document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203558>Question 1203558</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If A and B are events, show that:<BR>\n" );
document.write( "(a) P(A n B') = P(A) - P(A n B)<BR>\n" );
document.write( "(b) P(A u B) = i - P(A' n B')\r<BR>\n" );
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document.write( "note:<BR>\n" );
document.write( "A' and B' is complement </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839212 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=839212\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1\r<BR>\n" );
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document.write( "Draw a rectangle to represent the universal set (aka sample space). Inside the rectangle place partially overlapping circles A and B.\r<BR>\n" );
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document.write( "Shade circle A.<BR>\n" );
document.write( "If you are in region A, but outside B, then you are in region A n B' where the \"n\" refers to set intersection.\r<BR>\n" );
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document.write( "If you are in A and also in B, then you are located in set A n B.<BR>\n" );
document.write( "This is where the circles overlap.<BR>\n" );
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document.write( "If we started with set A, and kicked out members of set A n B, then we're left with members of set A n B' only.\r<BR>\n" );
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document.write( "This is an informal way to prove that<BR>\n" );
document.write( "P(A n B') = P(A) - P(A n B)\r<BR>\n" );
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document.write( "Another way to prove this would be to look at the law of total probability.<BR>\n" );
document.write( "P(A) = P(A n B') + P(A n B)<BR>\n" );
document.write( "which rearranges to<BR>\n" );
document.write( "P(A n B') = P(A) - P(A n B)\r<BR>\n" );
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document.write( "-----------------------\r<BR>\n" );
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document.write( "Problem 2\r<BR>\n" );
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document.write( "P(A u B) = 1 - P( (A u B)' )<BR>\n" );
document.write( "P(A u B) = 1 - P( A' n B' )\r<BR>\n" );
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document.write( "I used the complement rule P(A) = 1-P(A') on the first line.<BR>\n" );
document.write( "Then I used De Morgan's Law on the second line. \r<BR>\n" );
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document.write( "In terms of a Venn Diagram, region A u B is anywhere in the two circles. <BR>\n" );
document.write( "P(A' n B') is the probability of landing outside both circles.<BR>\n" );
document.write( "1 - P(A' n B') is the complement of this, and is the probability of landing in region A u B.<BR>\n" );
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