{"id":2561,"date":"2021-12-17T14:03:39","date_gmt":"2021-12-17T19:03:39","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=2561"},"modified":"2025-06-19T19:09:45","modified_gmt":"2025-06-19T23:09:45","slug":"3-11-review-questions","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/3-11-review-questions\/","title":{"raw":"3.11 Review Questions","rendered":"3.11 Review Questions"},"content":{"raw":"
    \r\n \t
  1. Roll five balanced dice,\r\n
      \r\n \t
    1. find the probability of rolling all 1s. Let A<\/em><\/span> be the event that the outcomes are all 1s.<\/li>\r\n \t
    2. find the probability that all the dice come up the same number.<\/li>\r\n \t
    3. Are the two events in parts (a) and (b) independent?<\/li>\r\n \t
    4. Are the two events in parts (a) and (b) mutually exclusive?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
    5. If events A<\/em><\/span> and B<\/em><\/span> are mutually exclusive, P<\/em>(A<\/em>)\u2004=\u20040.25,\u2006P<\/em>(B<\/em>)\u2004=\u20040.4<\/span>. Find the probability of each of the following events: not A<\/em><\/span>, (A<\/em> &B<\/em>)<\/span>, and (A<\/em> or B<\/em>)<\/span>.<\/li>\r\n \t
    6. A survey on 1000 employees about their gender and marital status gives the following data: 813 employees are male, 875 are married, and 572 married men. Is there anything wrong with these data? Explain why.<\/li>\r\n \t
    7. Suppose that STAT 151 has 8 sections, 3 students each randomly pick a section. Find the probability that\r\n
        \r\n \t
      1. they end up in the same section.<\/li>\r\n \t
      2. they are all in different sections.<\/li>\r\n \t
      3. nobody picks section 1.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      4. There are 10 students in our class. Assume that every student is equally likely to be born on any of the 365 days in a year. Find the probability that no two students in the class have the same birthday.<\/li>\r\n<\/ol>\r\n
        Show\/Hide Answer<\/summary>\r\n
          \r\n \t
        1. \u200c\r\n
            \r\n \t
          1. \r\n[latex]\\begin{aligned}\r\nP(\\mbox{all 1s})&=P(1 \\mbox{ for 1st die } \\& 1 \\mbox{ for 2st die } \\&1 \\mbox{ for 3rd die } \\&1 \\mbox{ for 4th die } \\&1 \\mbox{ for 5th die})\\\\\r\n&=P(1 \\mbox{ for 1st die})\\times P(1 \\mbox{ for 2st die}) \\times P(1 \\mbox{ for 3rd die})\\times P(1 \\mbox{ for 4th die})\\\\&\\times P(\\&1 \\mbox{ for 5th die}) \\\\\r\n&=\\frac{1}{6}\\times \\frac{1}{6}\\times\\frac{1}{6}\\times\\frac{1}{6}\\times\\frac{1}{6}=\\left(\\frac{1}{6}\\right)^5=0.000128.\\end{aligned}[\/latex]<\/span><\/li>\r\n \t
          2. \r\n[latex]\\begin{aligned}\r\nP(\\mbox{same number})&=P(\\mbox{all 1s or all 2s or all 3s or all 4s or all 5s or all 6s})\\\\\r\n&=P(\\mbox{all 1s})+P(\\mbox{all 2s})+P(\\mbox{all 3s})+P(\\mbox{all 4s})+P(\\mbox{all 5s})+P(\\mbox{all 6s})\\\\&=6\\times \\left(\\frac{1}{6}\\right)^5=\\left(\\frac{1}{6}\\right)^4=0.0007716.\\end{aligned}[\/latex]<\/span><\/li>\r\n \t
          3. Events A<\/em><\/span> and B<\/em><\/span> are not independent, since P<\/em>(A <\/em>& B<\/em>)\u2004=\u2004P<\/em>(A<\/em>)\u2004\u2260\u2004P<\/em>(A<\/em>)\u2005\u00d7\u2005P<\/em>(B<\/em>)<\/span>.<\/li>\r\n \t
          4. Events A<\/em><\/span> and B<\/em><\/span> are not mutually exclusive, since the overlap is event A<\/em><\/span> which is NOT an empty set.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n
