drink Coffee<\/th>\nTotals<\/th>\n<\/tr>\n
                            Drinks Tea<\/th>\n0.6<\/td>\n0.075<\/td>\n0.675<\/td>\n<\/tr>\n
                            Doesn’t Drink Tea<\/th>\n0.2<\/td>\n0.125<\/td>\n0.325<\/td>\n<\/tr>\n
                            Totals<\/th>\n0.8<\/td>\n0.2<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                            Question<\/strong><\/span>: Are drinking coffee and tea independent events?<\/p>\n

                            Solutions<\/strong><\/span>: Let’s start by writing out the values we will use from the above table:<\/p>\n

                              \n
                            • P(Drinks Coffee) = P(C) = 0.8<\/li>\n
                            • P(Drinks Tea) = P(T) = 0.675<\/li>\n
                            • P(Drinks Coffee and Drinks Tea) = P(C and T) = 0.6<\/li>\n<\/ul>\n

                              Let’s test, is P(C and T) = P(C)\u00d7P(T)?<\/p>\n

                                \n
                              • P(C)\u00d7P(T) = 0.8\u00d70.675 = 0.54<\/li>\n
                              • P(C and T) = 0.6 \u2260 0.54<\/li>\n<\/ul>\n

                                Conclusion<\/strong><\/span>: Because P(C and T) \u2260 P(C)\u00d7P(T), coffee and tea drinking are not independent events. Ie: whether someone drinks coffee has an effect on whether or not they choose to drink tea.<\/p>\n

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

                                Key Takeaways: Mutually Exclusive and Independent Events<\/p>\n

                                 <\/p>\n<\/header>\n

                                \n
                                \n