document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=468367>Question 468367</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi I was hoping someone could Please help me with this problem! Thank you so much!! :) \r<BR>\n" );
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document.write( "Movies A survey of 350 customers was taken at Regal Cinemas in Austin, Texas, regarding the type of movies customers liked. The following information was determined.<BR>\n" );
document.write( " 196 liked dramas.153 liked comedies.<BR>\n" );
document.write( " 88 liked science fiction.<BR>\n" );
document.write( " 59 liked dramas and comedies.<BR>\n" );
document.write( " 37 liked dramas and science fiction.<BR>\n" );
document.write( " 32 liked comedies and science fiction.<BR>\n" );
document.write( " 21 liked all three types of movies. \r<BR>\n" );
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document.write( "Of the customers surveyed, how many liked<BR>\n" );
document.write( " a)none of these types of movies?<BR>\n" );
document.write( " b)only dramas?<BR>\n" );
document.write( " c)exactly one of these types of movies?<BR>\n" );
document.write( " d)exactly two of these types of movies?<BR>\n" );
document.write( " e)dramas or comedies? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #321433 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=321433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The general formula for the union of 3 sets is:<BR>\n" );
document.write( "(A union B union C) = A + B + C - (A intersect B) - (A intersect C) - (B intersect C) + (A intersect B intersect C).<BR>\n" );
document.write( "We will assign the following:<BR>\n" );
document.write( "A = drama<BR>\n" );
document.write( "B = comedy<BR>\n" );
document.write( "C = science<BR>\n" );
document.write( "A intersect B = drama + comedy = AB<BR>\n" );
document.write( "A intersect C = drama + science = AC<BR>\n" );
document.write( "B intersect C = comedy + science = BC<BR>\n" );
document.write( "A intersect B intersect C = dramw + comedy + science (all 3) = ABC<BR>\n" );
document.write( "We will now assign numbers to each of these sets:<BR>\n" );
document.write( "A = 196<BR>\n" );
document.write( "B = 153<BR>\n" );
document.write( "C = 88<BR>\n" );
document.write( "AB = 59<BR>\n" );
document.write( "AC = 37<BR>\n" );
document.write( "BC = 32<BR>\n" );
document.write( "ABC = 21<BR>\n" );
document.write( "We plug these numbers into the formula and we get:<BR>\n" );
document.write( "A union B union C = 196 + 153 + 88 - 59 - 37 - 32 + 21<BR>\n" );
document.write( "This becomes:<BR>\n" );
document.write( "A union B union C = 330<BR>\n" );
document.write( "Since there are 350 customers who were surveyed, that leaves 20 who did not like any of the selections.<BR>\n" );
document.write( "We need to separate out the categories a little better in order to answer the rest of the questions.<BR>\n" );
document.write( "We'll start with ABC.<BR>\n" );
document.write( "There are 21 in that category.<BR>\n" );
document.write( "We'll then move up to AB, AC, and BC.<BR>\n" );
document.write( "Those categories each include ABC so we'll need to remove them from each of those categories.<BR>\n" );
document.write( "AB = 59 - 21 = 38 that are in AB only.<BR>\n" );
document.write( "AC = 37 - 21 = 16 that are in AC only.<BR>\n" );
document.write( "BC = 32 - 21 = 11 that are in BC only.<BR>\n" );
document.write( "We'll then move up to A, B, and C.<BR>\n" );
document.write( "Those categories each include AB, AC, BC, and ABC.<BR>\n" );
document.write( "We'll need to remove them to get A only, B only, C only. <BR>\n" );
document.write( "We need to remove AB, AC, and ABC from A to get:<BR>\n" );
document.write( "A  DISABLED_event_only=  196 - 38 - 16 - 21 = 121.<BR>\n" );
document.write( "We need to remove AB, BC, and ABC from B to get:<BR>\n" );
document.write( "B  DISABLED_event_only=  153 - 38 - 11 - 21 = 83<BR>\n" );
document.write( "We need to remove AC, BC, and ABC from C to get:<BR>\n" );
document.write( "C  DISABLED_event_only=  88 - 16 - 11 - 21 = 40<BR>\n" );
document.write( "We have now separated all these groups so there is no double counting in any of the groups.<BR>\n" );
document.write( "We have:<BR>\n" );
document.write( "A  DISABLED_event_only=  121<BR>\n" );
document.write( "B  DISABLED_event_only=  83<BR>\n" );
document.write( "C  DISABLED_event_only=  40<BR>\n" );
document.write( "A intersect B  DISABLED_event_only=  38<BR>\n" );
document.write( "A intersect C  DISABLED_event_only=  16<BR>\n" );
document.write( "B intersect C  DISABLED_event_only=  11<BR>\n" );
document.write( "A intersect B intersect C = 21<BR>\n" );
document.write( "NOT (A union B union C) = 20 (don't like any of the 3 selections).<BR>\n" );
document.write( "We add these all up and we get a total of 350 customers.<BR>\n" );
document.write( "A Venn Diagram of these sets is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/Venn_Diagram_1.jpg\" alt = \"***** picture not found *****\"/ ><BR>\n" );
document.write( "There are a total of 330 customer in this Venn Diagram.<BR>\n" );
document.write( "The 20 customer who did not have a preference are not shown.<BR>\n" );
document.write( "This checks out with what we were told up front, so we probably did it right.<BR>\n" );
document.write( "In removing the double counting, you have to work from the bottom layer up.<BR>\n" );
document.write( "The bottom layer was A intersect B intersect C<BR>\n" );
document.write( "The next bottom layer was A intersect B, A intersect C, B intersect C.<BR>\n" );
document.write( "The top layer was A, B, and C<BR>\n" );
document.write( "The higher layers include elements in the lower layers.<BR>\n" );
document.write( "A intersect B intersect C was the lowest layer  because it didn't include any sets that were part of it.<BR>\n" );
document.write( "A intersect B, A intersect C, B intersect C were above ABC because they included ABC as part of their sets.<BR>\n" );
document.write( "etc.<BR>\n" );
document.write( "Your questions should be able to be answered now.<BR>\n" );
document.write( "Those questions are:<BR>\n" );
document.write( "a)none of these types of movies?<BR>\n" );
document.write( "20<BR>\n" );
document.write( "b)only dramas?<BR>\n" );
document.write( "121<BR>\n" );
document.write( "c)exactly one of these types of movies?<BR>\n" );
document.write( "This would be A only plus B only plus C  DISABLED_event_only=  121 + 83 + 40 = 244<BR>\n" );
document.write( "d)exactly two of these types of movies?<BR>\n" );
document.write( "This would be AB only + AC only + BC  DISABLED_event_only=  65 <BR>\n" );
document.write( "e)dramas or comedies?<BR>\n" );
document.write( "This would be A only + B only + AB only + AC only + BC only + ABC = 121 + 83 + 38 + 16 + 11 + 21 = 290.<BR>\n" );
document.write( "It includes A only because they like drama.<BR>\n" );
document.write( "It include B only because they like comedy.<BR>\n" );
document.write( "It includes AB only because they like drama and comedy.<BR>\n" );
document.write( "It includes AC only because they like drama.<BR>\n" );
document.write( "It include BC only because they like comedy<BR>\n" );
document.write( "It includes ABC only because they like drama and comedy.<BR>\n" );
document.write( "The only category it does not include is C only.  Those customers only like Science.<BR>\n" );
document.write( " </SPAN>\n" );
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