              \r\n \t
            1. \r\n[latex]\\begin{aligned}\r\nP(\\mbox{not A})\\&=1-P(A)=1-0.25=0.75\\\\\r\nP(A \\mbox{ \\& } B)\\&=0 \\quad (\\mbox{mutually exclusive})\\\\\r\nP(A \\mbox{ or } B)\\&=P(A)+P(B)=0.25+0.4=0.65.\\end{aligned}[\/latex]<\/span><\/li>\r\n \t
            2. There are 1000-813=187 females. However, the number of married females is 875-572=303 which is larger than 187 and this is impossible.<\/li>\r\n \t
            3. \u200c\r\n
                \r\n \t
              1. Apply the basic counting rule:\r\n[latex]\\frac{8\\times 1\\times 1}{8\\times 8\\times 8}=\\frac{1}{64}=0.015625.[\/latex]<\/span><\/li>\r\n \t
              2. Apply the basic counting rule:\r\n[latex]\\frac{8\\times 7\\times 6}{8\\times 8\\times 8}=\\frac{42}{64}=0.65625.[\/latex]<\/span><\/li>\r\n \t
              3. Apply the basic counting rule:\r\n[latex]\\frac{7\\times 7\\times 7}{8\\times 8\\times 8}=\\left(\\frac{7}{8}\\right)^3=0.6699.[\/latex]<\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
              4. \u200c\r\n[latex]\\frac{\r\n\\left(\r\n\\begin{array}{@{ }c@{ }}\r\n365\\\\10\r\n\\end{array}\r\n\\right)\r\n\\times 10!}{365^{10}}=\\frac{365\\times 364\\times \\cdots\\times 356}{365^{10}}=0.88305.[\/latex]<\/li>\r\n<\/ol>\r\n<\/details>","rendered":"
                  \n
                1. Roll five balanced dice,\n
                    \n
                  1. find the probability of rolling all 1s. Let A<\/em><\/span> be the event that the outcomes are all 1s.<\/li>\n
                  2. find the probability that all the dice come up the same number.<\/li>\n
                  3. Are the two events in parts (a) and (b) independent?<\/li>\n
                  4. Are the two events in parts (a) and (b) mutually exclusive?<\/li>\n<\/ol>\n<\/li>\n
                  5. If events A<\/em><\/span> and B<\/em><\/span> are mutually exclusive, P<\/em>(A<\/em>)\u2004=\u20040.25,\u2006P<\/em>(B<\/em>)\u2004=\u20040.4<\/span>. Find the probability of each of the following events: not A<\/em><\/span>, (A<\/em> &B<\/em>)<\/span>, and (A<\/em> or B<\/em>)<\/span>.<\/li>\n
                  6. A survey on 1000 employees about their gender and marital status gives the following data: 813 employees are male, 875 are married, and 572 married men. Is there anything wrong with these data? Explain why.<\/li>\n
                  7. Suppose that STAT 151 has 8 sections, 3 students each randomly pick a section. Find the probability that\n
                      \n
                    1. they end up in the same section.<\/li>\n
                    2. they are all in different sections.<\/li>\n
                    3. nobody picks section 1.<\/li>\n<\/ol>\n<\/li>\n
                    4. There are 10 students in our class. Assume that every student is equally likely to be born on any of the 365 days in a year. Find the probability that no two students in the class have the same birthday.<\/li>\n<\/ol>\n
                      \nShow\/Hide Answer<\/summary>\n
                        \n
                      1. \u200c\n
                          \n

                        1. \n[latex]\\begin{aligned} P(\\mbox{all 1s})&=P(1 \\mbox{ for 1st die } \\& 1 \\mbox{ for 2st die } \\&1 \\mbox{ for 3rd die } \\&1 \\mbox{ for 4th die } \\&1 \\mbox{ for 5th die})\\\\ &=P(1 \\mbox{ for 1st die})\\times P(1 \\mbox{ for 2st die}) \\times P(1 \\mbox{ for 3rd die})\\times P(1 \\mbox{ for 4th die})\\\\&\\times P(\\&1 \\mbox{ for 5th die}) \\\\ &=\\frac{1}{6}\\times \\frac{1}{6}\\times\\frac{1}{6}\\times\\frac{1}{6}\\times\\frac{1}{6}=\\left(\\frac{1}{6}\\right)^5=0.000128.\\end{aligned}[\/latex]<\/span><\/li>\n

