document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201638>Question 1201638</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an experiment, A, B, C, and D are events with probabilities P[A∪B]=5/8, P[A]=3/8,<BR>\n" );
document.write( "P[C∩D]=1/3, and P[C]=1/2. Furthermore, A and B are disjoint, while C and D are independent.<BR>\n" );
document.write( "(20 points)<BR>\n" );
document.write( "a) Find P[A∩B], P[B], P[A∩Bc<BR>\n" );
document.write( "], P[AUBc<BR>\n" );
document.write( "].<BR>\n" );
document.write( "b) Are A and B independent?<BR>\n" );
document.write( "c) Find P[D], P[C∩Dc<BR>\n" );
document.write( "], P[Cc∩Dc<BR>\n" );
document.write( "], P[C|D].<BR>\n" );
document.write( "d) Are C and Dc<BR>\n" );
document.write( "independent?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837942 by </B><A HREF=/tutors/aboutme.mpl?userid=mathprof><B>mathprof(1)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathprof><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=837942\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First let's solve for P[B] and P[D], since P[A] and P[C] are given.\r<BR>\n" );
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document.write( "P[A∪B]= P[A] + P[B] = 5/8, since A and B are disjoint.<BR>\n" );
document.write( "      = 3/8 + P[B] = 5/8,  hence P[B] = 2/8 = 1/4\r<BR>\n" );
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document.write( "P[C∩D] = P[C]P[D] = 1/3, since C and D are independent.<BR>\n" );
document.write( "       = 1/2 * P[D] = 1/3, hence P[D] = 2/3\r<BR>\n" );
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document.write( "a)  <BR>\n" );
document.write( "P[A∩B] = 0, since A and B are disjoint.<BR>\n" );
document.write( "P[B] = 1/4 (see above)<BR>\n" );
document.write( "P[A∩Bc] = P[A] = 3/8, since A and B are disjoint, A is a subset of Bc.<BR>\n" );
document.write( "P[A∪Bc] = P[Bc] = 1 - 1/4 = 3/4, since A and B are disjoint, A is a subset of Bc\r<BR>\n" );
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document.write( "b) <BR>\n" );
document.write( "A and B are not independent, since P[A∩B]=0 does not equal P[A]P[B] = 3/8 * 1/4.\r<BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "P[D] = 2/3 (see above)<BR>\n" );
document.write( "P[C∩Dc] = 1/6, since P[C∩D] + P[C∩Dc] = P[C], i.e., 1/3 + P[C∩Dc] = 1/2.<BR>\n" );
document.write( "P[Cc∩Dc] = P[(C∪D)c] <BR>\n" );
document.write( "         = 1 - P[C∪D] = 1 - (P[C] + P[D] - P[C∩D]) <BR>\n" );
document.write( "         = 1 - (1/2 + 2/3 - 1/3) =  1/6<BR>\n" );
document.write( "P[C|D] = P[C] = 1/2, since C and D are independent.\r<BR>\n" );
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document.write( "d) C and Dc are independent, because P[C∩Dc] = P[C]P[Dc] = 1/2 * 1/3 = 1/6. </SPAN>\n" );
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