document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206109>Question 1206109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a large city, 37% of all restaurants accept both master and visa credit cards, and 50% accept master cards and 60% accept visa cards. A tourist visiting the city picks at random a restaurant at which to have lunch. Define the following events: \r<BR>\n" );
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document.write( "M\r<BR>\n" );
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document.write( "M = {the randomly chosen restaurant accepts master credit cards},\r<BR>\n" );
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document.write( "V\r<BR>\n" );
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document.write( "V = {the randomly chosen restaurant accepts visa credit cards}.\r<BR>\n" );
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document.write( "Which of the following shows independence between two events M\r<BR>\n" );
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document.write( "M and V\r<BR>\n" );
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document.write( "V? [CHECK ALL THAT APPLY] \r<BR>\n" );
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document.write( "A. P(V given M) is equal to P(V)\r<BR>\n" );
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document.write( "B. The events M and V are not disjoint\r<BR>\n" );
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document.write( "C. The events M and V are not disjoint\r<BR>\n" );
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document.write( "D.P(M given V) is equal to P(V)\r<BR>\n" );
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document.write( "e.P(M and V) is equal to P(M)×P(V)\r<BR>\n" );
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document.write( "F.P(M given V) is equal to P(M) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843344 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843344\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "It seems strange that answer choices B and C are identical. A typo perhaps?\r<BR>\n" );
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document.write( "M = restaurant accepts MasterCard<BR>\n" );
document.write( "V = restaurant accepts Visa\r<BR>\n" );
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document.write( "Given information<BR>\n" );
document.write( "P(V and M) = 0.37<BR>\n" );
document.write( "P(M) = 0.50<BR>\n" );
document.write( "P(V) = 0.60\r<BR>\n" );
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document.write( "We can then compute the following<BR>\n" );
document.write( "P(V given M) = P(V and M)/P(M)<BR>\n" );
document.write( "P(V given M) = 0.37/0.50<BR>\n" );
document.write( "P(V given M) = 0.74<BR>\n" );
document.write( "Note how this is <font color=red>not</font> the same value as P(V) = 0.60<BR>\n" );
document.write( "Therefore the equation P(V given M) = P(V) is false<BR>\n" );
document.write( "Consequently it means events M and V are <font color=red>not</font> independent. One event affects the other, or vice versa, or the two events are linked somehow.\r<BR>\n" );
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document.write( "Events M and V are <font color=red>not</font> disjoint since P(V and M) = 0.37 is nonzero.<BR>\n" );
document.write( "In other words, it's possible for both events to happen simultaneously. There is overlap between events.\r<BR>\n" );
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document.write( "Side note: An example of disjoint events would be \"getting heads\" and \"getting tails\" on the same coin on the same flip.\r<BR>\n" );
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document.write( "P(M given V) = P(M and V)/P(V)<BR>\n" );
document.write( "P(M given V) = 0.37/0.60<BR>\n" );
document.write( "P(M given V) = 0.61667 approximately<BR>\n" );
document.write( "This is not the same as P(M) = 0.50, so the equation P(M given V) = P(M) is false. This is more proof that events M and V are <font color=red>not</font> independent.\r<BR>\n" );
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document.write( "Because those events are <font color=red>not</font> independent, P(M and V) = P(M)*P(V) is false. <BR>\n" );
document.write( "Here is proof of such<BR>\n" );
document.write( "P(M)*P(V) = 0.50*0.60 = 0.30 which doesn't match with P(M and V) = 0.37\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Summary:<BR>\n" );
document.write( "If M and V were independent, then the following three equations would be true<BR>\n" );
document.write( "P(V given M) = P(V)<BR>\n" );
document.write( "P(M given V) = P(M)<BR>\n" );
document.write( "P(M and V) = P(M)*P(V)<BR>\n" );
document.write( "But we've shown that none of the equations are satisfied, so the events are <font color=red>not</font> independent.<BR>\n" );
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