                        2. \n[latex]\\begin{aligned} P(\\mbox{same number})&=P(\\mbox{all 1s or all 2s or all 3s or all 4s or all 5s or all 6s})\\\\ &=P(\\mbox{all 1s})+P(\\mbox{all 2s})+P(\\mbox{all 3s})+P(\\mbox{all 4s})+P(\\mbox{all 5s})+P(\\mbox{all 6s})\\\\&=6\\times \\left(\\frac{1}{6}\\right)^5=\\left(\\frac{1}{6}\\right)^4=0.0007716.\\end{aligned}[\/latex]<\/span><\/li>\n
                        3. Events A<\/em><\/span> and B<\/em><\/span> are not independent, since P<\/em>(A <\/em>& B<\/em>)\u2004=\u2004P<\/em>(A<\/em>)\u2004\u2260\u2004P<\/em>(A<\/em>)\u2005\u00d7\u2005P<\/em>(B<\/em>)<\/span>.<\/li>\n
                        4. Events A<\/em><\/span> and B<\/em><\/span> are not mutually exclusive, since the overlap is event A<\/em><\/span> which is NOT an empty set.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n
                            \n

                          1. \n[latex]\\begin{aligned} P(\\mbox{not A})\\&=1-P(A)=1-0.25=0.75\\\\ P(A \\mbox{ \\& } B)\\&=0 \\quad (\\mbox{mutually exclusive})\\\\ P(A \\mbox{ or } B)\\&=P(A)+P(B)=0.25+0.4=0.65.\\end{aligned}[\/latex]<\/span><\/li>\n
                          2. There are 1000-813=187 females. However, the number of married females is 875-572=303 which is larger than 187 and this is impossible.<\/li>\n
                          3. \u200c\n
                              \n
                            1. Apply the basic counting rule:
                              \n[latex]\\frac{8\\times 1\\times 1}{8\\times 8\\times 8}=\\frac{1}{64}=0.015625.[\/latex]<\/span><\/li>\n
                            2. Apply the basic counting rule:
                              \n[latex]\\frac{8\\times 7\\times 6}{8\\times 8\\times 8}=\\frac{42}{64}=0.65625.[\/latex]<\/span><\/li>\n
                            3. Apply the basic counting rule:
                              \n[latex]\\frac{7\\times 7\\times 7}{8\\times 8\\times 8}=\\left(\\frac{7}{8}\\right)^3=0.6699.[\/latex]<\/span><\/li>\n<\/ol>\n<\/li>\n
                            4. \u200c
                              \n[latex]\\frac{ \\left( \\begin{array}{@{ }c@{ }} 365\\\\10 \\end{array} \\right) \\times 10!}{365^{10}}=\\frac{365\\times 364\\times \\cdots\\times 356}{365^{10}}=0.88305.[\/latex]<\/li>\n<\/ol>\n<\/details>\n","protected":false},"author":19,"menu_order":12,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2561","chapter","type-chapter","status-publish","hentry"],"part":327,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2561","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":21,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2561\/revisions"}],"predecessor-version":[{"id":5588,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2561\/revisions\/5588"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/327"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/2561\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=2561"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=2561"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=2561"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=2561"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}